Parabolic equations on conic domains]
Sobolev space theory  and Hölder estimates for   the stochastic partial  differential equations with measurable coefficients
on conic and polygonal domains

The first and  third authors were  supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2020R1A2C1A01003354)
The second author was  supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2019R1F1A1058988)

Kyeong-Hun Kim
Kyeong-Hun Kim, Department of Mathematics, Korea University, Anam-ro 145, Sungbuk-gu, Seoul, 02841, Republic of Korea

Kijung Lee
Kijung Lee, Department of Mathematics, Ajou University, Worldcup-ro 206, Yeongtong-gu, Suwon, 16499, Republic of Korea

jinsol Seo
Jinsol Seo, Department of Mathematics, Korea University, Anam-ro 145, Sungbuk-gu, Seoul, 02841, Republic of Korea

[2010]60H15, 35R60, 35R05

We establish existence, uniqueness, and  Sobolev and Hölder regularity results for the stochastic partial differential equation

du=\left(\sum_{i,j=1}^d a^{ij}u_{x^ix^j}+f^0+\sum_{i=1}^d f^i_{x^i}\right)dt+\sum_{k=1}^{\infty}g^kdw^k_t, \quad t>0, \,x\in \cD

given with non-zero initial data. Here $\{w^k_t: k=1,2,\cdots\}$ is a family of independent Wiener processes defined on a probability space $(\Omega, \bP)$, $a^{ij}=a^{ij}(\omega,t)$ are merely  measurable functions on $\Omega\times (0,\infty)$, and  $\cD$ is either a polygonal domain in $\bR^2$ or an arbitrary dimensional conic domain of the type

\begin{equation}
\label{conic}
\cD(\cM):=\left\{x\in \bR^d :\,\frac{x}{|x|}\in \cM\right\}, \quad \quad \cM\in S^{d-1},  \quad (d\geq 2)
\end{equation}

where $\cM$ is an open subset of $S^{d-1}$ with $C^2$ boundary.
We measure the Sobolev and Hölder  regularities of arbitrary  order derivatives of the solution  using a system of mixed weights consisting of appropriate powers of the distance to the vertices and of the distance to the boundary.
The    ranges of admissible powers of the distance to the vertices and  to the boundary are sharp.

The goal of this article is to present a  Sobolev space theory and  Hölder regularity results for  the stochastic partial differential equation (SPDE)

\begin{equation}\label{main equation in introduction}
d u =\left(\sum_{i,j}^d a^{ij}u_{x^ix^j}+f^0+\sum_{i=1}^d f^i_{x^i}\right)dt +\sum^{\infty}_{k=1} g^kdw_t^k, \quad t>0; \quad u(0,\cdot)=u_0
\end{equation}

defined on either  multi-dimensional conic domains $\cD(\cM)$ (see (<ref>)) or  two dimensional polygonal domains.   Here, $\cM$ is an open subset of $S^{d-1}$ with $C^2$ boundary, $\{w^k_t: k=1,2,\cdots\}$ is an infinite sequence of independent one dimensional Wiener processes, and the coefficients $a^{ij}$ are  merely measurable  functions of $(\omega,t)$ with the uniform parabolicity condition; see Assumption <ref> below.

To give the reader a flavor of our results in this article we state a particular one, an estimate, below:  Let $\cD=\cD(\cM)$ be a conic domain in $\bR^d$, $\rho(x):=dist(x,\partial \cD)$, and $\rho_{\circ}(x):=|x|$. Then for the solution $u$ of (<ref>) with zero boundary and zero initial conditions, the following holds for any $p\geq 2$:

\begin{eqnarray}
&&\bE \int^T_0 \int_{\cD} \left(|\rho^{-1}u|^p+ |u_x|^p\right) \rho_{\circ}^{\theta-\Theta}\rho^{\Theta-d}\, dx\,dt \nonumber\\
&\leq& N \bE
\int^T_0 \int_{\cD} \Big( |\rho f^0|^p+\sum_{i=1}^d|f^i|^p +|g|_{l_2}^p\Big)
\rho_{\circ}^{\theta-\Theta}\rho^{\Theta-d}\, dx\,dt  \label{main estimate simple}
\end{eqnarray}

with $d-1<\Theta<d-1+p$ accompanied with the sharp admissible  range of $\theta$; see (<ref>) below.  Also see (<ref>) for  higher order derivative estimates. Unlike the range of $\Theta$, the range of $\theta$ is affected by the shape of domain $\cD$, which is determined by $\cM$.

Estimate (<ref>), if $\rho_{\circ}$ is replaced by the distance to the set of vertices, also holds when  $\cD$ is a (bounded) polygonal domain in $\bR^2$.  Regarding Hölder regularity, we have for instance, if $1-\frac{d}{p}=\delta>0$,

|\rho^{-1+\frac{\Theta}{p}} \rho^{(\theta-\Theta)/p}_{\circ}u(\omega,t,\cdot)|_{ C(\cD)}+
[\rho^{-1+\delta+\frac{\Theta}{p}} \rho^{(\theta-\Theta)/p}_{\circ} u( \omega,t,\cdot)]_{C^{\delta}(\cD)}<\infty,

for a.e. $(\omega,t)$. In particular,

\begin{equation}
|u(\omega,t,x)|\leq N(\omega,t) \rho^{1-\frac{\Theta}{p}}(x) \rho^{(-\theta+\Theta)/p}_{\circ}(x)\quad \text{for all }x\in\cD \label{Holder simple}.
\end{equation}

Estimate (<ref>) shows how $\theta$ and $\Theta$ are involved in measuring the boundary behavior of the solution with respect to $\rho$ and $\rho_{\circ}$. See Theorem <ref> and Theorem <ref> for the full Hölder regularity results with respect to both space and time variables.

To position our results in the context of regularity theory of stochastic parabolic equations, let us provide a stream of historical remarks.

The $L_p$-theory ($p\geq 2$) of   equation (<ref>)  defined on the entire space $\bR^d$ was first introduced by N.V. Krylov [17, 22].  In these articles the author used  an analytic approach and proved the maximal regularity estimate

\begin{equation}
\label{krylov lp}
\|u_x\|_{\bL_p(T)}\leq N\Big(\|f^0\|_{\bL_p(T)}+\sum_{i=1}^d \|f^i\|_{\bL_p(T)}
+\||g|_{\ell_2}\|_{\bL_p(T)}\Big), \qquad p\geq 2,
\end{equation}

provided that $u(0,\cdot)\equiv 0$, where $\bL_p(T):=L_p(\Omega\times (0,T); L_p(\bR^d))$.

As for other approaches on Sobolev regularity theory,  the method based  on $H^{\infty}$-calculus is also available in the literature. This approach  was introduced in [5],  in which  the maximal regularity of $\sqrt{-A}u$ is obtained  for the stochastic convolution

u(t):=\int^t_0 e^{(t-s)A} g(s) dW_H (s).

Here,  $W_H(t)$ is a cylindrical Brownian motion on a Hilbert space $H$, and  the operator $-A$ is assumed to admit a bounded $H^{\infty}$-calculus of angle less than $\pi/2$ on $L^q(\cO)$, where $q\geq 2$ and $\cO$ is a domain in $\bR^d$.  The result of [5]  generalizes (<ref>) with $f^i=0$, $i=1,\ldots,d$ as one can take $A=\Delta$ and $\cO=\bR^d$.

One advantage of the approach based on $H^{\infty}$-calculus is that it provides a unified way of handling a class of differential operators satisfying the above mentioned condition. However this approach is not applicable  for SPDEs with operators depending on $(\omega,t)$, and even  the simplest  case $A=\Delta$, it is needed that $\partial \cO$ is regular enough, that is $\partial \cO \in C^2$. Compared to the approach based on $H^{\infty}$-calculus, Krylov's analytic approach works well for SPDEs with operators depending also on $(\omega,t)$, and it also provides the arbitrary order regularity of solutions without much extra efforts even  under  weaker smoothness condition on  domains.

Since the work of [17, 22] on $\bR^d$,  the analytic approach has been further used for the regularity theory of SPDEs  on half space [19, 20, 14] and on $C^1$-domains [13, 11, 10].
The major obstacle of studying SPDEs on domains is that, unless certain compatibility conditions (cf. [4]) are fulfilled, the second and higher order derivatives of solutions to SPDEs blow up near the boundary, and such blow-ups are inevitable even on $C^{\infty}$-domains. Hence, one needs appropriate weight system to understand the behavior of  solutions  near the boundary.

It is shown  in [19, 13, 11] that if domains satisfy $C^1$ boundary condition, then blow-ups of derivatives of solutions can be described very accurately by a  weight system introduced in [21, 13, 25]. This weight system is  based solely on the distance to the boundary. Surprisingly enough, under this weight system it is  irrelevant  whether domains have $C^{\infty}$-boundary or $C^1$-boundary, that is, the regularity of solutions is not affected by the smoothness of  the boundary provided that the boundary is at least of class $C^1$. To be more specific, let $\cO$ be a $C^1$-domain, $\rho(x)=dist (x,\partial \cO)$, then the following   holds [11, 13]:  for any $n=0,1,2,\cdots$, and $d-1<\Theta<d-1+p$,

\begin{eqnarray}
\label{lp domain}&&
\sum_{|\alpha|\leq n+1}\bE \int^T_0 \int_{\cO} |\rho^{|\alpha|-1}D^{\alpha}u|^p \rho^{\Theta-d}dt  \\
&\leq& N\sum_{|\beta|\leq n} \bE \int^T_0 \int_{\cO}\left( |\rho^{|\beta|+1} D^{\beta}f^0|+\sum_{i=0}^d |\rho^{|\beta|} D^{\beta}f^i|+|\rho^{|\beta|}D^{\beta}g|_{\ell_2}\right)^p \rho^{\Theta-d}dt.
\nonumber
\end{eqnarray}

In particular, if $n=0$, then this yields a version of (<ref>), that is

\begin{equation}
\label{eqn 9.3.1}
\bE \int^T_0 \int_{\cO}(|\rho^{-1}u|+ |u_x|)^p \rho^{\Theta-d}dt
\leq N\bE \int^T_0 \int_{\cO}\big( |\rho f^0|^p+\sum_{i=1}^d |f^i|^p+|g|^p_{\ell_2} \big)^p \rho^{\Theta-d}dt.
\end{equation}

The condition $\Theta \in (d-1, d-1+p)$  is sharp and is not affected by further smoothness of  $\partial \cO$  as long as  $\partial \cO\in C^1$. Note that estimate (<ref>) (or (<ref>)) with smaller $\Theta$ gives better decay of solutions near the boundary than that with larger $\Theta$. In particular, we  have $u(\omega,t,\cdot)\in W^{1,p}_{0}(\cO)$ from (<ref>) if $\Theta\leq d$.

Now,  what if $\partial \cO \not\in C^1$ ? In this case, only few fragmentary results are known.  It turns out  that (<ref>) holds true  on general Lipschitz domains   if $\Theta \approx d-2+p$ (see [9]), and hence  the case $\Theta=d$ is not included in general if $p>2$.    An example in [9] also shows that if  $\Theta<p/2$, then estimate (<ref>) fails  to hold even on  simple wedge domains of the type

\begin{equation}\label{angular domain1}
\cD^{(\kappa)}=\{(r\cos \eta, r\sin \eta)\in \bR^2: r>0, \eta\in (-\kappa/2, \kappa/2)\}, \quad \kappa < 2\pi.
\end{equation}

The vertex $0$ makes the boundary non-smooth and changes the game.

Our interest on conic and polygonal domains arises from such question which, in particular, ask if estimates similar to (<ref>) hold on such simple Lipschitz domains.

There is a saying that, to have a successful regularity result, one has to fully make use of the equation itself rather than a solution representation if any.  It turns out that this is a right precept for  (<ref>) with $C^1$-domains. However,  for conic domains in $\bR^d$, even with angular domains (<ref>) in $\bR^2$, we  got lost in the attempt to obtain a decent estimate, meaning that without big loss of the range of $\Theta$, for SPDEs with strategy using equation and, for a while, we did not know how to handle the behavior of solutions near the pointed part, the vertex $0$, although a conic domain has $C^{\infty}$ boundary except the vertex.

Then we found  a PDE result [15] (also see [26, 28])  which is similar to (<ref>), without the term $g=(g^1, g^2,\cdots)$ of course, that addresses the behavior of solutions to the deterministic heat equation near the vertex. It uses the weight based on the distance to the vertex. Interestingly and then reluctantly enough, the estimate is based on the solution representation using Green's function and quite delicate yet elementary, if we are allowed to say, homogeneity arguments. From that on, the game is to have stochastic counterpart of the result. The main difficulty appears at the very beginning as we use Burkholder-Davis-Gundy inequality, which is the first step toward the analytic approach to SPDEs and makes us lose some information. Hence, the path to the aimed estimate has to be more delicate. A work  on SPDE using a weight system based only on the distance to the vertex is introduced in [3]
(also see [2]), in which we studied the model case of  $d=2$ and $a^{ij}=\delta_{ij}$ for a starter of the program.  Although this work does not use the distance to the boundary, the power of the distance to the vertex used in the weight system has its own sharp range. The important thing is that the range in [3]   is not shrunken compared to the one  [15] for PDEs.

Even for the model case considered in [2, 3]  we struggled to have higher order derivative estimate and left the problem  as the future work. The main issue is to include the distance to the boundary  in our weight system to have a satisfactory regularity relation between solutions and the inputs. In fact, there was an omen of aforementioned difficulty that is implied in the Green's function estimate used in  [3]  and [2]. The estimate dominating Green's function does not vanish at the boundary although it does at the vertex. We need more refined Green's function estimate for the starter of a satisfactory regularity result.

We then set a program of three steps: (i) preparing a refined $d$-dimensional Green's function estimate for  operators with measurable coefficients  (ii) preparing PDE result (iii) establishing SPDE result addressing the higher order derivative estimates. First two steps are done in [7] and [8],  and this article fulfills the last step.  In [7] the refined Green's function estimate involves both the distance to the vertex and the distance to the boundary and  it now vanishes at all the points on the boundary with informative decay rate near the boundary. The work [8] fully makes use of what we prepared in [7] and it is designed to serve this article well.  For instance, in this article we frequently consider the difference of two slightly different SPDEs, which yields a PDE and then the results in [8]  intervene. To make [8] well-prepared for SPDEs, it is important for us to embrace both the strategy using equation itself and the one using a solution representation with Green's function.

Let us explain, in details, the work done in [8] and the role of it in relation with this article.
In [8] we studied the (deterministic) parabolic equation on conic domains, using a mixed weight system involving both the distance to the vertex and the boundary. Now, we have two different ranges of powers of the distances. In  [8] we aim that the range for the power of the distance to the boundary is the same as the $C^1$-domain result and the one for the power of the distance to the vertex embraces the results of the model case in [3]  and [2]. The weight system in [8] is more natural and appropriate compared to the one in [3]  and [2], and leads us to a significant improvement of  the result therein at least with PDEs.

As we proceed to the stochastic counterpart of it, stochastic parabolic equations, we want to use  the same framework, the mixed weight system, to deduce a decent regularity theory.  The effect of stochastic noise term under the Dirichlet boundary condition requires us to design an appropriate weight system for establishing a regularity relation between $f^0$, $f^i$, $i=1,\ldots,d$, $g^k$, $k=1,2,\ldots$ and the solution $u$. For this matter, we want to point out that the weight system in [8] in fact has two purposes; see the introduction in [8]. One is to be a main ingredient of a natural and fine regularity result for deterministic parabolic equation, of course. The other, which related to this article, is to be a still very important ingredient for the stochastic parabolic equations, too.  Let us explain this. Recall
\rho_{\circ}(x):=|x| \quad  \text{and} \quad  \rho(x):=d(x,\partial \cD),
which denote the distance from $x$ to vertex and to the boundary of the conic domain $\cD=\cD(\cM)$, respectively.  We prove that for any $p\ge 2$ and $n=0,1,2,\cdots$, the estimate

\begin{eqnarray}
&&\bE \int^T_0 \int_{\cD} \left(|\rho^{-1}u|^p+|u_x|^p+\cdots+ |\rho^{n}D^{n+1}u|^p\right) \rho_{\circ}^{\theta-\Theta}\rho^{\Theta-d}\, dx\,dt \nonumber
\\
&\leq& N \bE
\int^T_0 \int_{\cD} \Big( |\rho f^0|^p+\cdots+|\rho^{n+1}D^nf^0|^p \nonumber
\\
&&\quad \quad \quad \quad \quad \quad+\sum^d_{i=1}|f^i|^p+\cdots+\sum_{i=1}^d|\rho^{n}D^nf^i|^p \nonumber \\
&&\quad \quad \quad \quad \quad \quad  +|g|_{\ell_2}^p+\cdots+|\rho^{n}D^{n}g|_{\ell_2}^p\Big)
\rho_{\circ}^{\theta-\Theta}\rho^{\Theta-d}\, dx\,dt  \label{main estimate intro}
\end{eqnarray}

holds for the solution $u=u(\omega,t,x)$ to equation (<ref>) with zero initial condition, provided  that

\begin{equation}
\label{theta con intro}
d-1<\Theta<d-1+p, \quad\,\,  p(1-\lambda^+_c)<\theta<p(d-1+\lambda^-_c).
\end{equation}

Here, $\lambda^+_c$ and $\lambda^-_c$ are positive constants  which depend on  $\cM$   and are defined in   Definition <ref> below (also see Proposition <ref> and Remark <ref>). The same estimate holds for polygonal domains in $\bR^2$.  Estimate (<ref>) with  condition (<ref>) is indeed an (seamless) extension of [8] to SPDEs, and what is satisfactory is that the ranges of  $\Theta$ and $\theta$ in (<ref>) are not shrunken smaller than the ranges for the deterministic parabolic equation. For this however very delicate computation is required and providing the work done successfully is one of main purposes of this article.

Regarding Hölder regularity, for the solution $u$ satisfying  (<ref>),  the following is in particular available: if $n+1-\frac{d}{p}= m+\delta$, where $m\in \bN_0$ and $\delta\in (0,1]$, then  for any $0\le k\leq m$

|\rho^{k-1+\frac{\Theta}{p}} \rho^{(\theta-\Theta)/p}_{\circ} D^{k}u(\omega,t,\cdot)|_{C(\cD)}+
[\rho^{m-1+\delta+\frac{\Theta}{p}} \rho^{(\theta-\Theta)/p}_{\circ} D^{m}( \omega,t,\cdot)]_{C^{\delta}(\cD)}<\infty

holds for a.e. $(\omega,t)$, implying

\begin{equation*}
|D^k u(\omega,t,x)|\leq N(\omega,t) \rho^{1-k-\frac{\Theta}{p}}(x) \rho^{(-\theta+\Theta)/p}_{\circ}(x)\quad \text{for all }x\in\cD.
\end{equation*}

; actually, Theorem <ref> and Theorem <ref> provide much general Hölder regularity results with respect to both space and time variables.

Finally, we want to summarize the improvement in this article over the results in [3]  and [2]. Our domains $\cD(\cM)$ in $\bR^d$, $d\ge 2$, generalize two dimensional angular domains (<ref>); the choice of $\cM$ is much richer when $d>2$. Our operator $\sum_{i,j}a^{ij}(\omega,t)D_{ij}$ far generalizes Laplacian operator $\Delta$ delt in [3]  and [2]. These generalizations make computation much more involved, especially, for the stochastic part of the solution. Also, thanks to the mixed weight system, we can now study the higher order derivatives in an appropriate manner and implementing it requires quite a work.  Moreover, in this article we do not pose zero initial condition and hence we propose right function spaces for the initial condition in terms of regularity relations between inputs and output, where the initial condition is one of inputs. This result is new  even with PDEs on conic domains. Hölder regularity results based on aforementioned improvements are also new even for PDEs on conic domains.

This article is organized as follows. In Section 2 we introduce some properties of weighted Sobolev spaces and  present our main results on conic domains, including Hölder regularity results.  In Section 3 we estimate weighed $L_p$ norm of the zero-th order derivative of the solution  on conic domains based on the solution representation via Green's function and elementary but highly involved computations. The estimates of the derivatives of the solution on conic domains are obtained in Section 4 and the proof of the main results on conic domains are posed there, too. In section 5 we establish a regularity theory on polygonal domains in $\bR^2$.

* We use $:=$ to denote a definition.

* For a measure space $(A, \cA, \mu)$, a Banach space $B$ and $p\in[1,\infty)$, we write $L_p(A,\cA, \mu;B)$ for the collection of all $B$-valued $\bar{\cA}$-measurable functions $f$ such that

\|f\|^p_{L_p(A,\cA,\mu;B)}:=\int_{A} \lVert f\rVert^p_{B} \,d\mu<\infty.

Here, $\bar{\cA}$ is the completion of $\cA$ with respect to $\mu$.  The Borel $\sigma$-algebra on a topological space $E$ is denoted by $\cB(E)$. We will drop $\cA$ or $\mu$ or even $B$ in $L_p(A,\cA, \mu;B)$   when they   are obvious in the context.

* $\bR^d$ stands for the $d$-dimensional Euclidean space of points $x=(x^1,\cdots,x^d)$, $B_r(x):=\{y\in \bR^d: |x-y|<r\}$,
$\bR^d_+:=\{x=(x^1,\ldots,x^d): x^1>0\}$, and $S^{d-1}:=\{x\in \bR^d: |x|=1\}$.

* For  a domain $\domain \subset \bR^d$, $B^{\domain}_R(x):=B_R(x)\cap \domain$ and  $Q^{\domain}_R(t,x):=(t-R^2,t]\times B^{\domain}_R(x)$.

* $\bN$ denotes the natural number system, $\bN_0=\{0\}\cup \bN$,  and   $\bZ$ denotes the set of integers.

* For $x$, $y$ in $\bR^d$,  $x\cdot y :=\sum^d_{i=1}x^iy^i$ denotes the standard inner product.

* For a domain $\domain$ in $\bR^d$, $\partial \domain$ denotes the boundary of $\domain$.

* For  any multi-index $\alpha=(\alpha_1,\ldots,\alpha_d)$, $\alpha_i\in \{0\}\cup \bN$,

f_t=\frac{\partial f}{\partial t}, \quad f_{x^i}=D_if:=\frac{\partial f}{\partial x^i}, \quad D^{\alpha}f(x):=D^{\alpha_d}_d\cdots D^{\alpha_1}_1f(x).

We denote $|\alpha|:=\sum_{i=1}^d \alpha_i$.  For the second order derivatives we denote $D_jD_if$ by $D_{ij}f$. We often use the notation
$|gf_x|^p$ for $|g|^p\sum_i|D_if|^p$ and $|gf_{xx}|^p$ for $|g|^p\sum_{i,j}|D_{ij}f|^p$.  We also use $D^m f$ to denote arbitrary partial derivatives  of order $m$ with respect to the space variable.

* $\Delta_x f:=\sum_i D_{ii}f$, the Laplacian for $f$.

* For $n\in \{0\}\cup \bN$,  $W^n_p(\domain):=\{f: \sum_{|\alpha|\le n}\int_{\domain}|D^{\alpha}f|^p dx<\infty\}$, the Sobolev space.

* For a domain $\domain\subseteq\bR^d$ and a Banach space $X$ with the norm $|\cdot|_X$, $C(\domain;X)$ denotes the set of $X$-valued continuous functions $f$ in $\domain$ such that $|f|_{C(\domain;X)}:=\sup_{x\in\domain}|f(x)|_X<\infty$. Also, for $\alpha\in (0,1]$, we define the Hölder space
$C^{\alpha}(\domain;X)$ as the set of all $X$-valued functions $f$ such that

with the semi-norm $[f]_{C^{\alpha}(\domain;X)}$ defined by

[f]_{C^{\alpha}(\domain;X)}=\sup_{x\neq y\in \domain} \frac{|f(x)-f(y)|_X}{|x-y|^{\alpha}}.

In particular, $\domain$ can be an interval in $\bR$.

* For a domain $\domain\subseteq\bR^d$, $\cC^{\infty}_c(\domain)$ is the the space of infinitely differentiable functions with compact support in $\domain$. $supp(f)$ denotes the support of the function $f$. Also, $\cC^{\infty}(\domain)$ denotes the the space of infinitely differentiable functions in $\domain$.

* If $f$ is a distribution with the reference domain $\domain$, then the expression $(f,\varphi)$, $\varphi\in \cC^{\infty}_c(\domain)$, will denote the evaluation of $f$ with the test function $\varphi$.

* For functions $f=f(\omega,t,x)$ depending on $\omega\in\Omega$, $t\geq 0$ and $x\in\bR^d$, we usually drop the argument $\omega$  and  just write $f(t,x)$ when there is no confusion.

* Throughout the article, the letter $N$denotes a finite positive constant which may have different values along the argument  while the dependence  will be informed;  $N=N(a,b,\cdots)$, meaning that  $N$ depends only on the parameters inside the parentheses.

* $A\sim B$ means that there is a constant $N$ independent of $A$ and $B$ such that  $A\leq N B$ and $B\leq N A$.

* $d(x,\domain)$ stands for the distance between a point $x$ and a set $\domain\in\bR^d$.

* $a \vee b =\max\{a,b\}$, $a \wedge b =\min\{a,b\}$.

* $1_U$ the indicator function on $U$.

* We will use the following sets of functions (see [15]).

$\mathcal{V}(Q^{\mathcal{\domain}}_R(t_0,x_0))$ : the set of functions $u$ defined at least on $Q^{\mathcal{\domain}}_R(t_0,x_0)$ and satisfying

\begin{equation*}
\sup_{t\in(t_0-R^2,t_0]}\|u(t,\cdot)\|_{L_2(B^{\domain}_{R}(x_0))} +\|\nabla u\|_{L_2(Q^{\domain}_{R}(t_0,x_0))}<\infty.\nonumber
\end{equation*}

$\mathcal{V}_{loc}(Q^{\domain}_R(t_0,x_0))$ : the set of functions $u$ defined at least on $Q^{\domain}_R(t_0,x_0)$ and satisfying

\begin{equation*}
u\in \mathcal{V}(Q^{\domain}_r(t_0,x_0)),  \quad \forall r\in (0,R).\nonumber
\end{equation*}

SPDE on $d$-dimensional conic domains

Throughout this article we assume $d\ge 2$. Let $\cM$ be a nonempty open set  in $S^{d-1}:=\left\{x\in \bR^d\,:\,|x|=1\right\}$.  We assume $\overline{\cM}^S\neq S^{d-1}$,  where
$\overline{\cM}^S$ denotes the closure of $\cM$ in $S^{d-1}$, and  define the $d$-dimensional conic domain $\mathcal{D}$ by

\mathcal{D}=\cD(\cM):=\Big\{x\in\mathbb{R}^d\setminus\{0\} \ \Big|  \ \ \frac{x}{|x|}\in \mathcal{M} \Big\}.

When $d=2$, the shapes of conic domains are quite simple.  For instance, with a fixed angle $\kappa$ in the range of $\left(0,2\pi\right)$ we can consider

\begin{equation}\label{wedge in 2d}
\mathcal{D}=\mathcal{D}^{(\kappa)}:=\left\{(r\cos\eta,\ r\sin\eta)\in\mathbb{R}^2 \mid r\in(0,\ \infty),\ -\frac{\kappa}{2}<\eta<\frac{\kappa}{2}\right\}.
\end{equation}

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(1.1,1.1) – (3.3,3.3);
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[fill=white][dashed] (1.1,0.55) .. controls (1.4,0.7) and (1.2,1) .. (1.1,1.1) .. controls (1.0,1.2) and (0.8,1.25) .. (0.7,1.3) .. controls (0.5,1.4) and (0.33,1.32) .. (0.3,1.2) .. controls (0.23,0.92) and (0.5,0.85) .. (0.6,0.7) .. controls (0.7,0.55) and (1,0.5) .. (1.1,0.55);

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(0,-3) node $d=3$;

Cases of $d=2$ and $d=3$

Let  $\{w_{t}^{k}\}_{k\in\mathbb{N}}$ be a family of independent one-dimensional
Wiener processes  defined on a complete probability space $(\Omega,\mathscr{F},\bP)$ equipped with an  increasing filtration of
$\sigma$-fields $\mathscr{F}_{t}\subset\mathscr{F}$, each of which
contains all $(\mathscr{F},\bP)$-null sets.  By $\cP$  we  denote the predictable $\sigma$-field  on $\Omega \times (0,\infty)$ generated by $\mathscr{F}_{t}$.

In this article we study the  regularity theory of  the stochastic  partial differential equation

\begin{equation}\label{stochastic parabolic equation}
d u =\left( \cL u+f^0+\sum_{i=1}^d f^i_{x^i}\right)dt +\sum^{\infty}_{k=1} g^kdw_t^k,\quad   t>0, \;x\in \cD(\cM)
\end{equation}

under the zero Dirichlet boundary condition.  Here

\begin{equation}\label{random operator}
\cL := \sum_{i,j=1}^d a^{ij}(\omega,t) D_{ij}.
\end{equation}

- Each of the stochastic integrals  in (<ref>) is understood as an Itô stochastic integral against the given Wiener process.

- The infinite sum of stochastic integral is understood as the limit in probability (uniformly in $t$) of the finite sums of stochastic integrals. See Remark <ref>.

Here are our assumptions on  $\cM$ and the diffusion coefficients.

The boundary $\partial \cM $ of $\cM$  in  $S^{d-1}$ is of class $C^2$.

The diffusion coefficients $a^{ij}$, $i,j=1,\cdots,d$,  are  real-valued $\cP$-measurable  functions of $(\omega,t)$, symmetric; $a^{ij}=a^{ji}$,  and satisfy the uniform parabolicity condition, i.e. there exist  constants $\nu_1, \nu_2>0$ such that for any $t\in\mathbb{R}$, $\omega\in \Omega$ and $\xi=(\xi^1,\ldots,\xi^d)\in\mathbb{R}^d$,

\begin{equation}
\nu_1 |\xi|^2\le \sum_{i,j}a^{ij}(\omega, t)\xi_i\xi_j\le \nu_2 |\xi|^2.  \label{uniform parabolicity}
\end{equation}

To explain our main result in the frame of weighted Sobolev regularity, we  introduce some  function spaces (c.f. [2, 8]). These spaces collect the functions  whose weak derivatives can be measured by the help of appropriate weights consisting of powers of the distance to the vertex  and of the distance to the boundary.  Let  us define

\rho_{\circ}(x)=\rho_{\circ,\cD}:=|x|,\quad \quad \rho(x)=\rho_{\cD}(x):=d(x,\partial\cD)

; $\rho_{\circ}(x)$ denotes the distance from a point $x\in\cD$ to the vertex $0$, the tip point of domain $\cD$, and $\rho(x) $ denotes the distance from a point $x\in\cD$ to the boundary of $\cD$.
For $p\in(1,\infty)$, $\theta\in\bR$ and $\Theta\in \bR$,  we define

and for $m\in \bN_0$ define

K^m_{p,\theta,\Theta}(\cD):=\{f\, :  \rho^{|\alpha|} D^{\alpha}f\in L_{p,\theta,\Theta}(\cD),  \, \,|\alpha|\leq m \}.

The norm in  $K^m_{p,\theta,\Theta}(\cD)$ is defined by

\begin{equation}
\|f\|_{K^m_{p,\theta,\Theta}(\cD)}
=\sum_{|\alpha|\leq m} \left(\int_{\cD} |\rho^{|\alpha|}D^{\alpha}f|^p  \rho_{\circ}^{\theta-\Theta}\rho^{\Theta-d}\,dx\right)^{1/p}. \label{space K norm}
\end{equation}

The space $K^m_{p,\theta,\Theta}(\cD)$ is related to the weighted Sobolev space  $H^m_{p,\Theta}(\cD)$ introduced in [21, 25, 13]  as follows:

% \quad H^n_{p,\Theta}(\cD;l_2):=K^n_{p,\Theta,\Theta}(\cD;l_2), \quad n\in \bN_0.

whose norm is given by

\begin{equation}
\label{eqn 8.9.5}
\|f\|_{H^m_{p,\Theta}(\cD)}:=\sum_{|\alpha|\leq m} \left(\int_{\cD} |\rho^{|\alpha|}D^{\alpha}f|^p \rho^{\Theta-d}\,dx\right)^{1/p}, \quad m\in \bN_0.
\end{equation}

Note that the weight of $H^m_{p,\Theta}(\cD)$ is based only on the distance to the boundary.
Using the fact that for any $\mu\in \bR$ and multi-index $\alpha$

\begin{equation}
\label{eqn 8.28.4}
\sup_{x\in \cD} \rho^{|\alpha|-\mu}_{\circ} |D^{\alpha} \rho^{\mu}_{\circ}(x)|\leq N(\mu,\alpha)<\infty,
\end{equation}

one can easily check

$$f\in K^m_{p,\theta,\Theta}(\cD) \quad \text{ if and only if} \quad  \rho^{(\theta-\Theta)/p}_{\circ}f\in H^m_{p,\Theta}(\cD),
and the norms in their corresponding spaces are equivalents, that is,

\begin{equation}
\label{eqn 8.9.7}
\|f\|_{K^m_{p,\theta,\Theta}(\cD)}\sim \|\rho^{(\theta-\Theta)/p}_{\circ}f\|_{H^m_{p,\Theta}(\cD)}, \quad n\in \bN_0.
\end{equation}

Below we use
relation (<ref>) to define $K^{\gamma}_{p,\theta,\Theta}(\cD)$ for all $\gamma \in \bR$.

Let   $\psi=\psi_{\cD}$  be a smooth function in $\cD$ (see e.g. <cit.>)

such that for any $m\in \bN_0$,

\begin{equation}
\label{eqn 8.9.1}
\psi_{\cD}(x)\sim  \rho_{\cD}(x),\quad \rho^{m}_{\cD}|D^{m+1}\psi_{\cD}|\leq
\end{equation}

Actually, such $\psi$ exists on any domains. Indeed, let $\cO$ be an arbitrary domain, and put $\rho_{\cO}(x)=d(x,\partial \cO)$, and

\begin{equation}
\label{eqn 8.25.1}
\cO_{n,k}:=\{x\in \cO: e^{-n-k}<\rho_{\cO} (x)<e^{-n+k}\}.
\end{equation}

Then mollifying $1_{\cO_{n,2}}$ one can easily construct $\xi_n$ such that

\xi_n \in C^{\infty}_c(\cO_{n,3}), \quad |D^m \xi_n|\leq N(m)e^{mn}, \quad \sum_{n\in \bZ} \xi_n(x) \sim 1,

and then one  can take

\begin{equation}\label{eqn 8.25.2}
\psi=\psi_{\cO}=\sum_{n\in \bZ} e^{-n}\xi_n(x).
\end{equation}

It is easy to check that $\psi=\psi_{\cO}$ satisfies (<ref>) with $\rho_{\cO}$  in place of

Next we choose a nonnegative function $\zeta\in C^{\infty}_{0}(\bR_{+})$ such that  $\zeta>0$ on $[e^{-1},e]$. Then, by the periodicity,

\begin{equation}
\label{11.4.1}
\sum_{n=-\infty}^{\infty}\zeta(e^{n+t})>c>0,\quad\forall\; t\in\bR.
\end{equation}

For  $p\in(1,\infty)$ and $\gamma\in \bR$,  by $H^{\gamma}_p=H^{\gamma}_p(\bR^d)$ we  denote the space of Bessel potential with the norm

\|u\|_{H^{\gamma}_p}:=\|(1-\Delta)^{\gamma/2}u\|_{L_p(\bR^d)}:=\|\cF^{-1}[(1+|\xi|^2)^{\gamma/2} \cF(u)(\xi)]\|_{L_p(\bR^d)}.

In case $\gamma\in \bN_0$, $H^{\gamma}_p(\bR^d)$ coincides with $W^{\gamma}_p(\bR^d)$. The spaces of Bessel potentials enjoys the property

\|u\|_{H^{\gamma_1}_p}\le \|u\|_{H^{\gamma_2}_p},\quad  \gamma_1\le \gamma_2.

Especially, we have $\|u\|_{L_p}\le \|u\|_{H^{\gamma}_p}$ for any $\gamma\ge 0$.
For  $\ell_2$-valued functions $g$  we also define

\|g\|_{H^{\gamma}_p(\ell_2)}:=\||(1-\Delta)^{\gamma/2}g|_{\ell_2}\|_{L_p(\bR^d)}.

Moreover, for $\bR^d$-valued functions $\tbf=(f^1,\ldots,f^d)$  we define

\|\tbf\|_{H^{\gamma}_p(d)}:=\|\,|(1-\Delta)^{\gamma/2}\tbf|\,\|_{L_p(\bR^d)}.

From now on, if a function  defined on  a domain $\cO$ vanishes near the boundary of $\cO$, then by a trivial extension we consider it as a function defined on $\bR^d$. In particular, for any $k\in \bZ$ and a function $f$ on $\cO$, the function $\zeta(e^{-k}\psi_{\cO}(x))f(x)$ has a compact support in $\cO$ and  can be considered as a function on $\bR^d$.

Let $p\in(1,\infty), \Theta, \gamma\in \bR$, and $\cO$ be a domain in $\bR^d$.  By $H^{\gamma}_{p,\theta}(\cO)$ we denote the class of all distributions $f$ on $\cO$ such that

\begin{equation}
\label{eqn 8.10.14}
\|f\|^p_{H^{\gamma}_{p,\Theta}(\cO)}:= \sum_{n\in \bZ} e^{n\Theta} \|\zeta(e^{-n}\psi(e^n\cdot))f(e^{n}\cdot)\|^p_{H^{\gamma}_p(\bR^d)}<\infty,
\end{equation}

where $\psi=\psi_{\cO}$ is taken from (<ref>).  Similarly, $H^{\gamma}_{p,\theta}(\cO;\ell_2)$ is the set of  $\ell_2$-valued functions $g$ such that

\begin{equation}
\label{eqn ell_2}
\|g\|^p_{H^{\gamma}_{p,\Theta}(\cO;\ell_2)}:= \sum_{n\in \bZ} e^{n\Theta} \|\zeta(e^{-n}\psi(e^n\cdot))g(e^{n}\cdot)\|^p_{H^{\gamma}_p(\bR^d;\ell_2)}<\infty.
\end{equation}

It turns out  (see <cit.> or <cit.>) that the new norm in (<ref>) is equivalent to the norm  in  (<ref>) if $\gamma\in \bN_0$. In other words,
for $\gamma \in \bN_0$,

\begin{equation}
\label{eqn 8.9.8}
\sum_{n\in \bZ} e^{n\Theta} \|\zeta(e^{-n}\psi_{\cO} (e^n\cdot))f(e^{n}\cdot)\|^p_{H^{\gamma}_p} \quad \sim \quad  \sum_{|\alpha|\leq \gamma} \int_{\cO} |\rho^{|\alpha|}D^{\alpha}f|^p \rho^{\Theta-d}\,dx,
\end{equation}

and the equivalence relation depends only on $p,\gamma, \Theta,d, n,\zeta, \psi$ and $\cO$.

Now we use equivalence relations (<ref>) and (<ref>),  and  define $K^{\gamma}_{p,\theta,\Theta}(\cD)$ for any chosen $\gamma\in \bR$.

Let $p\in (1,\infty), \theta, \Theta, \gamma \in \bR$, and $\cD$ be a conic domain in $\bR^d$.

We write $f\in K^{\gamma}_{p,\theta,\Theta}(\cD)$ if and only if $\rho^{(\theta-\Theta)/p}_{\circ} f\in H^{\gamma}_{p,\Theta}(\cD)$, and define

\begin{equation}
\label{eqn 8.10.1}
\|f\|_{ K^{\gamma}_{p,\theta,\Theta}(\cD)} := \|\rho^{(\theta-\Theta)/p}_{\circ} f\|_{H^{\gamma}_{p,\Theta}(\cD)}.
\end{equation}

The space $K^{\gamma}_{p,\theta,\Theta}(\cD;\ell_2)$ and its norm are defined similarly.  Also we write
$\tbf=(f^1,f^2,\cdots,f^d)\in K^{\gamma}_{p,\theta,\Theta}(\cD; \bR^d)$ if

\|\tbf\|_{K^{\gamma}_{p,\theta,\Theta}(\cD; \bR^d)}:=\sum_{i=1}^d \|f^i\|_{K^{\gamma}_{p,\theta,\Theta}(\cD)}<\infty.

Note that the new norm of the space $K^{\gamma}_{p,\theta,\Theta}(\cD)$ is equivalent to the previous one if $\gamma\in \bN_0$.
Below we collect some  basic properties of the space $K^{\gamma}_{p,\theta,\Theta}(\cD)$.

Let $p\in(1,\infty)$ and $\theta, \Theta, \gamma \in\bR$.

(i) For a domain $\cO$ and  $\eta\in C^{\infty}_c(\bR_+)$,

\begin{equation}
\label{eqn 4.24.5}
\sum_{n \in\bZ}
e^{n\Theta} \|\eta(e^{-n}\psi_{\cO} (e^n\cdot))f(e^{n}\cdot)\|^p_{H^{\gamma}_p} \leq N(p,\Theta,d,\gamma, \eta, \cO) \|f\|_{H^{\gamma}_{p,\Theta}(\cO)}^{p}.
\end{equation}

The reverse inequality  also holds if $\eta$ satisfies (<ref>).
Moreover, the same statements hold for $\ell_2$-valued functions.

(ii) $\cC_c^{\infty}(\cD)$ is dense in $K^{\gamma}_{p,\theta,\Theta}(\cD)$.

(iii) For any $\mu \in \bR$,

\begin{equation}
\label{eqn 8.19.81}
\|\psi^{\mu}f\|_{K^{\gamma}_{p,\theta,\Theta}(\cD)}\sim \|f\|_{K^{\gamma}_{p,\theta+\mu p, \Theta+\mu p}(\cD)},
\end{equation}

where $\psi$ satisfies (<ref>).  The same statement holds for $\ell_2$-valued functions.

(iv) (Pointwise multiplier) Let $\gamma\in \bR$, $n\in \bN_0$ with $|\gamma|\leq n$. If  $|a|^{(0)}_n:=\sup_{\cD} \sum_{|\alpha|\leq |n|} \rho^{|\alpha|}|D^{\alpha}a|<\infty$, then

\begin{equation}
\label{eqn 8.19.11}
\|af\|_{K^{\gamma}_{p,\theta,\Theta}(\cD)}\leq N(n,p,d)|a|^{(0)}_n \|f\|_{K^{\gamma}_{p,\theta,\Theta}(\cD)}.
\end{equation}

(v) The  operator $D_i:K^{\gamma}_{p,\theta,\Theta}(\cD)\to K^{\gamma-1}_{p,\theta+p,\Theta+p}(\cD)$ is  bounded for any $i=1,\ldots,d$.  In genereal,  for any multi-index $\alpha$  we have

\begin{align}
\label{eqn 4.16.1}
\|D^{\alpha}f\|_{K^{\gamma-|\alpha|}_{p,\theta+|\alpha|p,\Theta+|\alpha|p}(\cD)}\leq N \|f\|_{K^{\gamma}_{p,\theta,\Theta}(\cD)}.
\end{align}

The same statement holds for $\ell_2$-valued functions.

(vi) (Sobolev-Hölder embedding) Let $\gamma-\frac{d}{p}\geq n+\delta$, where $n\in \bN_0$ and $\delta\in (0,1]$.
Then for any $f\in K^{\gamma}_{p,\theta-p,\Theta-p}(\cD)$,

\begin{eqnarray}
&&\sum_{k\leq n} |\rho^{k-1+\frac{\Theta}{p}} \rho^{(\theta-\Theta)/p}_{\circ} D^{k}f|_{C(\cD)}  \nonumber \\
&&\quad + [\rho^{n-1+\delta+\frac{\Theta}{p}} \rho^{(\theta-\Theta)/p}_{\circ} D^{n} f]_{C^{\delta}(\cD)} \leq N \|f\|_{K^{\gamma}_{p,\theta-p,\Theta-p}(\cD)},
\label{eqn 8.21.1}
\end{eqnarray}

where $N=N(d,\gamma,p,\theta,\Theta,\cM)$.

All the results  follow from  Definition <ref> and properties of the weighted Sobolev space $H^{\gamma}_{p,\Theta}(\cO)$ (cf. [25, 21, 16, 13]). See e.g. <cit.>  for (i)-(iii) and see <cit.> for (iv).

To prove (v),  we put $\xi=\rho^{(\theta-\Theta)/p}_{\circ}$. Then, using $\xi Df=D(\xi f)-\xi (\xi^{-1}D \xi) f$ and  (<ref>), we get

\|Df\|_{K^{\gamma-1}_{p,\theta+p,\Theta+p}(\cD)}\leq \|D(\xi f)\|_{H^{\gamma-1}_{p,\Theta+p}(\cD)}+\|(\xi^{-1}D \xi) f\|_{K^{\gamma-1}_{p,\theta+p,\Theta+p}(\cD)}.

By <cit.>,

$$\|D(\xi f)\|_{H^{\gamma-1}_{p,\Theta+p}(\cD)} \leq N  \|\xi f\|_{H^{\gamma}_{p,\Theta}(\cD)}=N\|f\|_{K^{\gamma}_{p,\theta,\Theta}(\cD)}.

Using (<ref>), one can check $|\psi \xi^{-1}D\xi|^{(0)}_m<\infty$ for any $m\in \bN$. Thus, by (<ref>) and (<ref>),

\begin{eqnarray*}
\|(\xi^{-1}D \xi) f\|_{K^{\gamma-1}_{p,\theta+p,\Theta+p}(\cD)}
&\leq&N\| (\psi \xi^{-1}D \xi)  f\|_{K^{\gamma-1}_{p,\theta,\Theta}(\cD)}\leq N \|f\|_{K^{\gamma-1}_{p,\theta,\Theta}(\cD)}.
\end{eqnarray*}

Thus  (v) is proved.

Finally  we prove (vi). Put $v=\xi u$. Then by <cit.>,

\begin{equation}
\label{eqn 8.28.8}
\sum_{k\leq n} |\rho^{k-1+\frac{\Theta}{p}}  D^{k}v|_{C(\cD)}
+ [\rho^{n-1+\delta+\frac{\Theta}{p}} D^{n} v]_{C^{\delta}(\cD)} \leq N \|v\|_{H^{\gamma}_{p,\Theta-p}(\cD)}.
\end{equation}

Hence, to prove (vi), it is enough to note that  the left hand side of (<ref>) is bounded by a constant times of the left hand side of (<ref>).  The lemma is proved.

Using the aforementioned spaces,  we  introduce the function spaces that will embrace  the  solutions $u$ to equation (<ref>) together with the inputs $f^0,\tbf$, and $g$. To make equation (<ref>) well-defined after all, we prescribe  $p\in [2,\infty)$; see Remark <ref> ($i$) below. With such $p$ and a fixed time $T\in(0,\infty)$  we first define

\begin{eqnarray*}\label{entire}
&&\bH^{\gamma}_{p}(T):=L_p(\Omega\times (0,T], \cP ; H^{\gamma}_p),\\
&&\bH^{\gamma}_{p}(T,\ell_2):=L_p(\Omega\times (0,T], \cP ; H^{\gamma}_p(\ell_2)).
\end{eqnarray*}

Next, for   $\theta, \Theta, \gamma \in\bR$  we define the function spaces

\begin{eqnarray*}%\label{space time K norm}
&&\bK^{\gamma}_{p,\theta,\Theta}(\cD,T)\,:=L_p(\Omega\times (0,T], \cP;K^{\gamma}_{p,\theta,\Theta}(\cD)),\\
&&\bK^{\gamma}_{p,\theta,\Theta}(\cD,T,d)\,:=L_p(\Omega\times (0,T], \cP;K^{\gamma}_{p,\theta,\Theta}(\cD; \bR^d)),\\
&&\bK^{\gamma}_{p,\theta,\Theta}(\cD,T, \ell_2)\,:=L_p(\Omega \times (0,T], \cP;K^{\gamma}_{p,\theta,\Theta}(\cD;\ell_2)),
\end{eqnarray*}

and denote

\bL_{p,\theta,\Theta}(\cD,T):=\bK^0_{p,\theta,\Theta}(\cD,T),\quad \bL_{p,\theta,\Theta}(\cD,T,d):=\bK^0_{p,\theta,\Theta}(\cD,T,d),

$$ \bL_{p,\theta,\Theta}(\cD,T, \ell_2):=\bK^0_{p,\theta,\Theta}(\cD,T, \ell_2).

Also, by  $\bK^{\infty}_c(\cD,T)$  we denote  the space of all functions $f$ of the form

\begin{equation}
\label{den fun}
\end{equation}

where $\tau_0\le \cdots\le \tau_m$ is a finite sequence of bounded  stopping times with respect to the filtration $(\rF_t)_{t\geq 0}$, and $f_i\in C^{\infty}_c(\cD)$, $i=1,\ldots,m$. Similarly, we define $\bK^{\infty}_c(\cD,T, \ell_2)$ as the space of $\ell_2$-valued functions $g=(g^1,g^2,
\ldots)$ such that  the first finite number of $g^k$ are in $\bK^{\infty}_c(\cD,T)$ and the rest are all identically zero. We also define $\bK^{\infty}_c(\cD,T, d)$ for  $\bR^d$-valued functions $\tbf=(f^1,\ldots,f^d)$ in the same manner.
Moreover, by $\bK^{\infty}_c(\cD)$ we denote  the space of all functions $f$ of the form

\begin{equation}
\label{defn space function}
\end{equation}

where $A_i\in\rF_0$ and $f_i\in C^{\infty}_c(\cD)$, $i=1,\ldots,m$.

For any $\theta, \Theta, \gamma \in \bR$,   $\bK^{\infty}_c(\cD,T)$ is dense in $\bK^{\gamma}_{p,\theta,\Theta}(\cD,T)$  and so is $\bK^{\infty}_c(\cD,T,\ell_2)$ in $\bK^{\gamma}_{p,\theta,\Theta}(\cD,T,\ell_2)$. Indeed, by the definition of $\cP$, any function $f\in \bK^{\gamma}_{p,\theta,\Theta}(\cD,T)$ can be approximated by functions of the type

\sum_{i=1}^m 1_{(\tau_i(\omega), \tau_{i+1}(\omega)]}(t) h_i(x),

where $\tau_m$ are bounded stopping times and $h_i\in K^{\gamma}_{p,\theta,\Theta}(\cD)$, $i=1,\ldots,m$.  Thus the claim follows from Lemma <ref> (ii).  Similarly, $\bK^{\infty}_c(\cD)$ is dense in $L_p(\Omega;K^{\gamma}_{p,\theta,\Theta}(\cD)):=L_p(\Omega, \rF_0,\bP;K^{\gamma}_{p,\theta,\Theta}(\cD))$.

From now on we will also use the notation

The following definition frames the spaces for the solutions of our SPDE.

Let $p\in[2,\infty)$ and  $\theta, \Theta,\gamma \in\bR$.   We write $u\in\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$ if
$u \in \bK^{\gamma+2}_{p,\theta-p,\Theta-p}(\mathcal{D},T)$, $u(0,\cdot)\in \bU^{\gamma+2}_{p,\theta,\Theta}(\cD):=L_p(\Omega,\rF_0,\bP;U^{\gamma+2}_{p,\theta,\Theta}(\cD))$, and there exists $(\tilde{f}, \tilde{g}) \in\bK^{\gamma}_{p,\theta+p,\Theta+p}(\mathcal{D},T)\times \bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T, \ell_2)$ such that

\begin{align}
du=\tilde{f}\,dt+\sum_k \tilde{g}^kdw^k_t,\quad t\in(0,T] \nonumber
\end{align}

in the sense of distributions on $\cD$, that is,  for any $\varphi\in \cC_c^{\infty}(\cD)$ the equality

\begin{align}
\label{eqn sol}
\end{align}

holds for all $t\in (0,T]$ (a.s.).
In this case we  write

\begin{align*}
\bD u:=\tilde{f}\quad\text{and}\quad \bS u:=\tilde{g}.
\end{align*}

The norm in $\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$  is given  by

\begin{align*}
\|u\|_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)}&=\|u\|_{\bK^{\gamma+2}_{p,\theta-p,\Theta-p}(\cD,T)}+\|\bD u\|_{\bK^{\gamma}_{p,\theta+p,\Theta+p}(\cD,T)}+\|\bS u\|_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,\ell_2)}\\
&\quad +\|u(0,\cdot)\|_{\bU^{\gamma+2}_{p,\theta,\Theta}(\cD)}.
\end{align*}

Let us go back to our main equation (<ref>).  Let  $f^0\in \bK^{\gamma}_{p,\theta+p,\Theta+p}(\cD,T)$,
$\tbf=(f^1,\cdots,f^d)\in \bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T, d)$, $g\in \bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T, \ell_2)$, $u(0,\cdot)\in \bU^{\gamma+2}_{p,\theta,\Theta}(\cD)$, and  $u$ belong to $ \bK^{\gamma+2}_{p,\theta-p,\Theta-p}(\cD,T)$ and be a solution to   equation (<ref>), that is, $u$ satisfies

d u =\left( \cL u+f^0+\sum_{i=1}^d f^i_{x^i}\right)dt +\sum^{\infty}_{k=1} g^kdw_t^k,\quad   t\in(0,T]

in the sense of distributions on $\cD$. Then by (<ref>)  in Lemma <ref> ($v$), we have

\cL u+f^0+\sum_{i=1}^d f^i_{x^i}\in \bK^{\gamma}_{p,\theta+p,\Theta+p}(\cD,T)

and consequently $u$ belongs to $\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$  with the accompanied inequality

\begin{eqnarray}
&& \|u\|_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)}\nonumber\\ &\leq& N \Big(\|u\|_{\bK^{\gamma+2}_{p,\theta-p,\Theta-p}(\cD,T)}+ \|f^0\|_{\bK^{\gamma}_{p,\theta+p,\Theta+p}(\cD,T)}+ \sum_{i=1}^d \|f^i\|_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T)} \nonumber \\
&&\quad \quad +\|g\|_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,\ell_2)}+\|u(0,\cdot)\|_{\bU^{\gamma+2}_{p,\theta,\Theta}(\cD)}\Big).  \label{eqiv norm}
\end{eqnarray}

(i)    Note that for any $m,n\in \bN$ with $m>n$, the quadratic variation of the continuous martingale $\sum_{k=n}^m \int^t_0(\tilde{g}^k, \varphi) dw^k_s$ is $\sum_{k=n}^m \int^t_0 (\tilde{g}^k(s), \varphi)^2ds$.   Following the lines in <cit.> and using the condition $p\geq 2$, one can easily check

\bE\sum_{k=1}^{\infty} \int^T_0 (\tilde{g}^k(t),\varphi)^2 dt
\leq N(\varphi,p,T) \|\tilde{g}\|^p_{\bL_{p,\theta,\Theta}(\cD,T,\ell_2)},

which implies  the infinite series $\sum_{k=1}^{\infty}\int^t_0 (\tilde{g}^k(s),\varphi)dw^k_s$ converges in $L_2\big(\Omega;\cC([0,T])\big)$ and in probability uniformly in $t\in [0,T]$. As a consequence, $(u(t,\cdot),\varphi)$ in (<ref>) is a continuous semi-martingale on $[0,T]$.

(ii) In Definition <ref>, $\bD u$ and $\bS u$ are uniquely determined. This can be seen by using the same arguments in <cit.>.

The space $\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$ we defined is complete;

For any $p\in [2,\infty)$ and  $\theta,\Theta, \gamma\in \bR$,  $\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$ is a   Banach space.

We only need to prove the completeness.  This can be proved by repeating argument in Remark 3.8 of [16], which treats the case $\theta=\Theta$ and
$\cD=\bR^d_+$. The argument in this proof is quite universal and,    without any changes,  works on any conic domain $\cD$ with any $\theta,\Theta\in \bR$.

The following theorem addresses important temporal properties of the functions in $\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$. See Section <ref> for the notations $[\cdot]_{C^{\alpha}}$ and $|\cdot|_{C^{\alpha}}$.

Let $p\in[2,\infty)$ and $\theta,\Theta, \gamma\in \bR$.

(i) If $2/p<\alpha<\beta \leq 1$, then for any $u\in  \cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$,

\begin{eqnarray}
&&\bE [\psi^{\beta-1}u]^p_{C^{\alpha/2-1/p}\left([0,T]; K^{\gamma+2-\beta}_{p,\theta,\Theta}(\cD)\right)}\leq N\,T^{(\beta-\alpha)p/2}\|u\|^p_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)},   \label{eqn 8.10.10}\label{Holder1}
\end{eqnarray}

and in addition, if $\psi^{\beta-1}u(0,\cdot)\in L_p(\Omega;K^{\gamma+2-\beta}_{p,\theta,\Theta}(\cD))$,

\begin{eqnarray}
\bE  |\psi^{\beta-1}u|^p_{C\left([0,T]; K^{\gamma+2-\beta}_{p,\theta,\Theta}(\cD)\right)}&\leq& N\bE\|\psi^{\beta-1}u(0,\cdot)\|^p_{K^{\gamma+2-\beta}_{p,\theta,\Theta}(\cD)}\nonumber\\
&& \quad+N T^{p\beta/2-1}\|u\|^p_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)},\label{Holder2}
\end{eqnarray}

where $\psi$ satisfies (<ref>) and constants $N$ are independent of $T$ and $u$.

(ii) For any $u\in  \cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$ with $u(0,\cdot)= 0$, $u$ belongs to $ L_p(\Omega; C([0,T]; K^{\gamma}_{p,\theta,\Theta}(\cD))$ and

\bE \sup_{t\leq T} \|u(t)\|^p_{K^{\gamma+1}_{p,\theta,\Theta}(\cD)}\leq N \|u\|^p_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)},

where $N=N(d,p,n,\theta,\Theta,\cD, T)$. In particular, for any $t\leq T$,

\begin{equation}
\label{eqn 8.25.31}
\|u\|^p_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,t)}\leq \int^t_0 \bE\sup_{r\leq s} \|u(r)\|^p_{K^{\gamma+1}_{p,\theta,\Theta}(\cD,r)} ds \leq
N\int^t_0 \|u\|^p_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,s)}ds.
\end{equation}

We follow the argument in  <cit.> (or  the proof of <cit.>), using <cit.>.

(i).   As usual, we suppress the argument $\omega$. Put $\xi(x)=|x|^{(\theta-\Theta)/p}$ and set  $v=\xi u$, $\bar{f}=\xi \bD u$, $\bar{g}=\xi \bS u$. Then we have

dv=\bar{f}dt+\sum_{k=1}^{\infty} \bar{g}^k dw^k_t, \quad t\in(0,T]

in the sense of distributions on $\cD$ with the initial condition $v(0,\cdot)=\xi u(0,\cdot)$.   By (<ref>) and Definition <ref>, we have

\begin{eqnarray}
I_1&:=&\bE\left[\psi^{\beta-1}u\right]^p_{C^{\alpha/2-1/p}([0,T],K^{\gamma+2-\beta}_{p,\theta, \Theta}(\cD))} \nonumber\\
&\sim&   \bE\left[v\right]^p_{C^{\alpha/2-1/p}([0,T],H^{\gamma+2-\beta}_{p,\Theta+p(\beta-1)}(\cD))}   \label{eqn 8.20.11}\\
&\leq& N\sum_n e^{n(\Theta+p(\beta-1))}\bE
\left[v(\cdot,e^n\cdot)\zeta(e^{-n}\psi(e^n\cdot))\right]^p_{C^{\alpha/2-1/p}([0,T];H^{\gamma+2-\beta}_{p})}.  \nonumber
\end{eqnarray}

Now, by assumption, the function $v_n(t,x):=v(t,e^nx)\zeta(e^{-n}\psi(e^nx))$ belongs to $\bH^{\gamma+2}_p(T)$ and  satisfies

\begin{equation}
\label{eqn 08.31.1}
dv_n=\bar{f}(t,e^nx)\zeta(e^{-n}\psi(e^nx))dt+ \sum_{k=1}^{\infty} \bar{g}^k(t,e^nx) \zeta(e^{-n}\psi(e^nx)) dw^k_t, \quad t>0
\end{equation}

on the entire space $\bR^d$. Then, by <cit.> and (<ref>), there exists a constant $N>0$,
independent of $T$ and $u$, so that for any constant $a>0$,

\begin{eqnarray*}
NT^{(\beta-\alpha)p/2}a^{\beta-1} \Big(a\|v(\cdot,e^n\cdot)\zeta(e^{-n}\psi(e^n\cdot))\|^p_{\bH^{\gamma+2}_{p}(T)}\\
&&\hspace{1cm}+\,a^{-1}\|\bar{f}(\cdot,e^n\cdot)\zeta(e^{-n}\psi(e^n\cdot))\|^p_{\bH^{\gamma}_{p}(T)}+\|\bar{g}^k(\cdot,e^n\cdot) \zeta(e^{-n}\psi(e^n\cdot)) \|^p_{\bH^{\gamma+1}_{p}(T,\ell_2)} \Big)
\end{eqnarray*}

holds. Taking $a=e^{-np}$, we note that
(<ref>) yields

\begin{eqnarray}
\nonumber
I_1 &\leq& NT^{(\beta-\alpha)p/2}\Big(\sum_n
e^{n(\Theta-p)}\|v(\cdot,e^n\cdot)\zeta(e^{-n}\psi(e^n\cdot))\|^p_{\bH^{\gamma+2}_p(T)}\\  \nonumber
&&\quad\quad+ \sum_n e^{n(\Theta+p)}
\|\bar{f}(\cdot,e^n\cdot)\zeta(e^{-n}\psi(e^n\cdot))\|^p_{\bH^{\gamma}_{p}(T)}\\
&&\quad \quad+\sum_n e^{n\Theta}\|\bar{g}^k(\cdot,e^n\cdot) \zeta(e^{-n}\psi(e^n\cdot)) \|^p_{\bH^{\gamma+1}_p(T,\ell_2)} \Big) \nonumber\\
&=& NT^{(\beta-\alpha)p/2} \left(\|u\|^p_{\bK^{\gamma+2}_{p,\theta-p, \Theta-p}(\cD,T)}
+\|\bD u\|^p_{\bK^{\gamma}_{p,\theta+p,\Theta+p}(\cD,T)} +\|\bS u\|^p_{\bK^{\gamma+1}_{p,\theta, \Theta}(\cD,T,\ell_2)}\right) \nonumber\\
&\le& N T^{(\beta-\alpha)p/2}\|u\|^p_{\cK^{\gamma+2}_{p,\theta, \Theta}(\cD,T)}.  \nonumber
\end{eqnarray}

Thus (<ref>) is proved.

If $\psi^{\beta-1}u(0,\cdot)\in L_p(\Omega;K^{\gamma+2-\beta}_{p,\theta,\Theta}(\cD))$, then we note that  $\psi^{\beta-1}u$ belongs to $C\left([0,T]; K^{\gamma+2-\beta}_{p,\theta,\Theta}(\cD)\right)$ now.  For  estimate (<ref>), we have

\begin{eqnarray}
I_2&:=&\bE  |\psi^{\beta-1}u|^p_{C\left([0,T]; K^{\gamma+2-\beta}_{p,\theta,\Theta}(\cD)\right)}\nonumber\\
&\leq& N\sum_n e^{n(\Theta+p(\beta-1))}\bE
|v(\cdot,e^n\cdot)\zeta(e^{-n}\psi(e^n\cdot))|^p_{C([0,T];H^{\gamma+2-\beta}_{p})}    \label{Holder2 proof}
\end{eqnarray}

and by <cit.> again, for any constant $a>0$,

\begin{eqnarray*}
&&+NT^{p\beta/2-1}a^{\beta-1} \Big(a\|v(\cdot,e^n\cdot)\zeta(e^{-n}\psi(e^n\cdot))\|^p_{\bH^{\gamma+2}_{p}(T)}\\
&&\hspace{0.7cm}+a^{-1}\|\bar{f}(\cdot,e^n\cdot)\zeta(e^{-n}\psi(e^n\cdot))\|^p_{\bH^{\gamma}_{p}(T)}+\|\bar{g}^k(\cdot,e^n\cdot) \zeta(e^{-n}\psi(e^n\cdot)) \|^p_{\bH^{\gamma+1}_{p}(T,\ell_2)} \Big).
\end{eqnarray*}

This, (<ref>), and the same argument above, especially the adjustment $a=e^{-np}$ for each $n$, lead us to (<ref>).

(ii).  We use the notations used in (i). Obviously,

\bE\sup_{t\leq
T}\|u(t)\|^p_{K^{\gamma+1}_{p,\theta,\Theta}(\cD)}\leq N
\sum_ne^{n\Theta} \bE \sup_{t\leq T}
\|v(t,e^n\cdot)\zeta(e^{-n}\psi(e^n\cdot))\|^p_{H^{\gamma+1}_{p}}.

By Remark 4.14 in [16] with $\beta=1$ there,  $v_n \in L_p(\Omega; C([0,T]; H^{\gamma+1}_p))$ and for any $a>0$,

\bE \sup_{t\leq T}
\|v(t,e^n\cdot)\zeta(e^{-n}\psi(e^n\cdot))\|^p_{H^{\gamma+1}_{p}}\leq
N \Big(a\|v(\cdot,e^n\cdot)\zeta(e^{-n}\psi(e^n\cdot))\|^p_{\bH^{\gamma+2}_{p}(T)}

+\|\bar{g}^k(\cdot,e^n\cdot) \zeta(e^{-n}\psi(e^n\cdot)) \|^p_{\bH^{\gamma+1}_{p}(T,\ell_2)}\Big).$$

Again, taking $a=e^{-np}$ and following the above arguments,  we get

\begin{eqnarray*}
&&\bE\sup_{t\leq T}\|u(t)\|^p_{K^{\gamma+1}_{p,\theta, \Theta}(\cD)} \\
N \left(\|u\|^p_{\bK^{\gamma+2}_{p,\theta-p, \Theta-p}(\cD,T)}
+\|\bD u\|^p_{\bK^{\gamma}_{p,\theta+p, \Theta+p}(\cD,T)}+\|\bS u\|^p_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,\ell_2)}\right)\\
&&= N\|u\|^p_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)}.
\end{eqnarray*}

The theorem is proved.

The additional condition  $\psi^{\beta-1}u(0,\cdot)\in L_p(\Omega;K^{\gamma+2-\beta}_{p,\theta,\Theta}(\cD))$ for (<ref>) does not follow  $u(0,\cdot)\in \bU^{\gamma+2}_{p,\theta,\Theta}(\cD)$, which is implied by the assumption $u\in\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$. This is because our domain $\cD=\cD(\cM)$ is unbounded and hence $\psi$ is unbounded. When the domain is bounded, this  additional condition is redundant.

In Definition <ref> and Theorem <ref> the ranges of $\theta$ and $\Theta$ are  open, but certain restrictions will be needed later for our regularity results on  equation (<ref>).
It turns out that the  admissible range of $\theta$ for our Sobolev-regularity theory of equation (<ref>) is affected by the shape of the conic domain $\cD=\cD(\cM)$, the uniform parabolicity of the diffusion coefficients, the space dimension $d$, and the summability parameter $p$. On the the hand, the admissible range of  $\Theta$ depends only on $d$ and  $p$,  that is, we assume

\begin{equation}
\label{eqn 8.10.114}
\end{equation}

Actually condition (<ref>) is used for the heat equation $u_t=\Delta u+f$  on a half space  and it  is sharp (see [21]). Our regularity theory also holds under the same condition on $\Theta$,  and  we believe this is  because the parameter $\Theta$ controls the behavior of solutions near the boundary points away from the vertex, and near such points conic domains just look  like a half space. What interests us, compared with the result of deterministic  equation on conic domain  in [8], is that the range of $\theta$  does not need to be narrowed down for our stochastic parabolic equation.

To explain the admissible range of $\theta$ for equation (<ref>) we need the following definitions. For some of the notations in them one can refer to   Section <ref>.

Let $L=\sum_{i,j=1}^d \alpha^{ij}(t)D_{ij}$ be a uniformly parabolic “deterministic"  operator with bounded coefficients $\alpha^{ij}$s.

(i)  By  $\lambda^+_{c,L}=\lambda^+_{c,L,\cD}$  we denote  the supremum of all $\lambda\geq 0$ such that  for some constant $K_0=K_0(\lambda, L,\cM)$ it holds that

\begin{equation} \label{eqn 8.17.10}
|v(t,x)|\le K_0 \left(\frac{|x|}{R}\right)^{\lambda}\sup_{Q^{\mathcal{D}}_{\frac{3R}{4}}(t_0,0)}\ |v|,
\quad
\forall \;(t,x)\in Q^{\mathcal{D}}_{R/2}(t_0,0)
\end{equation}

for any  $R>0$, $t_0$, and  the deterministic function $v=v(t,x)$ belonging to  $\mathcal{V}_{loc}(Q^{\mathcal{D}}_R(t_0,0))$  and satisfying

\begin{equation}
\label{eqn 8.17.14}
v_t=L v  \quad \text{in}\; Q^{\mathcal{D}}_R(t_0,0)\quad ; \;\quad
v(t,x)=0\quad\text{for}\;\; x\in\partial\mathcal{D}.
\end{equation}

(ii) By   $\lambda^-_{c,L}$  we denote   the supremum of  $\lambda \geq 0$ with above property  for the operator

\begin{equation}
\hat{L}:=\sum_{i,j}\alpha^{ij}(-t)D_{ij}.\label{another operator}
\end{equation}

Note that $K_0$ in (<ref>) may depend on the operator $L$.  Such dependency on $L$ is one of major obstacles when one handles SPDE having random coefficients,  since  it naturally involves infinitely many operators at the same time. To treat such case, which is in fact our case in this article, we design the following definition.

(i) By $\cT_{\nu_1,\nu_2}$ we denote the collection of all “deterministic" operators in the form $L=\sum_{i,j=1}^d \alpha^{ij}(t)D_{ij}$, where $\alpha^{ij}(t)$ are measurable in $t$ and satisfy Assumption <ref> with the fixed constants $\nu_1,\nu_2$ in the uniform parabolicity condition (<ref>).

(ii) For a fixed $\cD=\cD(\cM)$, by $\lambda_c(\nu_1,\nu_2)=\lambda_c(\nu_1,\nu_2,\cD)$ we denote the supremum of all $\lambda\geq 0$ such that for some constant $K_0=K_0(\lambda, \nu_1,\nu_2,\cM)$ it holds that for any operator $L \in \cT_{\nu_1,\nu_2}$, $R>0$ and $t_0$,

\begin{equation}
\label{eqn 8.17.11}
|v(t,x)|\le K_0 \left(\frac{|x|}{R}\right)^{\lambda}\sup_{Q^{\mathcal{D}}_{\frac{3R}{4}}(t_0,0)}\ |v|, \quad
\forall \;(t,x)\in Q^{\mathcal{D}}_{R/2}(t_0,0),
\end{equation}

provided that $v$ is a deterministic function in  $\mathcal{V}_{loc}(Q^{\mathcal{D}}_R(t_0,0))$ satisfying

\begin{equation*}
v_t=L v  \quad \text{in}\; Q^{\mathcal{D}}_R(t_0,0)\quad ; \;\quad
v(t,x)=0\quad\text{for}\;\; x\in\partial\mathcal{D}.
\end{equation*}

(i) Note that the dependency of $K_0$ in Definition <ref> is more explicit compared to that of Definition <ref>. By definitions, if $L$ is an operator in $\cT_{\nu_1,\nu_2}$,  then

$$\lambda^{\pm}_{c,L}\geq \lambda_c(\nu_1,\nu_2).

(ii) The values of $\lambda^{\pm}_{c,L}$ and $\lambda_c(\nu_1,\nu_2)$ do not change if one replaces $\frac{3}{4}$ in (<ref>) and (<ref>) by any number in $(1/2,1)$ (see <cit.>).  Following the proof of  <cit.>, one can also show that for any constant $\beta>0$

\lambda^{\pm}_{c,\beta L}= \lambda^{\pm}_{c,L}, \qquad  \lambda_c(\beta \nu_1,\beta \nu_2)= \lambda_c(\nu_1,\nu_2).

Below are some sharp estimates for $\lambda^{\pm}_{c,L}$ and $\lambda_c(\nu_1,\nu_2)$. See  [7, 15] for more informations.

(i)  If $L=\Delta_x$,  then

\begin{equation}\label{CUB1}
\lambda^{\pm}_{c,L}=-\frac{d-2}{2}+\sqrt{\Lambda+\frac{(d-2)^2}{4}} \, >0,
\end{equation}

where $\Lambda=\Lambda_{\cD}$ is the first eigenvalue of Laplace-Beltrami operator with the Dirichlet condition on  $\mathcal{M}$.
In particular, if $d=2$ and $\cD=\cD^{(\kappa)}$ (see (<ref>)), then

\begin{equation}
\label{kappa-1}
\lambda^{\pm}_{c,L}=\frac{\pi}{\kappa}.
\end{equation}

(ii)  Let $0<\nu_1\leq \nu_2<\infty$.     Then  we have $\lambda_{c}(\nu_1,\nu_2)>0$ and

\begin{equation}\label{CUB3}
\lambda_{c}(\nu_1,\nu_2) \geq  -\frac{d-2}{2}+\sqrt{\frac{\nu_1}{\nu_2}}\sqrt{\Lambda+\frac{(d-2)^2}{4}}.
\end{equation}

(i) follows from  <cit.>, and (<ref>) is a consequence of <cit.>.  Hence, we only need to prove $\lambda_{c}(\nu_1,\nu_2)>0$. In fact, the proof  is the same as the proof of  <cit.>, which gives $\lambda^{\pm}_{c,L}>N>0$ if $L\in \cT_{\nu_1,\nu_2}$, where the constant $N$ comes from the Hölder exponent  of solutions to equation (<ref>) and it can be chosen in the way that it depends only on $\nu_1,\nu_2$ and $\cM$. Hence, the constant $N$ works for all $L\in\cT_{\nu_1,\nu_2}$ and this ends the proof.

For   $\kappa\in (0,2\pi)$ and $\alpha\in [0,2\pi)$,  we consider

\cD=\mathcal{D}_{\kappa,\alpha}:=\left\{x=(r\cos\theta,\ r\sin\theta)\in\mathbb{R}^2 \,|\, r\in(0,\ \infty),\ -\frac{\kappa}{2}+\alpha<\theta<\frac{\kappa}{2}+\alpha\right\}

and  the constant operator

where $a,b,c$ are constants such that $a+c>0$ and $ac-b^2>0$.   Then, by  <cit.>, we have

\begin{align}
\lambda^{\pm}_{c,L}=\lambda^{\pm}_{c,L,\cD_{\kappa,\alpha}}=\frac{\,\pi\,}{\widetilde{\kappa}},   \label{eqn 8.28.9}
\end{align}

\widetilde{\kappa}=\pi-\arctan\Big(\,\frac{\bar{c}\,\cot(\kappa/2)+\bar{b}}{\sqrt{\det(A)}}\,\Big)-\arctan\Big(\,\frac{\bar{c}\,\cot(\kappa/2)-\bar{b}}{\sqrt{\det(A)}}\,\Big)

with constants $\bar{a}, \bar{b}, \bar{c}$ from the relation

\begin{pmatrix} \bar{a} & \bar{b}\\
\bar{b}& \bar{c} \end{pmatrix}
= \begin{pmatrix} \cos \alpha & \sin \alpha\\
-\sin \alpha & \cos \alpha \end{pmatrix}
\begin{pmatrix} a & b\\
b& c  \end{pmatrix}
\begin{pmatrix} \cos \alpha & - \sin \alpha\\
\sin \alpha & \cos \alpha \end{pmatrix}.

In particular, we have $\tilde{\kappa}=\pi$ if $\kappa=\pi$.

Now, let $\kappa\neq \pi$, $\alpha=0$ for $\cD$.  Also, let $b=0$ in $L$.  In this case we can take   $\nu_1=a\wedge c$ and $\nu_2=a\vee c$ in (<ref>). We note that $\tilde{\kappa}$ is determined by the simple relation

\begin{equation}\label{new kappa simpler case of operator}
\tan\Big(\frac{\widetilde{\kappa}}{\,2\,}\Big)=\sqrt{\frac{a}{c}}\tan\Big(\frac{\kappa}{\,2\,}\Big).
\end{equation}

Then, as the function $\tan x/x$ is an increasing function for $x\in (0,\pi)\setminus \{\frac{\pi}{2}\}$, (<ref>) implies

\begin{equation}\label{example for lambda}
\frac{\pi}{\tilde{\kappa}}\ge \sqrt{\frac{a}{c}}\, \frac{\pi}{\kappa}\ge \sqrt{\frac{\nu_1}{\nu_2}}\, \frac{\pi}{\kappa}\quad\textrm{and hence}\quad\lambda^{\pm}_{c,L}\ge \sqrt{\frac{\nu_1}{\nu_2}}\, \sqrt{\Lambda}.
\end{equation}

Interestingly, Proposition <ref> tells us that (<ref>) holds not only for the specified $L$, but also for all operators in $\cT_{\nu_1,\nu_2}$ that share the same constants $\nu_1,\nu_2$ in the uniform parabolicity condition (<ref>).

We are ready to pose our Sobolev  regularity results on conic domains.  We formulate them into two  theorems to handle random and non-random coefficients separately.  The proofs of them are located in Section <ref>.
Note that the admissible range of $\theta$ for non-random coefficients is relatively wider than that of random coefficients.

(SPDE on conic domains with non-random coefficients)

Let $\cL=\sum_{ij}a^{ij}(t)D_{ij}$ be non-random,  $p\in[2,\infty)$, and   $\gamma \geq -1$. Also assume that    Assumptions <ref> and  <ref> hold, and     $\theta, \Theta\in\bR$  satisfy

\begin{equation}
\label{theta}
p(1-\lambda^+_{c,\cL})<\theta<p(d-1+\lambda^-_{c,\cL}), \qquad  d-1<\Theta<d-1+p.
\end{equation}

Then for any $f^0\in\bK^{\gamma \vee 0}_{p,\theta+p,\Theta+p}(\cD,T)$, $\tbf=(f^1,\cdots,f^d)\in \bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,d)$, $g\in\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,l_2)$, and $u_0\in\bU^{\gamma+2}_{p,\theta,\Theta}(\cD)$,  equation (<ref>) has a unique solution $u$ in  the class $\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$ and moreover we have

\begin{eqnarray}\label{main estimate}
\|u\|_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)}
&\leq& N\big(\|f^0\|_{\bK^{\gamma \vee 0}_{p,\theta+p,\Theta+p}(\cD,T)}
+ \|\tbf\|_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,d)}+\|g\|_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,l_2)}\nonumber\\
\end{eqnarray}

where the constant $N$ depends only on $\cM,d,p,\theta,\Theta,\cL, \gamma$.  In particular, it is independent of $T$.

(i) A particular  result of the above theorem  is introduced in  [2] (cf. [3]).  More precisely,   the combination of
Theorem 2.8 and Corollary 2.11 in [2] covers the case

\cL=\Delta, \quad   \Theta=d=2, \quad \cD=\cD^{(\kappa)}\text{ of}\;\; \eqref{wedge in 2d}.

(ii) If $\gamma \geq 0$, the separation of two terms $f^0$ and $\tbf=(f^1,\cdots,f^d)$ in our equation is redundant and we simply pose $f\in\bK^{\gamma}_{p,\theta+p,\Theta+p}(\cD,T)$ instead. This is because, by (<ref>), we have
h^0+\sum_{i=1}^d h^i_{x^i}\in K^{\gamma}_{p,\theta+p,\Theta+p}(\cD)
$ for
h^0\in K^{\gamma}_{p,\theta+p,\Theta+p}(\cD),\;\; h^i \in K^{\gamma+1}_{p,\theta,\Theta}(\cD),\;i=1,\ldots,d.
The corresponding change in the estimate (<ref>) is clear.

(SPDE on conic domains with random coefficients)

Let $\cL=\sum_{ij}a^{ij}(\omega,t)D_{ij}$ be random,  $p\in[2,\infty)$, and   $\gamma \geq -1$. Also assume that    Assumptions <ref> and  <ref> hold,  $d-1<\Theta<d-1+p$, and

\begin{equation}
\label{theta}
\end{equation}

Then all the claims of Theorem <ref> hold with a constant $N=N(\cM,d,p,\gamma,\theta,\Theta,\nu_1,\nu_2)$.

By Proposition <ref>,  (<ref>) is fulfilled if

\begin{equation}
\label{eqn 8.28.10}
p\left(\frac{d}{2}-\sqrt{\frac{\nu_1}{\nu_2}}\sqrt{\Lambda_{\cD}+\frac{(d-2)^2}{4}}\right)<\theta< p\left(\frac{d}{2}+\sqrt{\frac{\nu_1}{\nu_2}}\sqrt{\Lambda_{\cD}+\frac{(d-2)^2}{4}}\right).
\end{equation}

By (<ref>),  inequality (<ref>)  yields (<ref>).  In particular, if $\gamma=-1$ and $u(0,\cdot)\equiv 0$, then we have

\begin{eqnarray*}
&&\bE \int^T_0 \int_{\cD} \left(|\rho^{-1}u|^p+|u_x|^p\right) \rho_{\circ}^{\theta-\Theta}\rho^{\Theta-d}\, dx\,dt \nonumber
\\
&\leq& N\, \bE
\int^T_0 \int_{\cD} \Big( |\rho f^0|^p+\sum_{i=1}^d |f^i|^p+|g|^p_{\ell_2}\Big)
\rho_{\circ}^{\theta-\Theta}\rho^{\Theta-d}\, dx\,dt.
\end{eqnarray*}

The solutions $u$ in Theorems <ref>  and <ref> satisfy zero Dirichlet boundary condition.  Indeed, under the assumption $d-1<\Theta<d-1+p$, <cit.> implies that the trace operator  is well defined for functions in $\bK^1_{p,\theta-p, \Theta-p}(\cD,T)$, and hence by Lemma <ref> (iv)  we have $u|_{\partial \cD}=0$.

Here comes  our  Hölder regularity properties of solutions  on conic domains.

Let $p\in [2,\infty)$, $\theta, \Theta\in \bR$, and $u\in \cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$
be the solution  taken from Theorem <ref> (or  from Theorem <ref>).

(i) If $\gamma+2-\frac{d}{p}\geq n+\delta$, where $n\in \bN_0$ and $\delta\in (0,1]$, then for any $0\le k\leq n$,

|\rho^{k-1+\frac{\Theta}{p}} \rho^{(\theta-\Theta)/p}_{\circ} D^{k}u(\omega,t,\cdot)|_{C(\cD)}+
[\rho^{n-1+\delta+\frac{\Theta}{p}} \rho^{(\theta-\Theta)/p}_{\circ} D^{n}( \omega,t,\cdot)]_{C^{\delta}(\cD)}<\infty

holds for a.e. $(\omega,t)$, in particular,

\begin{equation}
\label{eqn 8.10.21}
|u(\omega,t,x)|\leq N(\omega,t) \rho^{1-\frac{\Theta}{p}}(x) \rho^{(-\theta+\Theta)/p}_{\circ}(x)\quad \text{for all }x\in\cD.
\end{equation}

(ii) Let

2/p<\alpha<\beta\leq 1, \quad \gamma+2-\beta-d/p \geq m+\varepsilon,

where $m\in \bN_0$ and $\varepsilon\in (0,1]$.  Put $\eta=\beta-1+\Theta/p$. Then for any $0\le k\leq m$,

\begin{eqnarray}
\label{eqn 8.11.11}
&&\bE \sup_{t\ne s\leq T}  \frac {\left|\rho^{\eta+k}  \rho^{(\theta-\Theta)/p}_{\circ} \left(D^ku(t,\cdot)-D^ku(s,\cdot)\right)\right|^p_{C(\cD)}}
{|t-s|^{p\alpha/2-1}}<\infty, \\
&& \bE \sup_{t\ne s\leq T}  \frac {\left[\rho^{\eta+m+\varepsilon}  \rho^{(\theta-\Theta)/p}_{\circ} \left(D^mu(t,\cdot)-D^mu(s,\cdot)\right)\right]^p_{C^{\varepsilon}(\cD)}}
{|t-s|^{p\alpha/2-1}} <\infty.  \label{eqn 8.11.21}
\end{eqnarray}

(i) By definition, for almost all $(\omega, t)$, we have $u(\omega,t,\cdot)\in K^{\gamma+2}_{p,\theta-p,\Theta-p}(\cD)$. Thus (i) is a consequence of   (<ref>).  Similarly, the  claims  of (ii)  follow from  (<ref>)  (<ref>), and the observation

\begin{eqnarray*}
&&\bE \sup_{t\ne s\le T} \frac{\|\psi^{\beta-1}(u(t)-u(s))\|^p_{K^{\gamma+2-\beta}_{p,\theta,\Theta}(\cD)}}{|t-s|^{(\alpha/2-1/p)p}} \\
\bE \sup_{t\ne s\le T} \frac{\hspace{1cm}\|u(t)-u(s)\|^p_{K^{\gamma+2-\beta}_{p,\theta+\beta p-p,\Theta+\beta p-p}(\cD)}}{|t-s|^{(\alpha/2-1/p)p}}.
\end{eqnarray*}

(i)  (<ref>) tells how fast the solution  from Theorem <ref> (or  Theorem <ref>) vanishes near the boundary. Near boundary points away from the vertex, $u$ is controlled by   $\rho^{1-\Theta/p}$and, if $p>\Theta$, the decay near the vertex is not slower than   $\rho^{1-\theta/p}_{\circ}$.

(ii) In (<ref>) and (<ref>), $\alpha/2-1/p$ is the  Hölder exponent in time and $\eta$ is related to the decay rate near the boundary.
As  $\alpha/2-1/p\to 1/2-1/p$,   $\eta$ must  increase accordingly.

(iii) Suppose $\theta=d$ satisfies (<ref>), and  let $u\in \cK^{1}_{p,d,d}(\cD,T)$ be the solution from Theorem <ref>. Assume

\kappa_0:=1-\frac{(d+2)}{p}>0.

Then for any $\kappa\in (0,\kappa_0)$, we have

\begin{equation}
\label{eqn 8.11.12}
\bE \sup_{t\leq T} \sup_{x,y\in\cD} \left| \frac{|u(t,x)-u(t,y)|}{|x-y|^{\kappa}}\right|^p + \bE \sup_{t\ne s\leq T}\sup_{x\in \cD}\left| \frac{|u(t,x)-u(s,x)|}{|t-s|^{\kappa/2}}\right|^p <\infty.
\end{equation}

Indeed,  (<ref>) can be  obtained from (<ref>) and (<ref>) with appropriate choices of
$\alpha,\beta$. For the first part, to apply (<ref>)  we  take $\beta=\kappa_0-\kappa+2/p$ such that $2/p<\beta<1$, and take $\varepsilon=1-\beta-d/p=\kappa=-\eta$. For the second part, we use (<ref>) with $\alpha=\kappa+2/p, \beta=1-d/p$ so that $1-\alpha p/2=-p\kappa/2$.

Key  estimates on conic domains

In this section we consider the solutions  to SPDEs having a non-random operator induced from our random operator $\cL$ in (<ref>). Precisely, we  fix any $\omega_0\in \Omega$, and by $L_0$ we denote the deterministic operator

which obviously satisfies the uniform parabolicity condition (<ref>).
We will stimate the zeroth order derivative of the solution of the equation

\begin{equation}\label{one event equation}
d u =\left( L_0 u+f^0+\sum_{i=1}^d f^i_{x^i}\right)dt +\sum^{\infty}_{k=1} g^kdw_t^k,\quad   t>0, \;x\in \cD(\cM).
\end{equation}

The inputs $f^0$, $\bf{f}$, and $g$ can be still random as in (<ref>); we just freeze the operator at a random event $\omega_0$ and make the operator deterministic.  We design and consider equation (<ref>), aiming to obtain the key estimate for the proofs of both Theorem <ref> and Theorem <ref>. Note that  $L_0$ and $\cL$ are the same  if $a^{ij}(t)$ are non-random. We will access the case of $\cL$ being random via the method of continuity; see the proof of Theorem <ref>.
The reason we use  equation (<ref>) is that the operator $L_0$ is much simpler than $\cL$ when $\cL$ is random and most importantly it allows us to have a solution representation of (<ref>) since $L_0$ is deterministic after all.  The estimate of the zeroth-order derivative of the solution of (<ref>) is our key estimate. Based on this, the  full derivative estimates  of solutions of (<ref>) with either non-random $\cL$ or random one will be constructed in the next section.

Let  $G(t,s,x,y)$   denote the Green's function for the operator $\partial_t-L_0$ on $\cD=\cD(\cM)$. By definition (cf. <cit.>), $G$ is a nonnegative function such that  for any fixed $s\in \mathbb{R}$ and $y\in\mathcal{D}$,  the function $v(t,x)=G(t,s,x,y)$ satisfies

\begin{align}
&\Big(\partial_t -L_0\Big)v(t,x)=\delta(x-y)\delta(t-s) \quad  \text{in}\quad \bR \times \cD, \nonumber\\
& v(t,x)=0\quad \textrm{on}\quad  \bR \times\mathcal{\partial D} \; ; \quad  v(t,x)=0\quad \textrm{for} \quad t<s.\label{definition Green's function}
\end{align}

Now, for any given

f^0\in \bL_{p,\theta+p,\Theta+p}(\cD,T),  \quad \textbf{f}=(f^1,\cdots,f^d)\in  \bL_{p,\theta,\Theta}(\cD,T,d), \quad

g\in\bL_{p,\theta,\Theta}(\cD,T, \ell_2), \quad  u_0 \in L_p(\Omega; K^0_{\theta+2-p,\Theta+2-p}(\cD)),

we define  the function $\cR(u_0,f^0,\textbf{f},g)$ by

\begin{eqnarray}
&:=&\int_{\cD} G(t,0,x,y)u_0(y)dy\nonumber\\
\int^t_0\int_{\cD}G(t,s,x,y)f(s,y)dyds-\sum_{i=1}^d \int^t_0\int_{\cD}G_{y^i}(t,s,x,y)f^i(s,y)dyds\nonumber\\
&&+\sum_{k=1}^{\infty}\int^t_0\int_{\cD}G(t,s,x,y)g^k(s,y)dy\,dw^k_s.   \label{eqn 8.21.11}
\end{eqnarray}

One immediately notices that the function $\cR(u_0,f^0,\textbf{f},g)$  is a representation of a solution of (<ref>) with zero boundary condition and initial condition $u(0,\cdot)=u_0(\cdot)$; see Lemma <ref> in the next section.
Our main result of this section is about this representation and it is given in the following lemma.

Let $T<\infty$,  $p\in [2,\infty)$ and let $\theta\in\bR$, $\Theta\in\bR$ satisfy

p(1-\lambda^+_{c,L_0})<\theta<p(d-1+\lambda^-_{c,L_0})\quad\text{and}\quad d-1<\Theta<d-1+p.

If $f^0\in \bL_{p,\theta+p,\Theta+p}(\cD,T)$, $\tbf\in  \bL^d_{p,\theta,\Theta}(\cD,T,d)$, $g\in\bL_{p,\theta,\Theta}(\cD,T, \ell_2)$, and $u_0\in  L_p(\Omega; K^0_{\theta+2-p,\Theta+2-p}(\cD)):=L_p(\Omega,\rF_0; K^0_{\theta+2-p,\Theta+2-p}(\cD))$, then  $u:=\cR(u_0,f^0,\tbf,g)$ belongs to  $\bL_{p,\theta-p,\Theta-p}(\cD,T)$ and  the estimate

\begin{eqnarray}
\|u\|_{\bL_{p,\theta-p,\Theta-p}(\cD,T)}&\leq& N \Big(\|f^0\|_{\bL_{p,\theta+p,\Theta+p}(\cD,T)} + \|\tbf\|_{\bL_{p,\theta,\Theta}(\cD,T,d)}\nonumber \\
\|g\|_{\bL_{p,\theta,\Theta}(\cD,T,\ell_2)} +\|u_0\|_{ L_p(\Omega; K^0_{\theta+2-p,\Theta+2-p}(\cD))}\Big)\nonumber\\
\label{main inequality}
\end{eqnarray}

holds, where  $N=N(\cM,d,p,\theta,\Theta,L_0)$.   Moreover, if

\begin{equation}
\label{eqn 8.18.1}
p\left(1-\lambda_c(\nu_1,\nu_2)\right)<\theta< p\left(d-1+ \lambda_c(\nu_1,\nu_2)\right),
\end{equation}

then the constant $N$ depends only on $\cM,d,p,\theta,\Theta, \nu_1$ and $\nu_2$.

To prove Lemma <ref>, we use the following two results. Lemma <ref> gathers rather technical but important inequalities we keep using in this section.

(i) Let $\alpha+\beta>0,\ \beta > 0$, and $\gamma>0$. Then  for any $a\geq b>0$

\begin{align}\label{eq.lemma3.1}
\int^{\infty}_0 \frac{1}{\left(a+\sqrt{t}\right)^{\alpha}\left(b+\sqrt{t}\right)^{\beta+\gamma}t^{1-\frac{\gamma}{2}}}dt \leq \frac{N}{a^\alpha b^\beta},
\end{align}

where $N= N(\alpha,\beta,\gamma)$.

(ii) Let $\sigma>0,\ \alpha+\gamma>-d$, $\gamma>-1$ and $\beta,\ \nu \in \mathbb{R}$. Then   for any $x\in\cD$,

\begin{align*}
\int_{\mathcal{D}} \frac{|y|^{\alpha}}{\left(|y|+1\right)^{\beta}}\frac{\rho(y)^{\gamma}}{\left(\rho(y)+1\right)^{\nu}}\ e^{-\sigma |x-y|^2} dy \leq
N \left(|x|+1\right)^{\alpha-\beta}\left(\rho(x)+1\right)^{\gamma-\nu},
\end{align*}

where $N=N(\cM,d, \alpha, \beta, \gamma,\nu,\sigma)$.

See Lemma 3.2 and Lemma 3.7 in  [8].

For the operator $L_0$,  we  take the constants $K_0, \lambda^+_{c,L_0}, \lambda^-_{c,L_0}$ and the operator $\hat{L}_0$ from Definition <ref>.

Let $\lambda^+\in \big(0,\lambda^+_{c,L_0}\big)$ and  $\lambda^-\in\big(0,\lambda^-_{c,L_0}\big)$. Denote

K_0^+=K_0(L_0,\cM,\lambda^+), \quad K_0^-=K_0(\hat{L}_0,\cM,\lambda^-).$$

Then,  there exist positive constants
$N=N(\mathcal{M},\nu_1,\nu_2,\,\lambda^{\pm}, K_0^{\pm})$
and $\sigma=\sigma(\nu_1,\nu_2)$ such that for any $t>s$ and $x,y\in \mathcal{D}(\cM)$, the estimates

\begin{align*}
&(i)\quad G(t,s,x,y)\leq \frac{N}{(t-s)^{d/2}}J_{t-s,x}\,J_{t-s,y}\,R^{\lambda^+-1}_{t-s,x}\,R^{\lambda^--1}_{t-s,y}\,e^{-\sigma\frac{|x-y|^2}{t-s}}\\
&(ii)\quad \left|\nabla_y G(t,s,x,y)\right|\leq \frac{N}{(t-s)^{(d+1)/2}}J_{t-s,x}\,R^{\lambda^+-1}_{t-s,x}\,R^{\lambda^--1}_{t-s,y}\,e^{-\sigma\frac{|x-y|^2}{t-s}}
\end{align*}

hold, where

R_{t,x}:=\frac{\rho_{\circ}(x)}{\rho_{\circ}(x)+\sqrt{t}},\quad J_{t,x}:=\frac{\rho(x)}{\rho(x)+\sqrt{t}}.

In particular, if $\lambda^{\pm}\in (0,\lambda_c(\nu_1,\nu_2))$, then $N$ depends only on $\cM, \nu_1,\nu_2, \lambda^{\pm}$.

(i) See inequality (2.8) in [7].

(ii) Denote $\hat{G}(t,s,x,y)=G(-s,-t,y,x)$. Then $\hat{G}$ is  the Green's function of the operator $\partial_t-\hat{L}_0$,
where $\hat{L}_0:=\sum_{i,j}a^{ij}(\omega_0,-t)D_{ij}$.  Then by inequality (2.14) of [7] applied to $\hat{G}$,  for any $\lambda^+\in(0,\lambda_c^+)$ and $\lambda^-\in(0,\lambda_c^-)$, there exist constant $N, \sigma>0$, with the dependencies prescribed in the lemma,  such that

\begin{align*}
|\nabla_x \hat{G}(t,s,x,y)|\leq \frac{N}{(t-s)^{(d+1)/2}}J_{t-s,y}R^{\lambda^--1}_{t-s,x}R^{\lambda^+-1}_{t-s,y}e^{-\sigma\frac{|x-y|^2}{t-s}}
\end{align*}

for any $t>s$ and $x,\,y\in\cD$.  This and  the fact $\nabla_y G(t,s,x,y)=\nabla_x \hat{G}(-s,-t,y,x)$ prove (ii).

Since $\cR(u_0,f^0,\tbf,g)=\cR(u_0,0,0,0)+\cR(0,f^0,\tbf,0)+\cR(0,0,0,g)$ with $0$ as zero functions in their corresponding function spaces, we will treat these three parts  separately in  following three lemmas and then combine them to obtain the claim of Lemma <ref>. Especially, the stochastic part $\cR(0,0,0,g)$  is important in this article and  elaborated thoroughly in Lemma <ref>. The difficulty in handling this part is that we start the analytic investigation only after we apply Burkholder-Davis-Gundy inequality and hence we inevitably lose some subtle information at the very beginning and we should be very careful in every step to reach the goal that we guessed and set.

Let $p\in(1,\infty)$, and let $\theta\in\bR$, $\Theta\in\bR$ satisfy

p(1-\lambda^+_{c,L_0})<\theta<p(d-1+\lambda^-_{c,L_0})\quad\text{and}\quad d-1<\Theta<d-1+p.

If $u_0\in L_p(\Omega; K^0_{\theta+2-p,\Theta+2-p}(\cD))$, then $u=\cR(u_0,0,0,0)$ belongs to $\bL_{p,\theta-p,\Theta-p}(\cD,T)$ and

\begin{align}
\|u\|_{\bL_{p,\theta-p,\Theta-p}(\cD,T)}\leq N \|u_0\|_{L_p(\Omega; K^0_{\theta+2-p,\Theta+2-p}(\cD))}
\label{main inequality}
\end{align}

holds, where  $N=N(\cM,d,p,\theta,\Theta,L_0)$. Moreover, if

\begin{equation}\label{theta range restricted}
\end{equation}

then the constant $N$ depends only on $\cM,d,p,\theta,\Theta,\nu_1$ and $\nu_2$.

Green's function itself is not random. Hence, recalling  the definitions of $\cR(u_0,0,0,0)$ and $\bL=\bK^0$,   for simplicity we may assume that $u_0$ and hence $u$ are non-random and we just prove

\begin{align}
\int^T_0\int_{\cD}|\rho^{-1}u|^p\rho_{\circ}^{\theta-\Theta}\rho^{\Theta-d}dxdt\leq N\int_{\cD}|\rho^{-1+\frac{2}{p}}\,u_0|^p\rho_{\circ}^{\theta-\Theta}\rho^{\Theta-d}dx.
\label{main inequality3}
\end{align}

1. Let us denote $\mu:=-1+(\theta-d+2)/p$, $\alpha:=-1+(\Theta-d+2)/p$, and

Then the claimed estimate (<ref>) turns into a simpler form of

\begin{equation}\label{main inequality 3}
\Big\|\rho_{\circ}^{\mu-\alpha}\rho^{\alpha-\frac{2}{p}}u\Big\|_{L_p\left([0,T]\times \mathcal{D}\right)}\le N \|h\|_{L_p\left(\mathcal{D}\right)}.
\end{equation}

On the other hand, by the range of $\theta$ given in the condition, we can always find $\lambda^+\in\big(0,\lambda^+_{c,L_0}\big)$ and $\lambda^-\in\big(0,\lambda^-_{c,L_0}\big)$ satisfying

\begin{align}
-\frac{d-2}{p}-\lambda^+<\mu<\frac{d-2}{p}+\lambda^-.\label{inequality mu1}
\end{align}

Also, by the given range of $\Theta$  we have

\begin{align}
-1+\frac{1}{p}<\alpha<\frac{1}{p}.\label{inequality alpha1}
\end{align}

Hence, we can choose and fix the constants $\gamma$, $\beta$ satisfying

\begin{align*}
0<\gamma<\lambda^-+\frac{d-2}{p}-\mu\,,\,\quad 0<\beta<\frac{1}{p}-\alpha.
\end{align*}

Noting  $\frac{d-2}{p}<d-\frac{d}{p}$, $\frac1p<2-\frac1p$ which is due to condition $p\in(1,\infty)$,  we then have

\begin{align}
0<\gamma<\lambda^-+d-\frac{d}{p}-\mu\,,\,\quad 0<\beta<2-\frac{1}{p}-\alpha.\label{gamma beta chosen 1}
\end{align}

Moreover, as $\lambda^+\in (0,\lambda^+_{c,L_0})$ and $\lambda^-\in (0,\lambda^-_{c,L_0})$, by Lemma <ref> there exist constants $N=N(\cM,L_0,\nu_1,\nu_2, \lambda^{\pm}),\,\sigma=\sigma(\nu_1,\nu_2)>0$  such that

\begin{align}\label{20.09.15}
\nonumber G(t,0,x,y)&\leq N\,t^{-\frac{d}{2}}\,R^{\lambda^+-1}_{t,x}R^{\lambda^- -1}_{t,y}\,J_{t,x} J_{t,y}\,e^{-\sigma\frac{|x-y|^2}{t}}\\
& = N\,t^{-\frac{d}{2}}\,R^{\lambda^+-1}_{t,x}J_{t,x}\,R^{\gamma}_{t,y}\left(\frac{J_{t,y}}{R_{t,y}}\right)^{\beta}\, R^{\lambda^--\gamma}_{t,y}\left(\frac{J_{t,y}}{R_{t,y}}\right)^{1-\beta}\,e^{-\sigma\frac{|x-y|^2}{t}}
\end{align}

holds for all $t>s$ and $x,y\in\cD$. Let us prove estimate (<ref>).

2. Using Hölder inequality and (<ref>), we have

\begin{align*}
&\leq \int_{\mathcal{D}}G(t,0,x,y)|y|^{-\mu+\alpha}\rho(y)^{-\alpha}|h(y)|dy\\
&\leq N\cdot I_1(t,x)\cdot I_2(t,x),
\end{align*}

where $q=p/(p-1)$; $\frac1p+\frac1q=1$,

I_1(t,x)= \left( \int_{\mathcal{D}}t^{-\frac{d}{2}}\,e^{-\sigma\frac{|x-y|^2}{t}}\cdot R_{t,x}^{(\lambda^+-1)p}J_{t,x}^{\,p}\cdot  K_{1}(t,y)\cdot |h(y)|^pdy\right)^{1/p},

I_2(t,x)=\left(\int_{\mathcal{D}}t^{-\frac{d}{2}}\,e^{-\sigma\frac{|x-y|^2}{t}}\cdot K_{2}(t,y)\cdot |y|^{(-\mu+\alpha)q}\rho^{-\alpha q}(y)dy\right)^{1/q}

K_{1}(t,y) =R^{\gamma p}_{t,y}\left(\frac{J_{t,y}}{R_{t,y}}\right)^{\beta p},\quad \, K_{2}(t,y)=R^{(\lambda^--\gamma)q}_{t,y}\left(\frac{J_{t,y}}{R_{t,y}}\right)^{(1-\beta)q}.

3. We show that there exists a constant $N$ depending only on $\cM, d,p,\theta,\Theta, \nu_1, \nu_2$ and $\lambda^{\pm}$ such that

\begin{align*}
I_2(t,x)\leq  N \left(|x|+\sqrt{t}\right)^{-\mu+\alpha}\left(\rho(x)+\sqrt{t}\right)^{-\alpha}.
\end{align*}

This is done by Lemma <ref> (ii). Indeed, by change of variables $y/\sqrt{t}\to y$ and the fact $\rho(y)/\sqrt{t}=\rho(y/\sqrt{t})$, we have

\begin{align*}
I_2^q(t,x)&=t^{-\frac{d}{2}}\int_{\mathcal{D}}e^{-\sigma\frac{|x-y|^2}{t}}K_{2}(t,y)|y|^{(-\mu+\alpha)q}|\rho(y)|^{-\alpha q}dy\\
&=t^{-\mu q/2}\int_{\mathcal{D}}e^{-\sigma|\frac{x}{\sqrt{t}}-y|^2}\frac{|y|^{(\lambda^--\mu-\gamma-1+\alpha+\beta)q}}{(|y|+1)^{(\lambda^--\gamma-1+\beta)q}}\cdot\frac{\rho(y)^{(1-\alpha-\beta)q}}{(\rho(y)+1)^{(1-\beta)q}}dy,
\end{align*}

for which we can apply Lemma <ref> since (<ref>) implies $(\lambda^--\mu-\gamma)q>-d$ and $(1-\alpha-\beta)q>-1$. Thus we get constant $N=N(\cM,d,p,\theta,\Theta, \lambda^-,\sigma)$ such that

I_2^q(t,x) \leq  N \left(|x|+\sqrt{t}\right)^{(-\mu+\alpha)q}\left(\rho(x)+\sqrt{t}\right)^{-\alpha q}

holds for all $t,x$.

4. To prove estimate (<ref>), by Step 3 we first note

\begin{align*}
|x|^{\mu-\alpha}\rho(x)^{\alpha-\frac{2}{p}}\cdot |u(t,x)|&\leq  N\cdot |x|^{\mu-\alpha}\rho(x)^{\alpha-\frac{2}{p}} \cdot I_1(t,x)\cdot I_2(t,x)\\
& \leq N\,\rho(x)^{-2/p}R^{\,\mu-\alpha}_{t,x} J^{\,\alpha}_{t,x}\cdot I_1(t,x)
\end{align*}

for any $t,x$. Using this and Fubini's Theorem, we have

\begin{align*}
\|\rho_{\circ}^{\mu-\alpha}\rho^{\alpha-\frac2p}u\|^p_{L_p([0,T]\times\mathcal{D})}&\leq N \int_0^T\int_{\mathcal{D}}|\rho(x)|^{-2}\Big(R^{\,\mu-\alpha}_{t,x} J^{\,\alpha}_{t,x}\, I_1(t,x) \Big)^p\,dxdt\\
&=N\int_{\mathcal{D}}I_3(y)\cdot |h(y)|^pdy,
\end{align*}

\begin{align*}
I_3(y)&=\int^T_0t^{-\frac{d}{2}}K_{1}(t,y)\left(\int_{\mathcal{D}}e^{-\sigma\frac{|x-y|^2}{t}}\, R^{\,(\lambda^++\mu-\alpha-1)p}_{t,x} J^{\,(\alpha+1) p}_{t,x}\rho(x)^{-2}\,dx\right)dt.
\end{align*}

Since (<ref>) and (<ref>) imply $(\lambda^++\mu) p-2>-d$ and $(\alpha+1) p-2>-1$, by change of variables $x/\sqrt{t}\to x$, the fact $\rho(x)/\sqrt{t}=\rho(x/\sqrt{t})$, and Lemma <ref> (ii), we have

\begin{align*}
I_3(y)&=\int^T_0\frac{1}{t}K_{1}(t,y)\, \int_{\mathcal{D}}e^{-\sigma|x-\frac{y}{\sqrt{t}}|^2}\frac{|x|^{(\lambda^++\mu-\alpha-1)p}}{(|x|+1)^{(\lambda^++\mu-\alpha-1)p}}\frac{\rho(x)^{(\alpha+1) p-2}}{(\rho(x)+1)^{(\alpha+1) p}}\,dx\,dt\\
&\leq N\int_0^{\infty}K_{1}(t,y)\left(\rho(y)+\sqrt{t}\right)^{-2}dt\\
&= N\int_0^{\infty}\frac{|y|^{(\gamma-\beta)p}}{\left(|y|+\sqrt{t}\right)^{(\gamma-\beta)p}}\cdot\frac{\rho(y)^{\beta p}}{\left(\rho(y)+\sqrt{t}\right)^{\beta p+2}}dt.
\end{align*}

Lastly, owing to $\gamma p>0$, $\beta p> 0$, and the fact $|y|\ge \rho(y)$ in $\cD$, we can apply Lemma <ref> (i) and we obtain

\begin{align*}
I_3(y)\leq N(\cM,d,p,\theta,\Theta,\nu_1,\nu_2, \lambda^{\pm}).
\label{Last constant}
\end{align*}

Hence, there exists a constant $N$ having the dependency described in the lemma such that

\begin{align*}
\left\|\rho_{\circ}^{\mu-\alpha}\rho^{\alpha-\frac2p}u\right\|^p_{L_p([0,T]\times\mathcal{D})}\leq N \|h\|^p_{L_p(\mathcal{D})}.
\end{align*}

Estimate (<ref>) and the lemma are proved.

5. When $\theta$ obeys (<ref>),  we choose $\lambda^{\pm}$ in the interval $(0,\lambda_c(\nu_1,\nu_2))$. Then the constant $N$ of Green's function estimates in Lemma <ref> depends only on $\cM,\nu_1,\nu_2,\lambda^{\pm}$.  Therefore, in particular, constant $N$ in
(<ref>) does not depend on $L_0$.  Tracking the constants down through Steps 1, 2, 3, 4, we note that the constant in (<ref>) does not depend on the particular operator $L_0$. Rather, it depends on $\nu_1,\, \nu_2$ and hence $N=N(\cM,d,p,\theta,\Theta,\nu_1,\nu_2)$.

For $\gamma\ge 0$, $\|u\|_{L_p(\bR^d)}\le \|u\|_{H^{\gamma}_p(\bR^d)}$ is a basic property of the space of Bessel potentials. This with Lemma <ref> and Definition <ref>,  in the context of Lemma <ref>, yields

\begin{equation}
\|u_0\|_{L_p(\Omega; K^0_{\theta+2-p,\Theta+2-p}(\cD))}\le \|u\|_{L_p(\Omega; K^{1-2/p}_{\theta+2-p,\Theta+2-p}(\cD))}=\|u_0\|_{\bU^{1}_{p,\theta,\Theta}(\cD)}
\end{equation}

if $p\ge 2$.

Let $p\in (1,\infty)$ and let $\theta\in\bR$, $\Theta\in\bR$ satisfy

p\big(1-\lambda^+_{c,L_0}\big)<\theta<p\big(d-1+\lambda^-_{c,L_0}\big)\quad\text{and}\quad d-1<\Theta<d-1+p.

If $f^0\in \bL_{p,\theta+p,\Theta+p}(\cD,T)$, $\tbf\in  \bL^d_{p,\theta,\Theta}(\cD,T,d)$, then $u:=\cR(0,f^0,\tbf,0)$ belongs to  $\bL_{p,\theta-p,\Theta-p}(\cD,T)$ and the estimate

\begin{eqnarray*}
\|u\|_{\bL_{p,\theta-p,\Theta-p}(\cD,T)}\leq N \Big(\|f^0\|_{\bL_{p,\theta+p,\Theta+p}(\cD,T)}+ \|\tbf\|_{\bL_{p,\theta,\Theta}(\cD,T,d)}\Big)
\end{eqnarray*}

holds, where  $N=N(\cM,d,p,\theta,\Theta,L_0)$.   Moreover, if

\begin{equation*}
p\left(1-\lambda_c(\nu_1,\nu_2)\right)<\theta< p\left(d-1+ \lambda_c(\nu_1,\nu_2)\right),
\end{equation*}

then the constant $N$ depends only on $\cM,d,p,\theta,\Theta, \nu_1$ and $\nu_2$.

By the same reason explained in the beginning of the proof of  Lemma <ref>,  we can  assume $f^0, \tbf$, and hence $u$ are non-random and we just prove

\begin{align}
\label{main est 2.1}
\int_0^T  \int_{\cD}  |\rho^{-1}u|^p  \rho_0^{\theta-\Theta} \rho^{\theta-d} dxdt
\leq N  \int_0^T  \int_{\cD}  \left(|\rho\,f|^p  +   |\tbf|^p  \right)\rho_0^{\theta-\Theta}  \rho^{\theta-d}  dxdt .
\end{align}

Furthermore, when $\tbf=0$  estimate (<ref>) is already proved in <cit.>, the deterministic counterpart of this article.
Hence, we may assume $f^0=0$. Finally, for simplicity we further assume $f^2=\cdots=f^d=0$.

1. We denote $\mu:=(\theta-d)/p$ and $\alpha:=(\Theta-d)/p$ and set

h(t,x)=\rho_{\circ}^{\mu-\alpha}(x)\rho^{\alpha}(x) f^1(t,x).

Then (<ref>) turns into

\begin{equation}\label{main inequality 2}
\Big\|\rho_0^{\mu-\alpha}\rho^{\alpha-1}u\Big\|_{L_p\left([0,T]\times \mathcal{D}\right)}\le N \|h\|_{L_p\left([0,T]\times \mathcal{D}\right)}.
\end{equation}

We prepare a few things as we did in Step 1 of the proof of Lemma <ref>. By the range of $\theta$ given in the statement, we can find $\lambda^+\in(0,\lambda^+_{c,L_0})$ and $\lambda^-\in(0,\lambda^-_{c,L_0})$ satisfying

\begin{align*}
\end{align*}

Also, by the range of $\Theta$ given we have

\begin{align*}
\end{align*}

Then we can choose and fix the constants $\gamma_1$, $\gamma_2$, $\beta_1$ and $\beta_2$ satisfying

\begin{align}
-\frac{d-1}{p} < \gamma_1 < \lambda^+ - 1 + \mu+\frac{1}{p},&\qquad 0<\gamma_2<\lambda^-+d-1-\frac{d}{p}-\mu\nonumber\\
0<\beta_1<\alpha+\frac{1}{p},\qquad\qquad&\qquad 0<\beta_2<1-\frac{1}{p}-\alpha.\label{inequality gamma beta2}
\end{align}

Moreover, since $\lambda^+\in (0,\lambda^+_c)$ and $\lambda^-\in (0,\lambda^-_c)$, by Lemma <ref> there exist constants $N=N(\cM,L_0,\nu_1, \nu_2, \lambda^{\pm}),\,\sigma=\sigma(\nu_1,\nu_2)>0$ such that

\begin{align}
|\nabla_y G(t,s,x,y)|&\leq \frac{N}{(t-s)^{(d+1)/2}}e^{-\sigma\frac{|x-y|^2}{t-s}}\,J_{t-s,x}R^{\lambda^+-1}_{t-s,x}R^{\lambda^--1}_{t-s,y} \nonumber\\
&=\frac{N}{(t-s)^{(d+1)/2}}e^{-\sigma\frac{|x-y|^2}{t-s}}R^{\gamma_1}_{t-s,x}\left(\frac{J_{t-s,x}}{R_{t-s,x}}\right)^{\beta_1}R^{\gamma_2}_{t-s,y}\left(\frac{J_{t-s,y}}{R_{t-s,y}}\right)^{\beta_2} \nonumber\\
&\qquad \times R^{\lambda^+-\gamma_1}_{t-s,x}\left(\frac{J_{t-s,x}}{R_{t-s,x}}\right)^{1-\beta_1}R^{\lambda^--1-\gamma_2}_{t-s,y}
\left(\frac{J_{t-s,y}}{R_{t-s,y}}\right)^{-\beta_2}  \label{deriv}
\end{align}

holds for all $t>s$ and $x,y\in\cD$. Now, we start proving (<ref>).

By  Hölder inequality and (<ref>), we have

\begin{align}
|u(t,x)|&=\left|\int^t_0\int_{\mathcal{D}}G_{y^1}(t,s,x,y)f^1(s,y)dyds\right| \nonumber\\
&\leq \int^t_0\int_{\mathcal{D}}|\nabla_y G(t,s,x,y)|\cdot|y|^{-\mu+\alpha}\rho(y)^{-\alpha}|h(s,y)|dyds \nonumber\\
&\leq N\cdot I_1(t,x)\cdot I_2(t,x),  \label{I12}
\end{align}

where $q=p/(p-1)$,

I_2(t,x)=\left(\int^t_0\int_{\mathcal{D}}\frac{1}{(t-s)^{(d+1)/2}}\,e^{-\sigma\frac{|x-y|^2}{t-s}}K_{2,1}(t-s,x)K_{2,2}(t-s,y)|y|^{(-\mu+\alpha)q}\rho^{\alpha q}(y)dyds\right)^{1/q}

\begin{align*}
K_{1,1}(t,x)=R^{\gamma_1p}_{t,x}\left(\frac{J_{t,x}}{R_{t,x}}\right)^{\beta_1p},\quad &
K_{1,2}(t,y)=R^{\gamma_2 p}_{t,y}\left(\frac{J_{t,y}}{R_{t,y}}\right)^{\beta_2 p},\\
\end{align*}

3.  We   show that there exists a constant $N=N(\cM,d,p,\theta,\Theta,\nu_1,\nu_2)>0$  such that

\begin{equation}\label{I2}
I_2(t,x)\leq N |x|^{-\mu+\alpha}\rho(x)^{-\alpha+\frac{1}{q}}
\end{equation}

holds for all $t,x$; we note that the right hand side is independent of $t$.

First, by change of variables $y/\sqrt{t-s}\to y$ and Lemma <ref> (ii), which we can apply since (<ref>) gives $(\lambda^--1-\mu-\gamma_2)q>-d$ and $(-\alpha-\beta_2)q>-1$,  we have

\begin{align*}
&\quad\frac{1}{(t-s)^{(d+1)/2}}\int_{\mathcal{D}}e^{-\sigma\frac{|x-y|^2}{t-s}}K_{2,2}(t-s,y)|y|^{(-\mu+\alpha)q}|\rho(y)|^{-\alpha q}dy\\
&=(t-s)^{-(\mu q+1)/2}\int_{\mathcal{D}}e^{-\sigma|\frac{x}{\sqrt{t-s}}-y|^2}\frac{|y|^{(\lambda^--\mu-1-\gamma_2+\alpha+\beta_2)q}}{(|y|+1)^{(\lambda^--1-\gamma_2+
\beta_2)q}}\cdot\frac{\rho(y)^{(-\alpha-\beta_2)q}}{(\rho(y)+1)^{-\beta_2 q}}dy\\
&\leq  N (t-s)^{-1/2}\left(|x|+\sqrt{t-s}\right)^{(-\mu+\alpha)q}\left(\rho(x)+\sqrt{t-s}\right)^{-\alpha q},
\end{align*}

where $N=N(\cM,d,p,\theta,\Theta,\nu_1,\nu_2)$. Using this, we have

\begin{align*}
I_{2}^q(t,x)\,&\leq N\int^t_0K_{2,1}(t-s,x)\cdot(t-s)^{-1/2}\left(|x|+\sqrt{t-s}\right)^{(-\mu+\alpha)q}\left(\rho(x)+\sqrt{t-s}\right)^{-\alpha q}ds\\
&\le N\int^t_{-\infty}\frac{|x|^{(\lambda^+-1-\gamma_1+\beta_1)q}}{(|x|+\sqrt{t-s})^{(\lambda^+-1+\mu-\alpha-\gamma_1+\beta_1)q}}\cdot\frac{\rho(x)^{(1-\beta_1)q}}{(\rho(x)+\sqrt{t-s})^{(1+\alpha-\beta_1)q}}\cdot \frac{1}{(t-s)^{1/2}}ds.
\end{align*}

Then, the change of variable $t-s\to s$ and  Lemma <ref>  (i), which we can apply since we have $(\lambda^++\mu-\gamma_1)q >1 $ and $(1+\alpha-\beta_1)q> 1$ from (<ref>),  we further obtain

\begin{align*}
I_2^q(t,x)&\leq N |x|^{(-\mu+\alpha)q}\rho(x)^{-\alpha q+1},
\end{align*}

which is equivalent to (<ref>).

4. Now, by (<ref>) and   (<ref>)  we have

\begin{align*}
|u(t,x)|\leq N\, I_1(t,x)\cdot I_2(t,x)\leq N\, |x|^{-\mu+\alpha}\rho(x)^{-\alpha+\frac{1}{q}}\, I_1(t,x)
\end{align*}

and hence

\begin{align*}
\|\rho_0^{\mu-\alpha}\rho^{\alpha-1}u\|^p_{L_p([0,T]\times\mathcal{D})}&\leq N \int_0^T\int_{\mathcal{D}}|\rho(x)|^{-1}I_1^p(t,x)\,dxdt\\
&=N\int_0^T\int_{\mathcal{D}}I_3(s,y)\cdot |h(s,y)|^pdyds,
\end{align*}

\begin{align*}
\end{align*}

; we used Fubini's Theorem. By change of variables $t-s\to t$ followed by $x/\sqrt{t}\to x$ and Lemma <ref> (ii) with $\gamma_1p-1>-d$ and $\beta_1p-1>-1$ from (<ref>), we have

\begin{align*}
&\leq \int^{\infty}_0\frac{1}{t}K_{1,2}(t,y)\left(\int_{\mathcal{D}}\frac{|x|^{(\gamma_1-\beta_1)p}}{(|x|+1)^{(\gamma_1-\beta_1)p}}\frac{\rho(x)^{\beta_1p-1}}{(\rho(x)+1)^{\beta_1p}}e^{-\sigma|x-\frac{y}{\sqrt{t}}|^2}\,dx\right)dt\\
&\leq N\int_0^{\infty}K_{1,2}(t,y)\left(\rho(y)+\sqrt{t}\right)^{-1}t^{-1/2}dt\\
&= N\int_0^{\infty}\frac{|y|^{(\gamma_2-\beta_2)p}}{\left(|y|+\sqrt{t}\right)^{(\gamma_2-\beta_2)p}}\cdot\frac{\rho(y)^{\beta_2p}}{\left(\rho(y)+\sqrt{t}\right)^{\beta_2p+1}}\cdot\frac{1}{t^{1/2}}dt.
\end{align*}

Lastly, due to $\gamma_2p>0$ and $\nu_2p> 0$, Lemma <ref> (i)  yields

\begin{align*}
I_3(s,y)\leq N(\cM,d,p,\theta,\Theta, \nu_1,\nu_2).
\end{align*}

Hence, there exists a constant $N$ having the dependency described in the lemma  such that

\begin{align*}
\left\|\rho_{\circ}^{\mu-\alpha}\rho^{\alpha-1}u\right\|^p_{L_p([0,T]\times\mathcal{D})}\leq N \|h\|^p_{L_p([0,T]\times\mathcal{D})}.
\end{align*}

(<ref>) and the lemma are proved.

5.  The last part of the claim related to the range of $\theta$ holds by the same reason explained in Step 5 of the proof of Lemma <ref>.

Now, we move on to the stochastic part, the most important and involved one.

Let $p\in [2,\infty)$ and let $\theta\in\bR$, $\Theta\in\bR$ satisfy

p(1-\lambda^+_{c,L_0})<\theta<p(d-1+\lambda^-_{c,L_0})\quad\text{and}\quad d-1<\Theta<d-1+p.

If $g\in\bL_{p,\theta,\Theta}(\cD,T, \ell_2)$, then $u:=\cR(0,0,0,g)$ belongs to  $\bL_{p,\theta-p,\Theta-p}(\cD,T)$ and  the estimate

\begin{eqnarray}\label{main inequality2}
\|u\|_{\bL_{p,\theta-p,\Theta-p}(\cD,T)}\leq N \|g\|_{\bL_{p,\theta,\Theta}(\cD,T,\ell_2)}
\end{eqnarray}

holds, where  $N=N(\cM,d,p,\theta,\Theta,L_0)$.   Moreover, if

\begin{equation}
\label{eqn 8.18.1.2}
p\left(1-\lambda_c(\nu_1,\nu_2)\right)<\theta< p\left(d-1+ \lambda_c(\nu_1,\nu_2)\right),
\end{equation}

then the constant $N$ depends only on $\cM,d,p,\theta,\Theta, \nu_1$, and $\nu_2$.

1. Again, we denote $\mu:=(\theta-d)/p$ and $\alpha:=(\Theta-d)/p$.
We put $h(\omega,t,x)=\rho_{\circ}^{\mu-\alpha}(x) \rho(x)^\alpha g(\omega,t,x)$ and recall

$$\Omega_T=\Omega \times (0,T], \quad L_p(\Omega_T \times \cD):=L_p(\Omega_T\times \cD, d\bP dt dx).

Then (<ref>) is the same as

\begin{equation}\label{main inequality 2}
\left\|\rho_{\circ}^{\mu-\alpha}\rho^{\alpha-1}u\right\|^p_{L_p(\Omega_T\times\mathcal{D})}\le N \||h|_{\ell_2}\|^p_{L_p\left(\Omega_T\times \mathcal{D}\right)}.
\end{equation}

As we did in the proof of Lemma <ref>, we prepare few things.  By the range of $\theta$ given, we can find constants $\lambda^+\in(0,\lambda^+_{c,L_0})$ and $\lambda^-\in(0,\lambda^-_{c,L_0})$ satisfying

\begin{align*}
\end{align*}

Also, by the range of $\Theta$ we have

\begin{align*}
\end{align*}

Then we can choose and fix the constants $\gamma_1$, $\gamma_2$, $\beta_1$, and $\beta_2$ satisfying

\begin{align}
-\frac{d-2}{p} < \gamma_1 < \lambda^+ - 1 + \mu + \frac{2}{p},&\qquad 0<\gamma_2<\lambda^-+d-\frac{d}{p}-\mu\nonumber\\
\frac{1}{p}<\beta_1<\alpha+\frac{2}{p},\qquad\qquad & \qquad 0<\beta_2<2-\frac{1}{p}-\alpha.\label{inequality gamma beta3}
\end{align}

Further, by Lemma <ref> there exist constants $N=N(\cM,L_0,\nu_1, \nu_2, \lambda^{\pm}),\,\sigma=\sigma(\nu_1,\nu_2)>0>0$ such that for any $t>s$ and $x,y\in\cD$,

\begin{align}
\nonumber
G(t,s,x,y)\leq& \frac{N}{(t-s)^{d/2}}e^{-\sigma\frac{|x-y|^2}{t-s}} \,J_{t-s,x}\, J_{t-s,y}\,R^{\lambda^+-1}_{t-s,x}\,R^{\lambda^--1}_{t-s,y}\\   \nonumber
=&N\, (t-s)^{-d/2}e^{-\sigma\frac{|x-y|^2}{t-s}} R^{\gamma_1}_{t-s,x}\left(\frac{J_{t-s,x}}{R_{t-s,x}}\right)^{\beta_1}
& \times   R^{\lambda^+-\gamma_1}_{t-s,x}\left(\frac{J_{t-s,x}}{R_{t-s,x}}\right)^{1-\beta_1}
R^{\lambda^--\gamma_2}_{t-s,y}\left(\frac{J_{t-s,y}}{R_{t-s,y}}\right)^{1-\beta_2} \label{eqn 8.7.3}
\end{align}

2.  We first estimate the $p$-th moment  $\bE|u(t,x)|^p$ for any given $t$ and $x$. Using Burkholder-Davis-Gundy inequality and Minkowski's integral inequality, we have

\begin{align*}
\bE|u(t,x)|^p&=\bE\left|\sum_{k\in\bN}\int^t_0\int_{\mathcal{D}}G(t,s,x,y)g^k(s,y)dydw^k_s\right|^p\\
&\leq N\bE\left(\int_0^t\sum_{k\in\bN}\left(\int_{\cD}G(t,s,x,y)g^k(s,y)dy\right)^2 ds \right)^{p/2}\\
&\leq N\bE\left(\int_0^t\left(\int_{\cD}G(t,s,x,y)|g(s,y)|_{\ell_2}dy\right)^2 ds \right)^{p/2}\\
&=N\bE\left(\int_0^t\left(\int_{\cD}G(t,s,x,y)|y|^{-\mu+\alpha}\rho(y)^{-\alpha}|h(s,y)|_{\ell_2}dy\right)^2 ds \right)^{p/2}.
\end{align*}

We denote

I(\omega,t,x):=\left(\int_0^t\left(\int_{\cD}G(t,s,x,y)|y|^{-\mu+\alpha}\rho(y)^{-\alpha}|h(\omega,s,y)|_{\ell_2}dy\right)^2 ds \right)^{1/2}.

Then,  using  (<ref>) and applying Hölder inequality twice for $x$ and then $t$, we get

\begin{align}
I(\omega,t,x)&\leq\,N\left(\int_0^t\Big(\int_{\cD}I_1\cdot I_2\;dy\Big)^2ds\right)^{1/2}  \label{eqn 8.7.7}\\
&\leq N\|I_1(\omega,t,\cdot,x,\cdot)\|_{L_p((0,t)\times \cD, ds\,dy)}\Big\|\|I_2(t,\cdot,x,\cdot)\|_{L_{q}(\cD,dy)}\Big\|_{L_{r}((0,t),ds)}  \nonumber
\end{align}

where $q=\frac{p}{p-1}$, $r=\frac{2p}{p-2}\,(=\infty\text{ if }p=2)$,

\begin{align}
&I_1^p(\omega,t,s,x,y)    \label{I_1^p}\\
&= (t-s)^{-d/2}e^{-\sigma\frac{|x-y|^2}{t-s}} \left(R^{\gamma_1}_{t-s,x}\left(\frac{J_{t-s,x}}{R_{t-s,x}}\right)^{\beta_1}R^{\gamma_2}_{t-s,y}\left(\frac{J_{t-s,y}}{R_{t-s,y}}\right)^{\beta_2}\right)^p|h(\omega,s,y)|_{\ell_2}^p   \nonumber \\
&= (t-s)^{-d/2}e^{-\sigma\frac{|x-y|^2}{t-s}} \,\,K_{1,1}(t-s,x)\,K_{1,2}(t-s,y)\,\,|h(\omega,s,y)|_{\ell_2}^p,\label{I_1^p} \nonumber
\end{align}

\begin{align*}
&= (t-s)^{-d/2}e^{-\sigma\frac{|x-y|^2}{t-s}} \left(R^{\lambda_1-\gamma_1}_{t-s,x}\left(\frac{J_{t-s,x}}{R_{t-s,x}}\right)^{1-\beta_1}R^{\lambda_2-\gamma_2}_{t-s,y}\left(\frac{J_{t-s,y}}{R_{t-s,y}}\right)^{1-\beta_2}\right)^q|y|^{(-\mu+\alpha)q}\rho(y)^{-\alpha q}\\
&= (t-s)^{-d/2}e^{-\sigma\frac{|x-y|^2}{t-s}} \,\,K_{2,1}(t-s,x)\,K_{2,2}(t-s,y)\,\,|y|^{(-\mu+\alpha)q}\rho(y)^{-\alpha q},
\end{align*}

\begin{align*}
K_{1,2}(t,y)=R^{\gamma_2 p}_{t,y}\left(\frac{J_{t,y}}{R_{t,y}}\right)^{\beta_2 p},\\
\end{align*}

Note, by (<ref>) we have

\begin{eqnarray}
\label{eqn 8.7.8}
\bE |u(t,x)|^p &\leq& N \bE I^p(t,x)  \\
&\leq& N \Big\|\|I_2(t,\cdot,x,\cdot)\|_{L_{q}(\cD,dy)}\Big\|^p_{L_{r}((0,t),ds)}
\bE\|I_1(\omega,t,\cdot,x,\cdot)\|^p_{L_p((0,t)\times \cD, ds\,dy)}. \nonumber
\end{eqnarray}

3. In this step we will show that there exists a constant $N=N(\cM,d,p,\theta,\Theta,\nu_1,\nu_2)>0$ such that

\begin{equation}\label{I_2}
\big\|\|I_2(t,\cdot,x,\cdot)\|_{L_{q}(\cD,dy)}\big\|_{L_{r}((0,t),ds)}\leq N |x|^{-\mu+\alpha}\rho(x)^{-\alpha+1-2/p}.
\end{equation}

In particular, the right hand side  is independent of $t$.

Case 1. Assume $p=2$ (hence, $q=2$ and $r=\infty$). First, we consider

\begin{align*}
&\int_{\cD}I_2^2(t,s,x,y) \,dy\\
\end{align*}

Since $2(\lambda^--\mu-\gamma_2)>-d$ and $2(1-\alpha-\beta_2)>-1$ from (<ref>), by change of variables $y/\sqrt{t-s}\to y$ and Lemma <ref> (ii), we have

\begin{align*}
&\leq N\left(|x|+\sqrt{t-s}\right)^{2(-\mu+\alpha)}\left(\rho(x)+\sqrt{t-s}\right)^{-2\alpha}.
\end{align*}

Hence, we have

\begin{align*}
&\leq N\sup_{s\in[0,t]}\left(K_{2,1}(t-s,x)\cdot\left(|x|+\sqrt{t-s}\right)^{2(-\mu+\alpha)}\left(\rho(x)+\sqrt{t-s}\right)^{-2\alpha}\right)^{1/2}\\
&= N\sup_{s\in[0,t]}\left(\frac{|x|^{\lambda^+-1-\gamma_1+\beta_1}}{(|x|+\sqrt{t-s})^{\lambda^+-1+\mu-\gamma_1-\alpha+\beta_1}}\cdot\frac{\rho(x)^{1-\beta_1}}{(\rho(x)+\sqrt{t-s})^{\alpha+1-\beta_1}}\right)\\
&\leq N |x|^{-\mu+\alpha}\rho(x)^{-\alpha}
\end{align*}

due to $\lambda^+-1+\mu-\gamma_1>0$, $\alpha+1-\beta_1>0$ and
$0\leq J_{t-s,x}\leq R_{t-s,x}\leq 1$. Thus (<ref>) holds.

Case 2. Let $p>2$.
Again, since $(\lambda^--\mu-\gamma_2)q>-d$ and $(1-\alpha-\beta_2)q>-1$, by change of variables and Lemma <ref> (ii), we observe

\begin{align*}
&\frac{1}{(t-s)^{d/2}}\int_{\mathcal{D}}e^{-\sigma\frac{|x-y|^2}{t-s}}K_{2,2}(t-s,y)|y|^{(-\mu+\alpha)q}\rho(y)^{-\alpha q}dy\\
=&\,(t-s)^{-\mu q/2}\int_{\mathcal{D}}e^{-\sigma|\frac{x}{\sqrt{t-s}}-y|^2}\frac{|y|^{(\lambda^--\mu-\gamma_2-1+\alpha+\beta_2)q}}{(|y|+1)^{(\lambda^--\gamma_2-1+\beta_2)q}}\cdot\frac{\rho(y)^{(1-\alpha-\beta_2)q}}{(\rho(y)+1)^{(1-\beta_2)q}}dy\\
\leq&\,N \left(|x|+\sqrt{t-s}\right)^{(-\mu+\alpha)q}\left(\rho(x)+\sqrt{t-s}\right)^{-\alpha q}.
\end{align*}

Hence, we have

\begin{align*}
&\int_0^t\|I_2(t,s,x,\cdot)\|_{L_{q}(\cD,dy)}^r ds\\
\leq &\,N \int^t_0\left\{K_{2,1}(t-s,x)\cdot\left(|x|+\sqrt{t-s}\right)^{(-\mu+\alpha)q}\left(\rho(x)+\sqrt{t-s}\right)^{-\alpha q}\right\}^{r/q}ds\\
=&\,N \int^t_0\frac{|x|^{(\lambda^+-1-\gamma_1+\beta_1)r}}{(|x|+\sqrt{t-s})^{(\lambda^+-1+\mu-\gamma_1-\alpha+\beta_1)r}}\cdot\frac{\rho(x)^{(1-\beta_1)r}}{(\rho(x)+\sqrt{t-s})^{(\alpha+1-\beta_1)r}}ds\,.
\end{align*}

Moreover, since (<ref>) also gives $\left(\lambda^++\mu-\gamma_1\right)r>2$ and $\left(\alpha+1-\beta_1\right)r>2$,
using Lemma <ref> we again obtain

\begin{align*}
\big\|\|I_2(t,\cdot,x,\cdot)\|_{L_{q}(dy;\cD)}\big\|_{L_{r}((0,t),ds)}&=\left(\int_0^t\|I_2\|_{L_{q}(\cD,dy)}^r ds\right)^{1/r}\\
&\leq N |x|^{-\mu+\alpha}\rho(x)^{-\alpha+1-2/p}.
\end{align*}

4.  Now, we return to (<ref>). Then by (<ref>) and (<ref>) we have

\begin{align*}
&\leq N \big(|x|^{\mu-\alpha}\rho(x)^{\alpha-1}\big)^p \cdot \bE\,\int_0^t\int_{\cD}I_1^p(t,s,x,y) dy\,ds\cdot \big(|x|^{-\mu+\alpha}\rho(x)^{-\alpha+1-2/p}\big)^p\\
&= N\,\rho(x)^{-2}\;\bE\,\int_0^t\int_{\cD}I_1^p(t,s,x,y) dy\,ds.
\end{align*}

Therefore, taking integrations with respect to $x$ and $t$, using Fubini theorem and recalling (<ref>), we have

\begin{align}
\nonumber
\bE\,\|\rho_0^{\mu-\alpha}\rho^{\alpha-1}u\|^p_{L_p(\Omega_T\times\mathcal{D})}&\leq N \,\bE\int_0^T\int_{\mathcal{D}}\int^t_0\int_{\cD}|\rho(x)|^{-2}I_1^p\,dyds\;dxdt\\
&=N\,\bE\,\int_0^T\int_{\mathcal{D}}I_3(s,y)\cdot |h(s,y)|_{\ell_2}^pdyds,   \label{eqn 8.11.31}
\end{align}

\begin{align*}
\end{align*}

Since (<ref>) also implies $\gamma_1p-2>-d$ and $\beta_1p-2>-1$, by change of variables $T-t\to t$ followed by $x/\sqrt{t}\to t$ and Lemma <ref> (ii),  we have

\begin{align*}
&\leq \int^{\infty}_0\frac{1}{t}K_{1,2}(t,y)\left(\int_{\mathcal{D}}\frac{|x|^{(\gamma_1-\beta_1)p}}{(|x|+1)^{(\gamma_1-\beta_1)p}}\frac{\rho(x)^{\beta_1p-2}}{(\rho(x)+1)^{\beta_1p}}e^{-\sigma'|x-\frac{y}{\sqrt{t}}|^2}\,dx\right)dt\\
&\leq N\int_0^{\infty}K_{1,2}(t,y)\left(\rho(y)+\sqrt{t}\right)^{-2}dt\\
&= N\int_0^{\infty}\frac{|y|^{(\gamma_2-\beta_2)p}}{\left(|y|+\sqrt{t}\right)^{(\gamma_2-\beta_2)p}}\cdot\frac{\rho(y)^{\beta_2p}}{\left(\rho(y)+\sqrt{t}\right)^{\beta_2p+2}}\,dt.
\end{align*}

Hence, by Lemma <ref> (i) with the conditions $\gamma_2p>0$ and $\beta_2p> 0$, we finally get

\begin{align*}
I_3(s,y)\leq N(\cM,d,p,\theta,\Theta, \nu_1,\nu_2).
\end{align*}

This and (<ref>) lead to  (<ref>) and the lemma is proved.

5.  Again, the last part of the claim related to the range of $\theta$ holds by the same reason explained in Step 5 of the proof of Lemma <ref>.

Proof of Theorems <ref> and <ref>
In this section we prove Theorems <ref> and <ref>, following the strategy below:

1.  A priori estimate and the uniqueness:

Using Definition <ref> and a result on the entire space $\bR^d$ (Lemma <ref> below), we first prove that for any   solution $u\in \cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$ to equation (<ref>) equipped with the general operator $\cL=\sum_{i,j=1}^d a^{ij}(\omega,t)D_{ij}$, we have

\begin{eqnarray}
&& \|u\|_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)}
\leq N \big( \|u\|_{\bL_{p,\theta-p,\Theta-p}(\cD,T)} +\text{norms of the force terms}\nonumber\\
&&\hspace{4cm}+\text{norm of the initial condition}\big). \label{eqn 8.18.21}
\end{eqnarray}

-    If  $\cL$ is non-random, we can  estimate $\|u\|_{\bL_{p,\theta-p,\Theta-p}(\cD,T)}$ based on  Lemma <ref>.

- To treat the SPDE with random coefficients, we introduce a SPDE having non-random coefficients equipped with the same inputs $f^0$, $\tbf$, $g$, and initial condition.  The solution to this  new SPDE can be estimated   by the above two steps. Then we prepare a priori estimate for a package of SPDEs containing the original one and the new one. For this,
we consider
the difference between the new SPDE and any SPDE  in the package.

-  The uniqueness of solution to SPDEs follows from the uniqueness result of PDEs.

2. The existence:

-  If the coefficients of $\cL$ are non-random, we use the representation formula.

- For general case, we use the method of continuity with the help of the a priori estimate.

Now we start our proofs. The following lemma is what we meant in (<ref>).
We emphasize that the lemma holds for any $\theta,\Theta\in \bR$ and the condition $\partial \cM\in C^2$ is not  needed in the proof.

Let $p\in [2,\infty)$,    $\gamma, \mu, \theta, \Theta \in\bR$,  $\mu<\gamma$, and the diffusion coefficients $a^{ij}=a^{ij}(\omega,t)$ satisfy   Assumption  <ref>.  Assume that   $f^0\in\bK^{\gamma}_{p,\theta+p,\Theta+p}(\cD,T)$, $\tbf=(f^1,\cdots,f^d) \in\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,d)$,  $g\in\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T, \ell_2)$, $u(0,\cdot)\in \bU^{\gamma+2}_{p,\theta,\Theta}(\cD)$, and  $u\in\bK^{\mu+2}_{p,\theta-p,\Theta-p}(\cD,T)$ satisfies

\begin{align}
du=(\cL u+f^0+\sum_{i=1}^d f^i_{x^i})\,dt+\sum_{k=1}^{\infty} g^kdw^k_t,\quad t\in(0,T]   \label{eqn 8.19.1}
\end{align}

in the sense of distributions on $\cD$.  Then  $u\in\bK^{\gamma+2}_{p,\theta-p,\Theta-p}(\cD,T)$, hence $ u\in\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$, and  the estimate

\begin{align}
\|u\|_{\bK^{\gamma+2}_{p,\theta-p,\Theta-p}(\cD,T)}
&\leq  N\Big(\|u\|_{\bK^{\mu+2}_{p,\theta-p,\Theta-p}(\cD,T)}+\|f^0\|_{\bK^{\gamma}_{p,\theta+p,\Theta+p}(\cD,T)}\nonumber\\
&\quad \quad \quad +\|\tbf\|_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,d)}+\|g\|_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,\ell_2)}+\|u(0,\cdot)\|_{\bU^{\gamma+2}_{p,\theta,\Theta}(\cD)} \Big) \label{intermediate estimate}
\end{align}

holds with $N=N(\cM,p,n,\theta,\Theta,\nu_1,\nu_2)$.

The proof of Lemma <ref> is based on the following result on $\bR^d$.

Let $p\in[2,\infty)$,  $\gamma \in \bR$, and  Assumption  <ref> hold. Assume  $f\in \bH^{\gamma}_p(T)$,  $g\in \bH^{\gamma+1}_p(T,\ell_2)$, $u(0,\cdot)\in L_p(\Omega;H^{\gamma+2-2/p}_p)$, and $u\in \bH^{\gamma+1}_p(T)$ satisfies

\begin{align}
du=(\cL u+f)\,dt+\sum^{\infty}_{k=1}g^kdw^k_t,\quad t\in(0,T]\nonumber
\end{align}

in the sense of distributions on the whole space $\bR^d$.  Then  $u\in \bH^{\gamma+2}_p(T)$ and

\begin{eqnarray}
\|u\|_{\bH^{\gamma+2}_p(T)} &\leq& N\big(\|u\|_{\bH^{\gamma+1}_p(T)}\nonumber\\
+\|g\|_{\bH^{\gamma+1}_p(T,\ell_2)}+\|u(0,\cdot)\|_{L_p(\Omega;H^{\gamma+2-2/p}_p)}\big),\label{whole space estimate}
\end{eqnarray}

where $N=N(d,p,\nu_1,\nu_2)$ is independent of $T$.

1. First, we consider the case $u(0,\cdot) \equiv 0$. Then, by e.g. <cit.>,  $u\in \bH^{\gamma+2}_p(T)$ and

\|u_{xx}\|_{\bH^{\gamma}_p(T)}\leq N(d,p,\nu_1,\nu_2) (\|f\|_{\bH^{\gamma}_p(T)}+\|g\|_{\bH^{\gamma+1}_p(T,\ell_2)}).

This and the inequality

\|u\|_{\bH^{\gamma+2}_p(T)}\leq  (\|u_{xx}\|_{\bH^{\gamma}_p(T)}+\|u\|_{\bH^{\gamma}_p(T)} )

together with  the  inequality $\|u\|_{\bH^{\gamma}_p(T)} \le \|u\|_{\bH^{\gamma+1}_p(T)} $, which due to a basic property of the space of Bessel potentials,
yield the claim of the lemma.

2. For the case of general $u(0,\cdot)\not\equiv 0$, we use the solution $v=v(\omega,t,x)$ to the equation

dv=\cL v \,dt, \quad t\in(0,T]

with $v(\omega,0,\cdot)=u(\omega,0,\cdot)$ for all $\omega\in\Omega$ (see <cit.>).
From a classical  theory of PDE, which we apply for each $\omega$, we have

\|v\|_{\bH^{\gamma+2}_p(T)}\leq N  \|u_0\|_{L_p(\Omega;H^{\gamma+2-2/p}_p)}.

Then for the function $u-v$, which has zero initial condition, we can apply Step 1 and  we obtain estimate (<ref>) for $u$ simply by triangle inequality.

We first note that we only need to consider the case $\mu=\gamma-1$.  Indeed, suppose that the lemma holds true if $\mu=\gamma-1$. Now let $\mu=\gamma-n$, $n\in \bN$.
Then  applying the result  for $\mu'=\gamma-k$ and $\gamma'=\mu'+1$ with $k=n,n-1,\cdots, 1$ in order, we get the claim when $\mu=\gamma-n$.
Now suppose that the difference between $\gamma$ and $\mu$ is not an integer, i.e.  $\gamma-\mu=n+\delta$, $n=0,1,2,\cdots$ and $\delta\in (0,1)$.  Then, since $\mu>\gamma-(n+1)=:\mu'$ and $\|\cdot \|_{\bK^{\mu'+2}_{p,\theta-p,\Theta-p}(\cD,T)}\le \|\cdot \|_{\bK^{\mu+2}_{p,\theta-p,\Theta-p}(\cD,T)}$, we conclude that our assumption holds for $\mu'$, that is,  $u\in \bK^{\mu'+2}_{p,\theta-p,\Theta-p}(\cD,T)$.  Therefore, the case $\gamma-\mu \not\in \bN$  is also covered by what we just discussed.

Now we prove the lemma when $\mu=\gamma-1$, i.e. $u\in\bK^{\gamma+1}_{p,\theta-p,\Theta-p}(\cD,T)$.
As usual,  we omit the argument $\omega$ for the simplicity of presentation.

1. For $u\in\bK^{\gamma+1}_{p,\theta-p,\Theta-p}(\cD,T)$, put

$$\xi(x)=|x|^{(\theta-\Theta)/p},\quad  v:=\xi u, \quad  f:=f^0+\sum_{i=1}^{d}f^i_{x^i},\quad v_0:=\xi u_0.

Using  Definition  <ref>, Definition <ref>, and the change of variables $t\to e^{2n}t$, we  have

\begin{eqnarray}
&&\|u\|^p_{\bK^{\gamma+2}_{p,\Theta-p,\Theta-p}(\cD,T)}=\|v\|^p_{\bH^{\gamma+2}_{p,\Theta-p}(\cD,T)} \nonumber\\
&&= \sum_{n\in \bZ} e^{n(\Theta-p)}\|\zeta(e^{-n}\psi(e^n\cdot))v(\cdot,e^n\cdot)\|^p_{\bH^{\gamma+2}_p(T)} \nonumber\\
&&= \sum_{n\in \bZ} e^{n(\Theta-p+2)}\|\zeta(e^{-n}\psi(e^n\cdot))v(e^{2n}\cdot,e^n\cdot)\|^p_{\bH^{\gamma+2}_p(e^{-2n}T)}.  \label{eqn 4.23.1}
\end{eqnarray}

For each $n\in \bZ$, we denote

\quad

Then using equation (<ref>) and the product rule of differentiation, one can easily check that $v_n$ satisfies

d v_n=(\cL_n v_n +f_n)dt+ \sum_{k=1}^{\infty} g^k_n dw^{n,k}_t \quad  t\in(0,e^{-2n}T]

in the sense of distributions on $\bR^d$ with the initial condition $v_{n}(0,\cdot)=v_{0,n}(\cdot)$,

\cL_n:=\sum_{i,j} a^{ij}_n(t)D_{ij},\quad a^{ij}_n(t):=a^{ij}(e^{2n}t),

g^k_n(t,x):=e^{n} \zeta(e^{-n}\psi(e^nx))\xi(e^nx) g^k(e^{2n}t,e^nx), \qquad w^{n,k}_t:=e^{-n}w^k_{e^{2n}t},

and, with Einstein's summation convention with respect to $i,j$,

\begin{eqnarray*}
f_n(t,x)&:=&\quad e^{2n}\zeta(e^{-n}\psi(e^nx))\xi(e^nx) f(e^{2n}t,e^nx) \\
&&+e^{n} a^{ij}_n(t)D_iu(e^{2n}t,e^nx) \zeta'(e^{-n}\psi(e^nx)) D_j\psi(e^nx) \xi(e^nx) \\
&&+e^{2n}a^{ij}_n(t)D_iu(e^{2n}t,e^nx) \zeta(e^{-n}\psi(e^nx)) D_j \xi(e^nx) \\
&&+e^na^{ij}_n(t)u(e^{2n}t,e^nx)\zeta'(e^{-n}\psi(e^nx))  D_i\psi(e^nx)D_j\xi(e^nx)\\
&&+e^{2n}a^{ij}_n(t)u(e^{2n}t,e^nx)\zeta(e^{-n}\psi(e^nx))D_{ij}\xi(e^nx) \\
&&+a^{ij}_n(t)u(e^{2n}t,e^nx)\zeta''(e^{-n}\psi(e^nx))D_i\psi(e^nx) D_j\psi(e^nx) \xi(e^nx)\\
&&+ e^na^{ij}_n(t)u(e^{2n}t,e^nx)\zeta'(e^{-n}\psi(e^nx))  D_{ij}\psi(e^nx)\\
&=:& \sum_{l=1}^7 f^{l}_n(t,x)
\end{eqnarray*}

;  $\zeta'$ and $\zeta''$ denote the first and second derivative of $\zeta$, respectively.  We note that for each $n\in \bZ$, the operator $\cL_n$  still satisfies  the uniform parabolicity condition (<ref>) and $\{w^{n,k}_t: k\in \bN\}$ is a sequence of independent Brownian motions. Hence,
we can apply Lemma <ref> and from  (<ref>) we get

\begin{eqnarray}
\nonumber
\|u\|^p_{\bK^{\gamma+2}_{p,\theta-p,\Theta-p}(\cD,T)}
&\leq& N   \sum_{n\in \bZ} e^{n(\Theta-p+2)}\|v_n\|^p_{\bH^{\gamma+1}_p(e^{-2n}T)}\\
&&+N \sum_{l=1}^7  \sum_{n\in \bZ} e^{n(\Theta-p+2)}\|f^l_n\|^p_{\bH^{\gamma}_p(e^{-2n}T)} \nonumber\\
&&+N\sum_{n\in \bZ} e^{n(\Theta-p+2)}\|g_n\|^p_{\bH^{\gamma+1}_p(e^{-2n}T, \ell_2)}\nonumber\\
&&+ N  \sum_{n\in \bZ} e^{n(\Theta-p+2)}\|v_{0,n}\|^p_{L_p(\Omega;H^{\gamma+2-2/p}_p)}\label{eqn 8.19.21}
\end{eqnarray}

provided that

\begin{equation}\label{in fact true}
v_n\in \bH^{\gamma+1}_p(e^{-2n}T), \quad  f^l_n \in \bH^{\gamma}_p(e^{-2n}T),\quad
g_n \in \bH^{\gamma+1}_p(e^{-2n}T,\ell_2), \quad (l=1,\ldots,7).
\end{equation}

It turns out that the claims in (<ref>) holds true.
Indeed, the change of variable $e^{2n}t \to t$ and Definition <ref> yield

\begin{eqnarray}
&&   \sum_{k\in \bZ} e^{n(\Theta-p+2)}\|v_n\|^p_{\bH^{\gamma+1}_p(e^{-2n}T)} \nonumber\\
&&= \sum_{n\in \bZ} e^{n(\Theta-p)}\| \zeta(e^n\psi(e^n\cdot))v(\cdot,e^n\cdot) \|^p_{\bH^{\gamma+1}_p(T)}
=  \|u\|^p_{\bK^{\gamma+1}_{p,\theta-p,\Theta-p}(\cD,T)} \label{eqn 8.19.31}
\end{eqnarray}

\begin{eqnarray}
&&   \sum_{n\in \bZ} e^{n(\Theta-p+2)}\|g_n\|^p_{\bH^{\gamma+1}_p(e^{-2n}T,\ell_2)} \nonumber \\
&&= \sum_{n\in \bZ} e^{n\Theta}\|\zeta(e^n\psi(e^n\cdot))\xi(e^n\cdot)g(\cdot,e^n\cdot) \|^p_{\bH^{\gamma+1}_p(T,\ell_2)}=  \|g\|^p_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,\ell_2)}. \label{eqn 8.19.41}
\end{eqnarray}

In particular,

v_n \in \bH^{\gamma+1}_p(e^{-2n}T), \quad
g_n \in \bH^{\gamma+1}_p(e^{-2n}T, \ell_2), \quad \forall\, n\in \bZ.

Next, we show that $f^l_n$ belong to $\bH^{\gamma}_p(e^{-2n}T)$ in the following manner.
For $l=1$,  by Definition <ref> and the change of variables $e^{2n}t \to t$, we have

\begin{eqnarray*}
&&   \quad\sum_{n\in \bZ} e^{n(\Theta-p+2)}\|f^1_n\|^p_{\bH^{\gamma}_p(e^{-2n}T)}\\
&& =
\sum_{n\in \bZ} e^{n(\Theta+p)}\|\zeta(e^{-n}\psi(e^n\cdot))\xi(e^n\cdot)f(e^{2n}\cdot,e^n\cdot) \|^p_{\bH^{\gamma}_p(T)}
= \|f\|^p_{\bK^{\gamma}_{p,\theta+p,\Theta+p}(\cD,T)}.
\end{eqnarray*}

For $l=2$,  by Definition <ref> and  (<ref>),
we get

\begin{eqnarray*}
&&  \quad\sum_{n\in \bZ} e^{n(\Theta-p+2)}\|f^2_n\|^p_{\bH^{\gamma}_p(e^{-2n}T)}\\
&&\leq N
\sum_{n\in \bZ} \sum_{i,j}e^{n\Theta}\|D_iu(\cdot,e^n\cdot)\zeta'(e^{-n}\psi(e^n\cdot))\xi(e^n \cdot) D_j\psi(e^n\cdot)\|^p_{\bH^{\gamma}_p(T)} \\
&&\leq N \|\psi_x \xi u_x  \|^p_{\bH^{\gamma}_{p,\Theta}(\cD,T)}=N\|\psi_x  u_x  \|^p_{\bK^{\gamma}_{p,\theta,\Theta}(\cD,T)} \\
N \|u_x\|^p_{\bK^{\gamma}_{p,\theta, \Theta}(\cD,T)}\leq
N \|u\|^p_{\bK^{\gamma+1}_{p,\theta-p, \Theta-p}(\cD,T)},
\end{eqnarray*}

where the   last two inequalities are  due to    (<ref>), (<ref>), and (<ref>).  For $l=3$, by definitions of norms, we have

\begin{eqnarray}
&& \sum_{n\in \bZ} e^{n(\Theta-p+2)}\|f^2_n\|^p_{\bH^{\gamma}_p(e^{-2n}T)}  \nonumber \\
& \leq& N  \sum_{n\in \bZ} \sum_{i,j}e^{n(\Theta+p)}\|D_iu(\cdot,e^n\cdot) \zeta(e^{-n}\psi(e^n\cdot)) D_j\xi(e^n\cdot)\|^p_{\bH^{\gamma}_p(T)}  \nonumber  \\
\|u_x \xi_x\|^p_{\bH^{\gamma}_{p,\Theta+p}(\cD,T)}
= N
\|\xi \xi^{-1}\xi_xu_x\|^p_{\bH^{\gamma}_{p,\Theta+p}(\cD,T)} \nonumber \\
\|\xi^{-1}\xi_x u_x \|^p_{\bK^{\gamma}_{p,\theta+p,\Theta+p}(\cD,T)} \leq N\|\psi \xi^{-1}\xi_x u_x\|^p_{\bK^{\gamma}_{p,\theta,\Theta}(\cD,T)},  \label{eqn 8.20.1}
\end{eqnarray}

where the  last  inequality is due to (<ref>).  Now we note that for any $n\in \bN$,

|\psi \xi^{-1}\xi_x|^{(0)}_n+|\psi^2 \xi^{-1}\xi_{xx}|^{(0)}_n\leq N(n,\xi)<\infty.

Thus, by (<ref>) the last term in (<ref>) is bounded by

N\|u_x\|^p_{\bK^{\gamma}_{p,\theta,\Theta}(\cD,T)}  \leq N \|u\|^p_{\bK^{\gamma+1}_{p,\theta-p,\Theta-p}(\cD,T)}.

For other $l$s one can argue similarly and we gather the results:

\begin{eqnarray}
&&  \sum_{l=1}^7  \sum_{n\in \bZ} e^{n(\Theta-p+2)}\|f^l_n\|^p_{\bH^{\gamma}_p(e^{-2n}T)} \nonumber \\
&\leq& N \|u\|^p_{\bK^{\gamma+1}_{p,\theta-p,\Theta-p}(\cD,T)}+
N\|f\|^p_{\bK^{\gamma}_{p,\theta+p,\Theta+p}(\cD,T)}.  \label{eqn 8.19.51}
\end{eqnarray}

Consequently, coming back to (<ref>) and using (<ref>), (<ref>), and  (<ref>), we get

\begin{align*}
& \leq N\left(\|u\|^p_{\bK^{\gamma+1}_{p,\theta-p,\Theta-p}(\cD,T)}
+\|u_0\|^p_{\bU^{\gamma+2}_{p,\theta,\Theta}(\cD)} \right).
\end{align*}

This yields what we want to have since $\|f^i_{x^i}\|_{K^{\gamma}_{p,\theta+p,\Theta+p}(\cD)}\leq N \|f^i\|_{K^{\gamma+1}_{p,\theta,\Theta}(\cD)}$. The lemma is proved.

Now, from  Section 3 we recall the deterministic operator $L_0$ and the Green function $G$ related to $L_0$.  Also, recall the representation $\cR (u_0,f^0,\tbf,g)$ defined in (<ref>) in connection with $L_0$.

If  $f^0\in \bK^{\infty}_c(\cD,T)$, $\tbf\in \bK^{\infty}_c(\cD,T,d)$, $g\in \bK^{\infty}_c(\cD,T,\ell_2)$, and $u_0\in\bK^{\infty}_c(\cD)$, then  $u=\cR (u_0,f^0,\tbf,g)$ at least belongs to $\cK^0_{p,\theta,\Theta}(\cD,T)$ and satisfies

\begin{equation}
\label{eqn 8.21.13}
du=\left(L_0u+f^0+\sum_{i=1}^d f^i_{x^i}\right)dt+\sum_{k=1}^{\infty}g^k dw^k_t, \quad t\in (0,T]
\end{equation}

in the sense of distributions on $\cD$ with $u(0,\cdot)=u_0$.

First, we note that

\begin{eqnarray*}\cR (u_0,f^0,\tbf,g)&=&\cR (u_0,0,0,0)+\cR (0,f^0,\tbf,0)+\cR (0,0,0,g)\\
\end{eqnarray*}

By considering $v_1$ for each $\omega$ and by the definition of Green's function with the condition $u_0\in\bK^{\infty}_c(\cD)$,  we note that $v_1$ satisfies

dv_1=L_0v_1dt,\quad t>0\,; \quad v_1(0,\cdot)=u_0(\cdot)

in the sense of distributions on $\cD$. Then Lemma <ref> and the facts that $\bK^{\infty}_c(\cD)$ is dense in $L_p(\Omega;K^0_{p,\theta+2-p,\Theta+2-p}(\cD))$ and $\|u_0\|_{\bU^0_{p,\theta,\Theta}(\cD)}\le \|u_0\|_{L_p(\Omega;K^0_{p,\theta+2-p,\Theta+2-p}(\cD))}$ confirm $v_1\in \cK^0_{p,\theta,\Theta}(\cD,T)$. Similarly, $v_2$ satisfies

dv_2=(L_0v_2+f^0+\sum_{i=1}^d f^i_{x^i})dt,\quad t>0

in the sense of distributions on $\cD$ with zero initial condition and Lemma <ref> leads us to have $v_2\in \cK^0_{p,\theta,\Theta}(\cD,T)$.
The fact that $v_3$ satisfies

dv_3=L_0v_3dt+\sum_{k=1}^{\infty}g^k dw^k_t, \quad t>0

in the sense of distributions on $\cD$ with zero initial condition can be proved by the same way in the proof of <cit.>, which is also based on the representation formula. Then  Lemma <ref> gives $v_2\in \cK^0_{p,\theta,\Theta}(\cD,T)$.

Hence,  $u=v_1+v_2+v_3$ satisfies the assertions and the lemma is proved.

1.  Existence and    estimate (<ref>) :

Note  that,  by assumption, $\cL=L_0$.
First, we assume that  $f^0\in \bK^{\infty}_c(\cD,T)$, $\tbf\in \bK^{\infty}_c(\cD,T,d)$, $g\in \bK^{\infty}_c(\cD,T,\ell_2)$, and $u_0\in\bK^{\infty}_c(\cD)$.  Then by Lemma <ref>, $u=\cR(u_0,f^0,\tbf,g) \in \cK^0_{p,\theta,\Theta}(\cD,T)$ satisfies equation (<ref>)  in the sense of distributions on $\cD$ with initial condition $u_0$. Then, we use   Lemma <ref> with $\mu=-2$.  As $\gamma+2\ge 1$,  Lemma <ref> and Remark <ref>  leads us to have
$u\in \cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$ and estimate (<ref>).

The general case can be easily  handled by standard approximation argument. Indeed, take $f^0_n\in \bK^{\infty}_c(\cD,T)$, $\tbf_n\in \bK^{\infty}_c(\cD,T,d)$, $g_n\in \bK^{\infty}_c(\cD,T,\ell_2)$, and $u_{0,n}\in\bK^{\infty}_c(\cD)$
such that $f^0_n \to f^0$, $\tbf_n \to \tbf$, $g_n \to g$, and $u_{0,n}\to u_{0}$,   as $n\to \infty$,  in the corresponding spaces.  Now let $u_n:=\cR(u_0,f^0_n, \tbf_n,g_n)$.  Then, estimate  (<ref>) applied for $u_n-u_m$ shows that $\{u_n\}$ is a Cauchy sequence in $\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$. Taking $u$ as the limit of $u_n$ in $\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$, we find that $u$ is a solution to equation (<ref>).  Estimate (<ref>)  for $u$ also follows from those of $u_n$.

2. Uniqueness :

Let  $u \in \cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$ be a solution to  equation (<ref>) with $f^0\equiv 0$, $\tbf \equiv 0$, $g\equiv 0$, and $u_0\equiv 0$.  Due to $\gamma+2\geq 1$,  $u$ at least belongs to $\bL_{p,\theta-p,\Theta-p}(\cD,T)$, and therefore by Lemma <ref> we have $u\in \cK^2_{p,\theta,\Theta}(\cD,T)$ as all the inputs are zeros. Hence, for almost all $\omega\in \Omega$, $u^{\omega}:=u(\omega,\cdot,\cdot)\in L_p((0,T]; K^{2}_{p,\theta-p,\Theta-p}(\cD))$, and satisfies

u^{\omega}_t=\cL u^{\omega}, \quad t\in (0,T]\quad ; \quad u^{\omega}(0,\cdot)=0.

Hence, from  the uniqueness result for the deterministic parabolic equation  (see <cit.>),  we conclude $u^{\omega}=0$ for almost all $\omega$. This handles the uniqueness.

The approximation argument and uniqueness result in the above proof  show that if $\cL$ is non-random, then the  solution in Theorem <ref> is given by the formula

u=\cR(u_0,f^0,\tbf,g), \quad \text{where}\quad  \tbf=(f^1,\cdots,f^d).

1. The a priori estimate :

Having the method of continuity in mind, we consider the following operators. Fix any $\omega_0\in \Omega$ and denote $L_0(t)=\cL(\omega_0,t)$. Also for $\lambda \in [0,1]$, denote

\begin{eqnarray}
\label{Llambda}
\cL_{\lambda}&=&(1-\lambda)L_0+\lambda \cL\\
&=&\sum_{i,j=1}^d \left((1-\lambda)a^{ij}(\omega_0,t)+\lambda a^{ij}(\omega,t)\right)D_{ij}=:\sum_{i,j=1}^d a^{ij}_{\lambda}(\omega,t) D_{ij}. \nonumber
\end{eqnarray}

Note that $(a^{ij}_{\lambda})_{d\times d}$ still satisfies the uniform parabolicity condition (<ref>), and hence we have

\begin{equation}
\label{eqn 8.21.51}
\cL_{\lambda}(\omega,\cdot)\in \cT_{\nu_1,\nu_2}, \quad \forall \, \lambda\in [0,1], \, \omega\in \Omega.
\end{equation}

Now we prove that  the a priori estimate

\begin{eqnarray}
\|v\|_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)}&\leq& N\Big(\|f^0\|_{\bK^{\gamma \vee 0}_{p,\theta+p,\Theta+p}(\cD,T)}+ \|\tbf\|_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,d)} \nonumber \\
&&\quad\quad +\|g\|_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T,l_2)}+\|u_0\|_{\bU^{\gamma+2}_{p,\theta,\Theta}(\cD)}\Big),\label{the a priori}
\end{eqnarray}

holds with $N=N(\cM,d,p,\gamma,\theta,\Theta,\nu_1,\nu_2)$,  provided that  $\lambda\in [0,1]$ and
$v\in \cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$ is a solution to the equation

\begin{equation}
\label{method}
dv=\left(\cL_{\lambda} v+f^0+\sum_{i=1}^d f^i_{x^i}\right)dt+\sum_{k=1}^{\infty}g^k dw^k_t, \quad t\in(0,T]\,\,\,; \quad v(0,\cdot)=u_0(\cdot).
\end{equation}

To prove (<ref>), we take  $u\in  \cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$ from Theorem <ref>, which is the solution to equation (<ref>)  with the operator $L_0$ and the initial condition $u(0,\cdot)=u_0$.  Then $\bar{v}:=v-u\in \cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$ satisfies

\bar{v}_t=\cL_{\lambda} \bar{v}+\bar{f}=\cL_{\lambda}\bar{v}+\sum_{i=1}^d\bar{f}^i_{x^i}, \quad t\in(0,T]\quad;\quad \bar{v}(0,\cdot)=0

$$\bar{f}:=(L_0-\cL_{\lambda})u=\sum_{i=1}^d \left(\sum_{j=1}^d [a^{ij}(\omega_0,t)-a^{ij}(\omega,t)]u_{x^j}\right)_{x^i}=:\sum_{i=1}^d \bar{f}^i_{x^i}.

Hence,  Lemma <ref>  together with Lemma  <ref> yields

\|v-u\|_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)}= \|\bar{v}\|_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)} \leq N \sum_{i=1}^d \|\bar{f}^i\|_{\bK^{\gamma+1}_{p,\theta,\Theta}(\cD,T)}\leq N\|u\|_{\bK^{\gamma+2}_{p,\theta-p,\Theta-p}(\cD,T)},

where for the last inequality we used (<ref>) and the boundedness of $a^{ij}(\omega_0,t)-a^{ij}(\omega,t)$, which due to uniform parabolicity and the condition $a^{ij}=a^{ji}$.
This with estimate (<ref>) obtained for $u$, which may depend on our fixed operator $L_0$, finally gives (<ref>).

2. Existence, uniqueness and the estimate :

Estimate (<ref>) and  uniqueness result of solution are direct consequences of   a priori estimate (<ref>), for which  the constant $N$ is independent of $\cL$ and $\lambda$.  Thus we only need to prove the existence result.

Let $J$ denote the set of $\lambda\in [0,1]$ such that   for any given  $f^0,\tbf, g,u_0$ in their corresponding spaces,   equation (<ref>) with given $\lambda$ has a solution $v$ in $\cK^{\gamma+2}_{p,\theta,\Theta}(\cD,T)$. Then by Theorem  <ref>, $0\in J$.  Hence,  the method of continuity
(see e.g. proof of  <cit.>) and  a priori estimate (<ref>) together yield  $J=[0,1]$, and in particular $1\in J$. This proves the existence result.  The theorem is proved.

In the next section, we use the result of Theorem <ref> to study the regularity of SPDEs on  polygonal domains in $\bR^2$. With space dimension $d=2$, Proposition <ref> (ii) gives better information on $\lambda^{\pm}_c$ compared to the case of higher dimensions. We  take the advantage of this. We  also use the following result which helps us prove the existence of a solution on a  polygonal domain
with several vertices by its nature.

For $j=1,\,2$, let $p_j\geq 2$ and $\theta_j,\Theta_j\in\bR$, and $d-1<\Theta_j<d-1+p_j$. Also let $\theta_j$ ($j=1,2$)  satisfy

\begin{equation*}
p_j(1-\lambda^+_c)<\theta_j<p_j(d-1+\lambda^-_c) \quad \text{if $\cL$ is non-random},
\end{equation*}

p_j(1-\lambda_{c}(\nu_1,\nu_2))<\theta_j<p_j(d-1+\lambda_{c}(\nu_1,\nu_2))  \quad \text{if $\cL$ is random}.

Then, if $u\in\cK^1_{p_1,\theta_1,\Theta_1}(\cD,T)$ is a solution to equation (<ref>) with the initial condition $u(0,\cdot)=u_0(\cdot)$ and $f^0, \tbf=(f^1,\cdots,f^d)$, $g$, $u_0$ satisfying

\begin{align*}
&f^0\in\bL_{p_j,\theta_j+p_j,\Theta_j+p_j}(\cD,T), \quad \tbf \in\bL_{p_j,\theta_j,\Theta_j}(\cD,T,d ),
\end{align*}

g\in \bL_{p_j,\theta_j,\Theta_j}(\cD,T,\ell_2), \quad u_0\in\bU^1_{p,\theta,\Theta}(\cD)

for both  $j=1$ and $j=2$,  then  $u\in\cK^1_{p_2,\theta_2,\Theta_2}(\cD,T)$.

If $\cL$ is non-random, the lemma follows from  Remark <ref>.  In general, as before we fix $\omega_0\in \Omega$, and denote $L_0(t)=\cL(\omega_0,t)$ and  $v=\cR(u_0,f^0,\tbf,g)$.  Then, since $L_0$ is non-random,

\begin{equation}
\label{eqn 8.31.5}
v\in \cK^1_{p_1,\theta_1,\Theta_1}(\cD,T) \cap \cK^1_{p_2,\theta_2,\Theta_2}(\cD,T).
\end{equation}

Put $\bar{u}_1:=u-v$. Then $\bar{u}=\bar{u}_1$ satisfies

\begin{equation}
\label{eqn 8.23.1}
d\bar{u}=\left[\cL \bar{u}+\sum_{i=1}^d \Big(\sum_{j=1}^d [a^{ij}(\omega_0,t)-a^{ij}(\omega,t)]v_{x^j}\Big)_{x^i}\right]dt, \quad t\in(0,T].
\end{equation}

Also, due to (<ref>), equation (<ref>) has a solution $\bar{u}_2\in \cK^1_{p_2,\theta_2,\Theta_2}(\cD,T)$.   Now note that for each fixed
$\omega$, both $\bar{u}_1(\omega,\cdot,\cdot)$ and  $\bar{u}_2(\omega,\cdot,\cdot)$ satisfy equation  (<ref>), which we can consider as a deterministic equation with non-random operator.  By the above result for non-random operator we conclude

for almost all $\omega$. From this we conclude both $v$ and $u-v$ are in $\cK^1_{p_2,\theta_2,\Theta_2}(\cD,T)$, and therefore the lemma is proved.

§ SPDE ON POLYGONAL DOMAINS

In this section,  based on Theorem <ref>, we develop a regularity theory of the stochastic parabolic equations on polygonal domains in $\bR^2$.  This development is an enhanced version of the corresponding result  in [2] in which
$\cL=\Delta_x$ and  $\Theta=d$. Our generalization is as follows:

$\Delta \quad \rightarrow \quad \cL=\sum_{i,j}a^{ij}(\omega,t)D_{ij}$; operator with (random) predictable  coefficients

$\Theta=2  \quad \rightarrow  \quad 1<\Theta<1+p$

The restriction on  $\theta$ is weakened

Sobolev regularity  with $\gamma\in \{-1,0,\cdots\}$

$ \quad  \rightarrow \quad$ Sobolev  and Hölder regularities with  real number $ \gamma \geq -1$

Let $\cO\subset \bR^2$ be a bounded polygonal domain with a finite number of vertices $\{p_1,\ldots,p_M\}\subset \partial \cO$.
For any $x\in\cO$, we denote

\begin{align*}
%\min_{1\leq m\leq M}|x-p_m|,\quad
\rho(x):=\rho_{\cO}(x):=d(x,\partial\cO).
\end{align*}

In the polygonal domain,  the function  of $x$ defined by

\min_{1\leq m\leq M}|x-p_m|

will play the role of $\rho_{\circ, \cD}$, which is the distance to the vertex in an angular domain $\cD$. We  first construct a smooth version of the function $\min_{1\leq m\leq M}|x-p_m|$ as follows.  Consider the domain $V:=\bR^2\setminus \{p_1,\cdots,p_M\}$ and note that

$$\rho_V(x):=d(x,\partial V)=\min_{1\leq m\leq M}|x-p_m|.

Then, applying  (<ref>) and (<ref>) for $\rho_V$ and the domain $V$, we define $\psi_{V}$ and set

\rho_{\circ}=\rho_{\circ,\cO}:=\psi_{V}.

We can check that for any multi-index $\alpha$ and $\mu\in \bR$,

\rho_{\circ} \sim \min_{1\leq m\leq M}|x-p_m|, \quad
\sup_{\cO}\left| \rho^{|\alpha|-\mu}_{\circ}D^{\alpha}\rho^{\mu}_{\circ}\right|<\infty.

On the other hand, we also choose a smooth function $\psi=\psi_{\cO}$ such that $\psi\sim \rho_{\cO}$ and satisfies (<ref>) with $\rho_{\cO}$ in place of $\rho_{\cD}$.

Then, we recall the norms of the spaces $H^{\gamma}_{p,\Theta}(\cO)$ and $H^{\gamma}_{p,\Theta}(\cO; \ell_2)$ introduced in Definition <ref>;

\begin{equation}
\label{eqn 8.25.5}
\|f\|^p_{H^{\gamma}_{p,\Theta}(\cO)}:= \sum_{n\in \bZ} e^{n\Theta} \|\zeta(e^{-n}\psi(e^n\cdot))f(e^{n}\cdot)\|^p_{H^{\gamma}_p(\bR^d)},
\end{equation}

\begin{equation}
\label{eqn 8.25.55}
\|g\|^p_{H^{\gamma}_{p,\Theta}(\cO;\ell_2)}:= \sum_{n\in \bZ} e^{n\Theta} \|\zeta(e^{-n}\psi(e^n\cdot))g(e^{n}\cdot)\|^p_{H^{\gamma}_p(\bR^d;\ell_2)},
\end{equation}
where $\psi=\psi_{\cO}$. Using $\rho_{\circ,\cO}$  in place of
$\rho_{\circ,\cD}$,  and following Definition <ref>,  we define the function spaces

K^{\gamma}_{p,\theta,\Theta}(\cO),  \quad K^{\gamma}_{p,\theta,\Theta}(\cO;\bR^d), \quad K^{\gamma}_{p,\theta,\Theta}(\cO;\ell_2),

as well as the stochastic  spaces

$$\bK^{\gamma}_{p,\theta,\Theta}(\cO,T), \quad \bK^{\gamma}_{p,\theta,\Theta}(\cO,T,d ),\quad \bK^{\gamma}_{p,\theta,\Theta}(\cO,T,\ell_2),

\cK^{\gamma+2}_{p,\theta,\Theta}(\cO,T), \quad \bK^{\infty}_c(\cO,T), \quad \bK^{\infty}_c(\cO,T,\ell_2), \quad \bK^{\infty}_c(\cO).

More specifically,  we write $f\in K^{\gamma}_{p,\theta,\Theta}(\cO)$ if and only if $\rho^{(\theta-\Theta)/p}_{\circ} f\in H^{\gamma}_{p,\Theta}(\cO)$, and define

\|f\|_{ K^{\gamma}_{p,\theta,\Theta}(\cO)} := \|\rho^{(\theta-\Theta)/p}_{\circ} f\|_{H^{\gamma}_{p,\Theta}(\cO)}.

As in Section 2, if $\gamma\in \bN_0$, then we have

\begin{equation}
\label{eqn 8.25.8}
\|f\|^p_{ K^{\gamma}_{p,\theta,\Theta}(\cO)}  \sim \sum_{|\alpha|\leq \gamma}\int_{\cO} |\rho^{|\alpha|}D^{\alpha}f|^p \rho^{\theta-\Theta}_{\circ}\rho^{\Theta-d} dx.
\end{equation}

We  write $u\in\cK^{\gamma+2}_{p,\theta,\Theta}(\cO,T)$ if
$u \in \bK^{\gamma+2}_{p,\theta-p,\Theta-p}(\cO,T)$ and there exist
$(\tilde{f}, \tilde{g}) \in\bK^{\gamma}_{p,\theta+p,\Theta+p}(\cO,T)\times \bK^{\gamma+1}_{p,\theta,\Theta}(\cO,T, \ell_2)$ and $u(0,\cdot)\in \bU^{\gamma+2}_{p,\theta,\Theta}(\cO)$ satisfying

du=\tilde{f}\,dt+\sum_k \tilde{g}^kdw^k_t,\quad t\in(0,T]

in the sense of distributions on $\cO$. The norm is defined by

\begin{eqnarray*}
\|u\|_{\cK^{\gamma+2}_{p,\theta,\Theta}(\cO,T)}&:=&\|u\|_{\bK^{\gamma+2}_{p,\theta-p,\Theta-p}(\cO,T)}+\|\tilde{f}\|_{\bK^{\gamma}_{p,\theta+p,\Theta+p}(\cO,T)}
\end{eqnarray*}

With $\cD$ replaced by $\cO$, all the claims of Lemma <ref>, Remark <ref>, Theorem   <ref>, Theorem <ref>, and  Lemma <ref>  hold.