# Malliavin Calculus for Degenerate Diffusions

A. S. Üstünel

Abstract:Let $(W,H,\mu)$ be the classical Wiener space on ${\rm I\\!R}^{d}$.
Assume that $X=(X_{t}(x))$ is a diffusion process satisfying the stochastic
differential equation with diffusion and drift coefficients $\sigma:{\rm
I\\!R}^{n}\to{\rm I\\!R}^{n}\otimes{\rm I\\!R}^{d}$, $b:{\rm I\\!R}^{n}\to{\rm
I\\!R}^{n}$, $B$ is an ${\rm I\\!R}^{d}$-valued Brownian motion. We suppose
that $b$ and $\sigma$ are Lipschitz. Let $P(x)$ be the orthogonal projection
from ${\rm I\\!R}^{d}$ to its closed subspace $\sigma(x)^{\star}({\rm
I\\!R}^{n})$, assuming that $x\to P(x)$ is continuously differentiable, we
construct a covariant derivative $\hat{\nabla}$ on the paths of the diffusion
process, along the elements of the Cameron-Martin space and prove that this
derivative is closable on $L^{p}(\nu)$, where $\nu$ represents the law of the
above diffusion process, i.e., $\nu=X(x)(\mu)$, the image of the Wiener
measure under the function $w\to X_{\cdot}(w,x)$. We study the adjoint of this
operator and we prove several results: representation theorem for
$L^{2}(\nu)$-functionals, the logarithmic Sobolev inequality for $\nu$. As
applications of these results the proof of the Logarithmic Sobolev inequality
on the path space of Dyson’s Brownian motion is given by using the covariant
derivative. We then explain how to use this theory for deriving the functional
inequalities for the measures defined by the semigroups of the diffusion
process at the time $t=1$ and with fixed starting point. As an example we give
a very simple proof of the Poincaré and log-Sobolev inequalities for the
semigroup of the Brownian motion on the Heisenberg group with the optimal
constants.

2020 Mathematics Subject Classification. Primary 60H; Secondary 22E30, 22E66,
35A23, 35K08, 35R03, 60J65.

Keywords: Entropy, degenerate diffusions, martingale representation, relative
entropy, Dyson’s Brownian motion, Poincaré inequality, logarithmic Sobolev
inequality, intertwining, Heisenberg group, left and right invariant vector
fields, opératur carré du champs.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Conditional and Covariant Sobolev derivatives for degenerate diffusions’ functionals
  4. 4 Poincaré and Logarithmic Sobolev Inequalities
  5. 5 Examples and Further Applications
    1. 5.1 Classical Situation
  6. 6 Log-Sobolev for Dyson’s Brownian Motion
  7. 7 Intertwining Relations and Logarithmic Sobolev Inequality
  8. 8 An example: Poincaré and Log-Sobolev inequalities on the Heisenberg Group

##  1\. Introduction

In a preceding work we have calculated a martingale representation theorem for
the functionals of (degenerate) diffusion processes under the hypothesis of
their weak uniqueness following the the path breaking ideas originating from
[6]. While doing the calculations an important concept shows up: the behaviour
of the projection operator which is defined as the orthogonal projection from
${\rm I\\!R}^{d}$ to the closed subspace $\sigma^{\star}(t,X_{t})({\rm
I\\!R}^{n})$, where $\sigma$ is the diffusion coefficient. This operator gives
rise to the Itô representation theorem of the martingales adapted to the
filtration of the disffusion process. In fact we have shown that, for any
functional $F(X)$ defined on the paths of the diffusion process $X$, which is
square integrable w.r.t. to its law, there exists an ${\rm I\\!R}^{d}$-valued
process adapted to the filtration of $X$, say $\alpha(X)$, such that

$F(X)=E[F(X)]+\int_{0}^{1}P(X_{s})\alpha_{s}(X)\cdot dB_{s}\,,$

$\mu$-a.s. Although $\alpha$ is not unique, the process
$(P(X_{s})\alpha_{s}(X),s\in[0,1])$ is unique $ds\times d\mu$-a.s. In the case
of strong solutions, we have also chaos representation à la N. Wiener
(cf.[13]) for the functionals of $X$ using the basic
${\mathcal{F}}(X)$-martingale $m$ defined by

$m_{t}=\int_{0}^{t}P(X_{s})dB_{s}\,.$

To calculate the integrands in this chaos developement, as well as the process
$P(X)\alpha(X)$ above, we need to create some kind of differential calculus on
the paths of the diffusion process $X$. The first kind of derivative which
comes to one’s mind is some kind of extension of the $H$-derivative of Leonard
Gross (cf.[10, 14]), where $H$ denotes the Cameron-Martin space associated to
the ${\rm I\\!R}^{d}$-valued Brownian motion which is at the origin of $X$, as
it has been done in the case of flat Malliavin Calculus, cf. [18]. However
this approach fails since $H$-derivatives of diffusion’s functionals, even the
simplest ones, fail to remain measurable with respect to the sigma algebra
generated by the diffusion process. To circumvent this difficulty we propose a
“new derivation” operator which consists of smoothing the $H$-derivative
w.r.t. the final instant sigma algebra, namely
${\mathcal{F}}_{1}(X)=\sigma(X_{t},t\in[0,1])$, of the diffusion process. This
operator is denoted by $\hat{\nabla}$ with a hat at the top to indicate the
role of the conditional expectation in its definition. It is then easy to show
that $\hat{\nabla}$, restricted to cylindrical diffusion functionals
${\mathcal{S}}(X)=\\{f(X_{t_{1}},\ldots,X_{t_{n}}):t_{1}\leq
t_{2}\leq\ldots\leq t_{n}\leq 1,\,f\in{\mathcal{S}}({\rm I\\!R}^{n}),n\geq
1\\}$ is a derivation (under almost sure equality) and it is a closable
operator in $L^{p}(\nu)$ for $p>1$, where $\nu$ denotes the law of $X$. We
show several relations between the adjoint of this derivative operator
(adjoint taken w.r.t. the law of $X$), and the martingale representation
theorem; in particular we calculate explicitly the integrand for of this
representation theorem which extends the flat case, which is called in the
flate case Clark-Haussmann-Bismut-Ocone formula (cf. [16, 17]) . Let us note
nevertheless that in the case of degenerate diffusions the situation is quite
different, for instance the derivative constructed above is not even torsion
free. As immediate applications of the theory so developed, we prove the
Poincaré inequality and the logarithmic Sobolev inequality (associated to the
derivative $\hat{\nabla}$) with respect to $\nu$, which is the law of $X$. As
example, we prove these equalities for the path measure of a very singular
process, namely the Dyson Brownian motion: although, for each instant its
realization is a nice Wiener functional, its law as a probability on the path
space is singular with respect to the Wiener measure, hence this is a
degenerate situation. Afterwards we explain how to use all the theory
developed in this document to derive the poincaré and log-Sobolev inequalities
for the semigroup measures of the diffusions for fixed instants. Taking a
fixed instant increases the number of symmetries and variations to circumvent
the difficulties as explained in Section 5.3. We then apply these ideas to
give a new and very different proof of these inequalities in the case of the
Heisenberg group w.r.t. the law at $t=1$ of the non-linear Brownian motion..
As reader will realize, the method that we have undertaken is quite different
then those of these works, where, a constant consuming integration by parts
formula has been put forward. Meanwhile our approach is based on the
invariance of the Brownian motion under the Euclidean symmetries and they are
applied to both right and left Brownian paths who are starting from the origin
and/ or to their conditional expectations (with respect to the filtration of
the process). In particular, the right or left semi-groups applied to a
positive function give the same result at the trajectories of right or left
processes starting from zero element of the group. This implies a similar
invariance for “les opérateurs du carré du champs” of left and right processes
and the commutation relations between the semigroup and the right invariant
vector fields suffice to obtain the logaritmic Sobolev inequality on the
Heisenberg group with the best constants possible. This problem has been
suggested by  Leonard Gross, who has been present all along this work and he
has communicated the references [1], [8] and [15] through which we have been
able to learn the subject and this work would have never seen the daylight
without his encouragements.

##  2\. Preliminaries

In this section we recall the representation theorem and its consequences,
which have all been proven in [24]: Let $X=(X_{t},t\in[0,1])$ be a weak
solution of the following stochastic differential equation:

(2.1) $dX_{t}=b(t,X)dt+\sigma(t,X)dB_{t},\,X_{0}=x,$

where $B=(B_{t},t\in[0,1])$ is an ${\rm I\\!R}^{d}$-valued Brownian motion and
$\sigma:[0,1]\times C([0,1],{\rm I\\!R}^{n})\to L({\rm I\\!R}^{d},{\rm
I\\!R}^{n})$ and $b:[0,1]\times C([0,1],{\rm I\\!R}^{n})\to{\rm I\\!R}^{n}$
are measurable maps, adapted to the natural filtration of $C([0,1],{\rm
I\\!R}^{n})$ and of linear growth. Denote by
$({\mathcal{F}}_{t}(X),t\in[0,1])$ the filtration of $X$ and let us denote by
$K$ the set of ${\rm I\\!R}^{n}$-valued,
$({\mathcal{F}}_{t}(X),t\in[0,1])$-adapted processes $\alpha(X)$, s.t.

$E\int_{0}^{1}(a(s,X){\alpha}_{s}(X),{\alpha}_{s}(X))ds<\infty\,,$

where $a(s,w)=\sigma(s,w)\sigma^{\star}(s,w),\,s\in[0,1],\,w\in C([0,1],{\rm
I\\!R}^{n})$.

######  Theorem 1.

The set $\Gamma=\\{N\in
L^{2}({\mathcal{F}}_{1}(X)):\,N=E[N]+\int_{0}^{1}({\alpha}_{s}(X),\sigma(s,X)dB_{s}),\,{\alpha}\in
K\\}$ is dense in $L^{2}({\mathcal{F}}_{1}(X))$.

Theorem 1 permits us to show the following extension of the martingale
representation theorem to the functionals of degenerate diffusions:

######  Theorem 2.

Denote by $P_{s}(X)$ a measurable version of the orthogonal projection from
${\rm I\\!R}^{d}$ onto $\sigma(s,X)^{\star}({\rm I\\!R}^{n})\subset{\rm
I\\!R}^{d}$ and let $F\in L^{2}({\mathcal{F}}_{1}(X))$ be any random variable
with zero expectation. Then there exists a process $\xi(X)\in
L_{a}^{2}(dt\times dP;{\rm I\\!R}^{d})$, adapted to
$({\mathcal{F}}_{t}(X),t\in[0,1])$, such that

$F(X)=\int_{0}^{1}(P_{s}(X)\xi_{s}(X),dB_{s})_{{\rm
I\\!R}^{d}}=\int_{0}^{1}(\xi_{s}(X),P_{s}(X)dB_{s})_{{\rm I\\!R}^{n}}$

a.s.
Conversely, any stochastic integral of the form

$\int_{0}^{1}(P_{s}(X)\xi_{s}(X),dB_{s})\,,$

where $\xi(X)$ is an $({\mathcal{F}}_{t}(X),t\in[0,1])$-adapted, measurable
process with $E\int_{0}^{1}|P_{s}(X)\xi_{s}(X)|^{2}ds<\infty$, gives rise to
an ${\mathcal{F}}_{1}(X)$-measurable random variable.

Remark: Let $\eta$ be an adapted process such that $\eta_{s}$ belongs to the
orthogonal complement of $\sigma^{\star}({\rm I\\!R}^{n})$ in ${\rm
I\\!R}^{d}$ $ds\times dP$-a.s. Then $\eta+\xi$ can also be used to represent
$F(X)$. Hence $\xi(X)$ is not unique but $P(X)\xi(X)$ is always unique.

As a consequence of above theorems, we get

######  Theorem 3.

Let $\dot{u}\in L^{2}(dt\times dP,{\rm I\\!R}^{d})$ be adapted to the Brownian
filtration, then we have

(2.2)
$E\left[\int_{0}^{1}(\dot{u}_{s},dB_{s})|{\mathcal{F}}_{1}(X)\right]=\int_{0}^{1}(E[P_{s}(X)\dot{u}_{s}|{\mathcal{F}}_{s}(X)],dB_{s})$

almost surely.

######  Remark 1.

If $K$ is a process adapted to the Brownian filtration with values in ${\rm
I\\!R}^{m}\otimes{\rm I\\!R}^{d}$ which is in $L^{2}(dt\times d\mu,{\rm
I\\!R}^{m}\otimes{\rm I\\!R}^{d})$, then we obtain from Theorem 3

(2.3)
$E\left[\int_{0}^{1}K_{s}dB_{s}|{\mathcal{F}}_{1}(X)\right]=\int_{0}^{1}E[K_{s}|{\mathcal{F}}_{s}(X)]P(X_{s})dB_{s}$

$\mu$-almost surely.

######  Corollary 1.

Let $h\in H^{1}([0,1],{\rm I\\!R}^{d})$ (i.e., the Cameron-Martin space),
denote by $\rho(\delta h)$ the Wick exponential
$\exp(\int_{0}^{1}(\dot{h}_{s},dB_{s})-\frac{1}{2}\>\int_{0}^{1}|\dot{h}_{s}|^{2}ds)$,
then we have

$E[\rho(\delta
h)|{\mathcal{F}}_{1}(X)]=\exp\left(\int_{0}^{1}(P_{s}(X)\dot{h}_{s},dB_{s})-\frac{1}{2}\>\int_{0}^{1}|P_{s}(X)\dot{h}_{s}|^{2}ds\right)\,.$

######  Theorem 4.

  * •

Assume that ${\mathcal{F}}_{t}(X)\subset{\mathcal{F}}_{t}(B)$ for any
$t\in[0,1]$, where $({\mathcal{F}}_{t}(B),t\in[0,1])$ represents the
filtration of the Brownian motion. Define the martingale $m=(m_{t},t\in[0,1])$
as $m_{t}=\int_{0}^{t}P_{s}(X)dB_{s}$, then the set

$K=\\{\rho(\delta_{m}(h)):\,h\in H\\}$

is total in $L^{2}({\mathcal{F}}_{1}(X))$, where
$\rho(\delta_{m}(h))=\exp\left(\int_{0}^{1}(\dot{h}_{s},dm_{s})-\frac{1}{2}\>\int_{0}^{1}|P_{s}(X)\dot{h}_{s}|^{2}ds\right)\,$.
In particular, any element $F$ of $L^{2}({\mathcal{F}}_{1}(X))$ can be written
in a unique way as the sum

(2.4)
$F=E[F]+\sum_{n=1}^{\infty}\int_{C_{n}}(f_{n}(s_{1},\ldots,s_{n}),dm_{s_{1}}\otimes\ldots\otimes
dm_{s_{n}})$

where $C_{n}$ is the $n$-dimensional simplex in $[0,1]^{n}$ and $f_{n}\in
L^{2}(C_{n},ds^{\otimes n})\otimes({\rm I\\!R}^{d})^{\otimes n}$.

  * •

More generally, without the hypothesis
${\mathcal{F}}_{t}(X)\subset{\mathcal{F}}_{t}(B)$, for any $F\in
L^{2}({\mathcal{F}}_{1}(X))\cap L^{2}({\mathcal{F}}_{1}(B))$, the conclusions
of the first part of the theorem hold true.

######  Remark 2.

In this theorem we need to make the hypothesis
${\mathcal{F}}_{t}(X)\subset{\mathcal{F}}_{t}(B)$ for any $t\in[0,1]$ to
assure the non-symmetric chaos representation (2.4). Without this hypothesis,
although we have Theorem 2 and when we iterate it we have a similar
representation, but, consisting of a finite number of terms. It is not
possible to push this procedure up to infinity since we have no control at
infinity.

######  Remark 3.

Let us note that if there is no strong solution to the equation defining the
process $X$, the chaotic representation property may fail. For example, let
$U$ be a weak solution of

(2.5) $dU_{t}=\alpha_{t}(U)dt+dB_{t}\,,$

with $U_{0}$ given. Assume that (2.5) has no strong solution, as it may happen
in the famous example of Tsirelson ([12]), i.e., $U$ is not measurable w.r.t.
the sigma algebra generated by $B$, then we have no chaotic representation
property for the elements of $L^{2}({\mathcal{F}}_{1}(U))$ in terms of the
iterated stochastic integrals of deterministic functions on $C_{n},n\in{\rm
I\\!N}$ w.r.t. $B$; the contrary would imply the equality of
${\mathcal{F}}_{1}(U)$ and of ${\mathcal{F}}_{1}(B)$, which would contradict
the non-existence of strong solutions.

##  3\. Conditional and Covariant Sobolev derivatives for degenerate
diffusions’ functionals

In this ection, we shall work with the diffusion processes in the restricted
sense, i.e., there will be no explicit dependence on the chance parameter, the
coefficients will be $C^{1}$-Lipschitz functions (or $C^{k}$, with large
$k\in{\rm I\\!N}$) of linear growth, for notational simplicity, we shall also
assume that they are time independent. To keep track of the nonlinearity due
to the diffusion’s paths interactions, we need to develop a special derivation
w.r.t. the Cameron-Martin space of the defining Brownian motion. The next
result paves the road and it is of fundemental importance:

######  Theorem 5.

For $h\in H$, define
$\hat{\nabla}_{h}X_{t}=E[\nabla_{h}X_{t}|{\mathcal{F}}_{1}(X)]$, then it
satisfies the following relation:

$\hat{\nabla}_{h}X_{t}=\int_{0}^{t}\partial\sigma(X_{s})\hat{\nabla}_{h}X_{s}P(X_{s})dB_{s}+\int_{0}^{t}\partial
b(X_{s})\hat{\nabla}_{h}X_{s}ds+\int_{0}^{t}\sigma(X_{s})\dot{h}(s)ds$

$\mu$-almost surely. If $\sigma$ and $b$ are twice differentiable and bounded,
and if the projection map is differentiable and of linear growth, for the
iterated derivative $\hat{\nabla}_{h}^{2}X_{t}$ we have the following
expression:

$\displaystyle\hat{\nabla}_{h}^{2}X_{t}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\partial^{2}\sigma(X_{s})\hat{\nabla}_{h}X_{s}\otimes\hat{\nabla}_{h}X_{s}P(X_{s})dB_{s}+\int_{0}^{t}\partial\sigma(X_{s})\hat{\nabla}_{h}^{2}X_{s}P(X_{s})dB_{s}$
$\displaystyle+\int_{0}^{t}\partial\sigma(X_{s})\hat{\nabla}_{h}X_{s}\partial
P(X_{s})\hat{\nabla}_{h}X_{s}P(X_{s})dB_{s}+\int_{0}^{t}\partial\sigma(X_{s})\hat{\nabla}_{h}X_{s}(P(X_{s})+I_{{\rm
I\\!R}^{d}})\dot{h}(s)ds$
$\displaystyle+\int_{0}^{t}\partial^{2}b(X_{s})(\hat{\nabla}_{h}X_{s}\otimes\hat{\nabla}_{h}X_{s}+\hat{\nabla}^{2}_{h}X_{s})ds$

$\mu$-almost surely.

Proof: We have

(3.6)
$\nabla_{h}X_{t}=\int_{0}^{t}\partial\sigma(X_{s})\nabla_{h}X_{s}dB_{s}+\int_{0}^{t}\partial
b(X_{s})\nabla_{h}X_{s}ds+\int_{0}^{t}\sigma(X_{s})\dot{h}(s)ds\,,$

taking the conditional expectation of both sides of this relation, the term
with stochastic integral follows from Theorem 3 and the Lebesgue integral part
follows from the fact that

$E[\nabla_{h}X_{t}|{\mathcal{F}}_{1}(X)]=E[\nabla_{h}X_{t}|{\mathcal{F}}_{t}(X)]$

due to the fact that ${\mathcal{B}}_{t}=\sigma(B_{s},s\leq t)$ is independent
of the future increments of the Brownian motion (after $t$). For the second
derivative we just iterate the calculation of the first derivative.

Remark:  It is clear that $\hat{\nabla}_{h}^{2}X_{t}\neq
E[\nabla_{h}^{2}X_{t}|{\mathcal{F}}_{1}(X)]$.

######  Corollary 2.

Let ${\mathcal{S}}(X)$ be the set of functions on $W$ defined as

${\mathcal{S}}(X)=\left\\{f(X_{t_{1}},\ldots,X_{t_{n}}):0\leq
t_{1}<\ldots<t_{n},\,f\in{\mathcal{S}}({\rm I\\!R}^{n}),n\geq 1\right\\}\,,$

where ${\mathcal{S}}({\rm I\\!R}^{n})$ denotes the space of rapidly decreasing
smooth functions of Laurent Schwartz. Let $h\in H$, assume that
$(F_{k}(X),k\geq 1)\subset{\mathcal{S}}(X)$ converges to zero in
$L^{2}({\mathcal{F}}(X))$ and that $(\hat{\nabla}_{h}F_{k}(X),k\geq 1)$ is
Cauchy in $L^{2}({\mathcal{F}}(X))$, then

$\lim_{k\to\infty}\hat{\nabla}_{h}F_{k}(X)=0$

$\mu$-a.s. In other words $\hat{\nabla}_{h}$ is a closable operator on
$L^{2}(X(\mu))$, where $X(\mu)$ denotes the law of the diffusion process, or
the image of the Wiener measure $\mu$ under the map $X$ which is defined by
the diffusion process.

Proof: Let $\eta=\eta(X)$ be the limit 0f $(\hat{\nabla}_{h}F_{k}(X),k\geq
1)$, then using Theorem 3, we have, for any cylindrical
$G(X)\in{\mathcal{S}}(X)$,

$\displaystyle E[\eta(X)\,G(X)]$ $\displaystyle=$
$\displaystyle\lim_{k}E[\hat{\nabla}_{h}F_{k}(X)G(X)]=\lim_{k}E[\nabla_{h}F_{k}(X)G(X)]$
$\displaystyle=$ $\displaystyle\lim_{k}E[F_{k}(X)(-\nabla_{h}G(X)+G(X)\delta
h)]$ $\displaystyle=$
$\displaystyle\lim_{k}E[F_{k}(X)(-\hat{\nabla}_{h}G(X)+G(X)\delta({\mathbb{P}}(X)h))]=0\,,$

where $\delta({\mathbb{P}}(X)h)=\int_{0}^{1}P(X_{s})\dot{h}(s)dB_{s}$.

For $F\in{\mathcal{S}}(X)$, we define $\hat{\nabla}F$ as

$\hat{\nabla}F=\sum_{i=1}^{\infty}\hat{\nabla}_{e_{i}}F\,,$

where $(e_{i},i\geq 1)$ is an orthonormal basis in the Cameron-Martin space
$H$. The above theorem implies in particular that $\hat{\nabla}$ is a closable
operator from $L^{p}({\mathcal{F}}(X))$ into $L^{p}({\mathcal{F}}(X),H)$. We
shall denote its closure with the same notation. In particular, we denote by
${\rm I\\!M}_{p,1}$ the completion of ${\mathcal{S}}(X)$ w.r.t. the norm

$\|F(X)\|_{p,1}=\||\hat{\nabla}F(X)|_{H}\|_{L^{p}(\mu)}+\|F(X)\|_{L^{p}(\mu)}\,.$

For $p=2$ we use the following version:

$\|F(X)\|^{2}_{2,1}=E\left[|\hat{\nabla}F(X)|^{2}_{H}+|F(X)|^{2}\right]\,.$

If $K$ is a separable Hilbert space, we denote by ${\mathcal{S}}(X,K)$, the
set of $K$-valued cylindrical functions of the trajectories of $X$ by
replacing ${\mathcal{S}}({\rm I\\!R}^{n})$ above by ${\mathcal{S}}({\rm
I\\!R}^{n})\otimes K$, where the latter denotes the (projective) tensor
product.

######  Definition 1.

Let $\nu$ be the measure given as $X(\mu)$, i.e., the law of the diffusion
process as a probability measure on the path space. We denote the adjoint of
$\hat{\nabla}$ w.r.t. $\nu$ as $\delta_{\nu}$ (or sometimes as
$\hat{\nabla}^{\star}$). Namely, for $\xi(X)\in{\mathcal{S}}(X)\otimes H$,
i.e., the $H$-valued cylindrical, ${\mathcal{F}}(X)$-measurable functions and
$G(X)\in{\mathcal{S}}(X)$, we write

$\displaystyle E[\delta_{\nu}\xi(X)\,G(X)]$ $\displaystyle=$ $\displaystyle
E_{\nu}[\delta_{\nu}\xi\,G]$ $\displaystyle=$ $\displaystyle
E_{\nu}[(\xi,\hat{\nabla}G)_{H}]$ $\displaystyle=$ $\displaystyle
E[(\xi(X),\hat{\nabla}G(X))_{H}]\,.$

######  Theorem 6.

Let $\xi(X)\in L^{2}({\mathcal{F}}(X),H)$ be of the form
$\sum_{i<\infty}\xi_{i}(X)e_{i}$, where $\xi_{i}(X)\in{\mathcal{S}}(X)$ and
$e_{i}\in H$ for each $i\in{\rm I\\!N}$.

  1. (1)

Then

$\delta_{\nu}\xi(X)=\sum_{i}\left[-\hat{\nabla}_{e_{i}}\xi_{i}(X)+\xi_{i}(X)\delta({\mathbb{P}}(X)e_{i})\right]\,,$

where we define the action of $P(X)$ on the Cameron-Martin space $H$ as

$({\mathbb{P}}(X)h)(t)=\int_{0}^{t}P(X_{s})\dot{h}(s)ds\,,$

for $h\in H$. In particular, we have

(3.7) $E[\delta\xi(X)|{\mathcal{F}}_{1}(X)]=\delta_{\nu}\xi(X)\,.$

  2. (2)

If $\eta=\eta(X)\in{\mathcal{S}}(X,H)$ is adapted to the filtration
$({\mathcal{F}}_{t}(X),t\in[0,1])$, where ${\mathcal{S}}(X,H)$ denotes
$H$-valued version of ${\mathcal{S}}(X)$, then

(3.8) $\delta_{\nu}\eta=\delta{\mathbb{P}}(X)\eta$

$\mu$-a.s.

Proof: Let $g(X)\in{\mathcal{S}}(X)$, then, using Definition 1 and the
integration by parts formula for $\mu$, we get

$\displaystyle E[(\xi(X),\hat{\nabla}g(X))_{H}]$ $\displaystyle=$
$\displaystyle E\left[\sum_{i}\xi_{i}(X)\hat{\nabla}_{e_{i}}g(X)\right]$
$\displaystyle=$ $\displaystyle
E\left[\sum_{i}\xi_{i}(X)\nabla_{e_{i}}g(X)\right]$ $\displaystyle=$
$\displaystyle\sum_{i}E\left[-\nabla_{e_{i}}\xi_{i}(X)g(X)+g(X)\xi_{i}(X)\,\delta
e_{i}\right]$ $\displaystyle=$
$\displaystyle\sum_{i}E\left[-\nabla_{e_{i}}\xi_{i}(X)g(X)+g(X)\xi_{i}(X)\,E[\delta
e_{i}|{\mathcal{F}}_{1}(X)]\right]$ $\displaystyle=$
$\displaystyle\sum_{i}E\left[-\hat{\nabla}_{e_{i}}\xi_{i}(X)g(X)+g(X)\xi_{i}(X)\,\delta({\mathbb{P}}(X)e_{i})\right]$
$\displaystyle=$ $\displaystyle
E\left[\left(\sum_{i}\left(-\hat{\nabla}_{e_{i}}\xi_{i}(X)+\xi_{i}(X)\,\delta({\mathbb{P}}(X)e_{i})\right)\right)g(X)\right]\,,$

since $E[\delta e_{i}|{\mathcal{F}}_{1}(X)]=\delta({\mathbb{P}}(X)e_{i})$ as
shown in Theorem 3. The relation (3.7) holds then true for the cylindrical
case, the general case follows by passing to the limit in ${\rm
I\\!M}_{2,1}(H)$ (i.e., the space ${\rm I\\!M}_{2,1}\otimes H$, completed
Hilbert-Schmidt tensor product).

To prove the second part, it follows already from the relation (3.7) that
$\delta{\mathbb{P}}(X)\eta=E[\delta\eta|{\mathcal{F}}_{1}(X)]$, let now
$G=G(X)\in{\mathcal{S}}(X)$, then

$\displaystyle E[\delta_{\nu}\eta(X)\,G(X)]$ $\displaystyle=$ $\displaystyle
E\left[(\eta(X),\hat{\nabla}G(X))_{H}\right]=E\left[(\eta(X),\nabla
G(X))_{H}\right]$ $\displaystyle=$ $\displaystyle
E[\delta\eta(X)\,G(X)]=E\left[E[\delta\eta(X)|{\mathcal{F}}_{1}(X)]\,G(X)\right]$
$\displaystyle=$ $\displaystyle
E\left[\delta{\mathbb{P}}(X)\eta\,G(X)\right]\,,$

as, by Theorem 3, $\delta{\mathbb{P}}(X)\eta$ is
${\mathcal{F}}_{1}(X)$-measurable, the proof follows.

######  Corollary 3.

Let $F(X),G(X)\in{\mathcal{S}}(X)$ and let $h\in H$, then we have the
integration by parts formula

(3.9)
$E[\hat{\nabla}_{h}F(X)\,G(X)=E\left[F(X)\,(-\hat{\nabla}_{h}G(X)+G(X)\delta{\mathbb{P}}(X)h)\right]\,.$

Proof: As $E[\hat{\nabla}_{h}F(X)\,G(X)]=E[\nabla_{h}F(X)\,G(X)]$. from the
Gaussian integration by parts formula, we have

$\displaystyle E[\hat{\nabla}_{h}F(X)\,G(X)]$ $\displaystyle=$ $\displaystyle
E[F(X)(-\nabla_{h}G(X)+G(X)\delta h)]$ $\displaystyle=$ $\displaystyle
E\left[F(X)E\left[(-\nabla_{h}G(X)+G(X)\delta
h)|{\mathcal{F}}_{1}(X)\right]\right]$ $\displaystyle=$ $\displaystyle
E[F(X)(-\hat{\nabla}_{h}G(X)+G(X)\delta{\mathbb{P}}(X)h)]\,.$

######  Corollary 4.

For $a(X)\in{\rm I\\!M}_{2,1}$ and $\xi(X)\in{\mathcal{S}}(X,H)$, we have

(3.10)
$\delta_{\nu}(a(X)\xi(X))=a(X)\delta_{\nu}\xi(X)-(\hat{\nabla}a(X),\xi(X))_{H}\,.$

Moreover, if $\dot{\xi}(X)$ is adapted to the filtration
$({\mathcal{F}}_{t}(X),t\in[0,1])$, then

(3.11) $\delta_{\nu}\xi(X)=\delta({\mathbb{P}}(X)\xi(X))\,.$

In particular, for any $h,k\in H$, we have

(3.12)
$E[\delta({\mathbb{P}}(X)\nabla_{h}{\mathbb{P}}(X)k)|{\mathcal{F}}_{1}(X)]=\delta_{\nu}({\mathbb{P}}(X)\hat{\nabla}_{h}{\mathbb{P}}(X)k)$

$\mu$-a.s.

Proof: The only claim which is not immediate is (3.12) and it follows from
Theorem 3.

######  Lemma 1.

For any $h,k\in H$, we have

$\hat{\nabla}_{k}(\delta{\mathbb{P}}(X)h)=({\mathbb{P}}(X)h,k)_{H}+\delta_{\nu}(\hat{\nabla}_{k}{\mathbb{P}}(X)h)$

$\mu$-a.s.

Proof: By definition

$\displaystyle\hat{\nabla}_{k}(\delta{\mathbb{P}}(X)h)$ $\displaystyle=$
$\displaystyle E[\nabla_{k}\delta{\mathbb{P}}(X)h|{\mathcal{F}}_{1}(X)]$
$\displaystyle=$
$\displaystyle({\mathbb{P}}(X)h,k)_{H}+E[\delta\nabla_{k}{\mathbb{P}}(X)h|{\mathcal{F}}_{1}(X)]$
$\displaystyle=$
$\displaystyle({\mathbb{P}}(X)h,k)_{H}+\delta{\mathbb{P}}(X)\hat{\nabla}_{k}{\mathbb{P}}(X)h$
$\displaystyle=$
$\displaystyle({\mathbb{P}}(X)h,k)_{H}+\delta_{\nu}\hat{\nabla}_{k}{\mathbb{P}}(X)h$

by Theorem 3 and Corollary 4.

######  Theorem 7.

Let $\xi=\xi(X),\,\eta=\eta(X)$ be in ${\mathcal{S}}(X,H)$, then the following
identity holds true:

$\displaystyle E[\delta_{\nu}\xi\,\delta_{\nu}\eta]$ $\displaystyle=$
$\displaystyle E[({\mathbb{P}}(X)\xi,\eta)_{H}+{\,\,\rm
trace\,\,}(\hat{\nabla}\xi\cdot\hat{\nabla}\eta)]$ $\displaystyle+E[{\,\,\rm
trace\,\,}(\hat{\nabla}{\mathbb{P}}(X)\xi\cdot\hat{\nabla}\eta)+{\,\,\rm
trace\,\,}(\hat{\nabla}{\mathbb{P}}(X)\eta\cdot\hat{\nabla}\xi)]$

Proof: Writing
$E[\delta_{\nu}\xi\,\delta_{\nu}\eta]=E[(\xi,\hat{\nabla}\delta_{\nu}\eta)_{H}]$,
then we shall calculate $\hat{\nabla}\delta_{\nu}\eta$. Let $k\in H$ and let
$(e_{i},i\geq 1)$ be an orthonormal basis of $H$, then

$\displaystyle\hat{\nabla}\delta_{\nu}\eta$ $\displaystyle=$
$\displaystyle\hat{\nabla}_{k}\left[\sum_{i}\eta_{i}\delta{\mathbb{P}}(X)e_{i}-\hat{\nabla}_{e_{i}}\eta_{i}\right]$
$\displaystyle=$
$\displaystyle\sum_{i}\left[\hat{\nabla}_{k}\eta_{i}\delta{\mathbb{P}}(X)e_{i}+\eta_{i}\hat{\nabla}_{k}\delta{\mathbb{P}}(X)e_{i}-\hat{\nabla}_{k}\hat{\nabla}_{e_{i}}\eta_{i}\right]$
$\displaystyle=$
$\displaystyle\sum_{i}\left[\hat{\nabla}_{k}\eta_{i}\delta{\mathbb{P}}(X)e_{i}+\eta_{i}({\mathbb{P}}(X)e_{i},k)_{H}\right]$
$\displaystyle+\sum_{i}\left[\eta_{i}\delta_{\nu}\hat{\nabla}_{k}{\mathbb{P}}(X)e_{i}-\hat{\nabla}_{k}\hat{\nabla}_{e_{i}}\eta_{i}\right]$

Taking the expectations after replacing the vector $k$ with $e_{k}$ and using
the above equality, we get

$\displaystyle E[\delta_{\nu}\xi\,\delta_{\nu}\eta]$ $\displaystyle=$
$\displaystyle
E[(\xi,\hat{\nabla}\delta_{\nu}\eta)_{H}]=E\left[\sum_{i}\eta_{i}\delta_{\nu}(\delta{\mathbb{P}}(X)e_{i})+({\mathbb{P}}(X)\xi,\eta)_{H}\right]$
$\displaystyle+E\left[\sum_{k,i}\eta_{i}(\delta_{\nu}\hat{\nabla}_{e_{k}}{\mathbb{P}}(X)e_{i})\xi_{k}-\xi_{k}\hat{\nabla}_{e_{k}}\hat{\nabla}_{e_{i}}\eta_{i}\right]\,,$

where $\eta_{i}=(\eta,e_{i})_{H},\xi_{i}=(\xi,e_{i})_{H}$. Using the relation

$\hat{\nabla}_{\xi}\hat{\nabla}_{e_{i}}\eta_{i}=\hat{\nabla}_{e_{i}}\hat{\nabla}_{\xi}\eta_{i}-\hat{\nabla}_{\hat{\nabla}_{e_{i}}\xi}\eta_{i}\,,$

we have

$\displaystyle E[\delta_{\nu}\xi\,\delta_{\nu}\eta]$ $\displaystyle=$
$\displaystyle
E\left[\sum_{i}\hat{\nabla}_{\xi}\eta_{i}\delta{\mathbb{P}}(X)e_{i}+({\mathbb{P}}(X)\xi,\eta)_{H}\right]$
$\displaystyle+E\left[\sum_{k,i}\eta_{i}\delta_{\nu}(\hat{\nabla}_{e_{k}}{\mathbb{P}}(X)e_{i})\xi_{k}-\hat{\nabla}_{e_{i}}\hat{\nabla}_{\xi}\eta_{i}+{\,\,\rm
trace\,\,}(\hat{\nabla}\eta\cdot\hat{\nabla}\xi)\right]$ $\displaystyle=$
$\displaystyle
E\left[\delta_{\nu}\hat{\nabla}_{\xi}\eta+({\mathbb{P}}(X)\xi,\eta)_{H}+{\,\,\rm
trace\,\,}(\hat{\nabla}\eta\cdot\hat{\nabla}\xi)+\sum_{k,i}\eta_{i}\delta_{\nu}(\hat{\nabla}_{e_{k}}{\mathbb{P}}(X)e_{i})\xi_{k}\right]\,.$

Using the identity

$\delta_{\nu}(g\,\tau)=g\delta_{\nu}\tau-(\tau,\hat{\nabla}g)_{H}\,,$

for $g\in{\mathcal{S}}(X),\,\tau\in{\mathcal{S}}(X,H)$, we have

$\displaystyle\sum_{k,i}E\left[\eta_{i}\delta_{\nu}(\hat{\nabla}_{e_{k}}{\mathbb{P}}(X)e_{i})\xi_{k}\right]$
$\displaystyle=$
$\displaystyle\sum_{k,i}E\left[\eta_{i}\delta_{\nu}(\xi_{k}\hat{\nabla}_{e_{k}}{\mathbb{P}}(X)e_{i})+\eta_{i}(\hat{\nabla}\xi_{k},\hat{\nabla}_{e_{k}}{\mathbb{P}}(X)e_{i})_{H}\right]$
$\displaystyle=$ $\displaystyle
E\left[\sum_{i}\eta_{i}\delta_{\nu}(\hat{\nabla}_{\xi}{\mathbb{P}}(X)e_{i})+\sum_{k}(\hat{\nabla}\xi_{k},\hat{\nabla}_{e_{k}}{\mathbb{P}}(X)\eta-{\mathbb{P}}(X)\hat{\nabla}_{e_{k}}\eta)_{H}\right]$

and since

(3.15)
$\eta_{i}\delta_{\nu}(\hat{\nabla}_{\xi}{\mathbb{P}}(X)e_{i})=\delta_{\nu}(\eta_{i}\hat{\nabla}_{\xi}{\mathbb{P}}(X)e_{i})+(\hat{\nabla}_{\xi}{\mathbb{P}}(X)e_{i},\hat{\nabla}\eta_{i})_{H}\,,$

we obtain by substituting (3.15) in (3)

$\sum_{k,i}E\left[\eta_{i}\delta_{\nu}(\hat{\nabla}_{e_{k}}{\mathbb{P}}(X)e_{i})\xi_{k}\right]=I+II\,,$

where

$\displaystyle I$ $\displaystyle=$
$\displaystyle\sum_{i}E\left[\delta_{\nu}(\eta_{i}\hat{\nabla}_{\xi}{\mathbb{P}}(X)e_{i})+(\hat{\nabla}_{\xi}{\mathbb{P}}(X)e_{i},\hat{\nabla}\eta_{i})_{H}\right]$
$\displaystyle=$ $\displaystyle E[{\,\,\rm
trace\,\,}(\hat{\nabla}_{\xi}{\mathbb{P}}(X)\cdot\hat{\nabla}\eta)]$

and

$\displaystyle II$ $\displaystyle=$
$\displaystyle\sum_{k,i}E[\eta_{i}(\hat{\nabla}\xi_{k},\hat{\nabla}_{e_{k}}{\mathbb{P}}(X)e_{i})_{H}]$
$\displaystyle=$
$\displaystyle\sum_{k}E[(\hat{\nabla}\xi_{k},(\hat{\nabla}_{e_{k}}{\mathbb{P}}(X))\eta)_{H}$
$\displaystyle=$ $\displaystyle E[{\,\,\rm
trace\,\,}(\hat{\nabla}\xi\cdot(\hat{\nabla}{\mathbb{P}}(X))\eta)]$

and this completes the proof.

The next result shows the local character of the derivative operator
$\hat{\nabla}$, which is well-known in the flat case:

######  Proposition 1.

Assume that $F=F(X)$ is in ${\rm I\\!M}_{2,1}$, then $\hat{\nabla}F=0$ on the
set $\\{F=0\\}$ $\nu$-almost surely.

Proof: Let $\theta$ be a positive, smooth function of compact support on ${\rm
I\\!R}$ with $\theta(0)=1$, denote by $\zeta$ its primitive and let
$\zeta_{\varepsilon}(t)=\varepsilon\zeta(t/\varepsilon)$. Clearly
$\zeta_{\varepsilon}\circ F$ belongs to ${\rm I\\!M}_{2,1}(X)$. If
$u\in{\mathcal{S}}(X,H)$, from the definitions

$\displaystyle E[\zeta_{\varepsilon}(F)\,\delta_{\nu}u]$ $\displaystyle=$
$\displaystyle E[(\hat{\nabla}\zeta_{\varepsilon}(F),u)_{H}]$ $\displaystyle=$
$\displaystyle E[\theta(F/\varepsilon)(\hat{\nabla}F,u)_{H}]\to
E[1_{\\{F=0\\}}]\,.$

On the other hand, from the Dominated Convergence Theorem, we have
$\lim_{\varepsilon\to 0}E[\zeta_{\varepsilon}(F)\delta_{\nu}u]=0$ and as
${\mathcal{S}}(X,H)$ is dense in ${\rm I\\!M}_{2,1}(X,H)$, the proof follows.

The following result shows that $\hat{\nabla}$ defines a Markovian Dirichlet
form on the path space of the diffusion process:

######  Proposition 2.

Assume that $f,\,g\in{\rm I\\!M}_{2,1}$, then $f\wedge g\in{\rm I\\!M}_{2,1}$,
where $\wedge$ denotes the minimum.

Proof: Let $(f_{n},n\geq 1)$ and $(g_{n},n\geq 1)$ be two sequences of
cylindrical functions approximating $f$ and $g$ in ${\rm I\\!M}_{2,1}$
respectively. We have

$\displaystyle\hat{\nabla}(f_{n}\wedge g_{n})$ $\displaystyle=$ $\displaystyle
E[\nabla f_{n}\,1_{\\{f_{n}\geq g_{n}\\}}+\nabla
g_{n}\,1_{\\{f_{n}<g_{n}\\}}|{\mathcal{F}}_{1}(X)]$ $\displaystyle=$
$\displaystyle\hat{\nabla}f_{n}1_{\\{f_{n}\geq
g_{n}\\}}+\hat{\nabla}g_{n}1_{\\{f_{n}<g_{n}\\}}\,,$

Hence

$|hna(f_{n}\wedge
g_{n})|_{H}\leq|\hat{\nabla}f_{n}|_{H}+|\hat{\nabla}g_{n}|_{H}\,.$

Consequently, $(f_{n}\wedge g_{n},n\geq 1)$ is bounded in ${\rm I\\!M}_{2,1}$,
then it has a subsequence which converges weakly. This implies that $f\wedge
g\in{\rm I\\!M}_{2,1}$ and that $|\hat{\nabla}(f\wedge
g)|_{H}\leq|\hat{\nabla}f|_{H}+|\hat{\nabla}g|_{H}$ almost surely.

While making calculations we need also a second covariant derivation on the
paths of the diffusion process, this time this will be the smoothed derivative
w.r.t. the state space elements, namely we define

$\hat{\partial}_{\xi}X_{t}(\xi)=E[\partial_{\xi}X_{t}(\xi)|{\mathcal{F}}_{1}(X(\xi))]$

where $\partial_{\xi}$ denotes the derivative of the diffusion process at the
instant $t\in[0,1]$ with respect to its initial starting point $\xi\in{\rm
I\\!R}^{n}$, similarly, for smooth functions on ${\rm I\\!R}^{n}$, we define

$\hat{\partial}_{\xi}f(X_{t}(\xi))=E[\partial
f(X_{t}(\xi))\partial_{\xi}X_{t}(\xi)|{\mathcal{F}}_{1}(X(\xi))]=\partial
f(X_{t}(\xi))\hat{\partial}_{\xi}X_{t}(\xi),$

in such a way that $\hat{\partial}$ behaves as a derivative w.r.t. state space
variable of the functionals of the paths of the diffusion process. We have the
immediate result which follows from Theorem 2.2

######  Theorem 8.

The state space covariant derivative $\hat{\partial}_{x}X_{t}(x)=J_{t}(x)$
satisfies the following stochastic differential equation:

(3.16) $J_{t}=I_{{\rm
I\\!R}^{n}}+\int_{o}^{t}\partial\sigma(X_{s}(x))J_{s}(x)P(X_{s}(x))dB_{s}+\int_{0}^{t}\partial
b(X_{s}(x)J_{s}(x)ds\,,$

$P$-a.s., for any $t\in[0,1]$.

It follows from Theorem 8 and from the Itô formula that

######  Proposition 3.

$(J_{t},t\in[0,1])$ is a semimartingale with values in $GL(n,{\rm I\\!R})$
(non-singular matrices) and its inverse, denoted as $(K_{t},t\in[0,1])$ has
the following representation:

$\displaystyle dK_{t}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{d}K_{t}\partial\sigma^{i}(X_{t}(x))dm^{i}_{t}$
$\displaystyle+K_{t}\left[\sum_{i,j=1}^{d}\partial\sigma^{i}(X_{t}(x))\partial\sigma^{j}(X_{t}(x))P_{i,j}(X_{t}(x))-\partial
b(X_{t}(x))\right]dt\,,$

where $dm_{t}=P(X_{t}(x))dB_{t}$, $m_{t}=(m^{1}_{t},\ldots,m^{d}_{t})$, and
$\sigma^{i}$ is the $i$-th column vector of the $n\times d$-matrix $\sigma$.

Proof: The equation (3) is a linear stochastic differential equation with
bounded coefficients, hence it has a unique solution without explosion. To
prove the claim it suffices to show that $J_{t}K_{t}=K_{t}J_{t}=I_{{\rm
I\\!R}^{n}}$ almost surely, where $I_{{\rm I\\!R}^{n}}$ denotes the identity
map of ${\rm I\\!R}^{n}$, and this last equation follows from the Itô formula.

An immediate consequence of these calculations, combined with tha variation of
constants’ method, is the following

######  Corollary 5.

For any $t<\tau\in[0,1]$, we have

$\hat{D}_{t}X_{\tau}(x)=J_{\tau}K_{t}\sigma(X_{t}(x))$

almost surely.

##  4\. Poincaré and Logarithmic Sobolev Inequalities

Here are some applications of the results of the preceding section. Let us
begin with a non-trivial extension of Itô-Clark-Hausmann-Bismut-Ocone formula:

######  Theorem 9.

Assume that $F(X)\in{\rm I\\!M}_{2,1}$, then it can be represented as

$F(X)=E[F(X)]+\int_{0}^{1}P(X_{s})E[\hat{D}_{s}F(X)|{\mathcal{F}}_{s}(X)]dB_{s}\,,$

where $\hat{D}_{s}F(X)$ is defined as
$\hat{\nabla}F(X)(\cdot)=\int_{0}^{\cdot}\hat{D}_{s}F(X)ds$.

Proof: We know from Theorem 2 that $F(X)$ can be represented as

$F(X)=E[F(X)]+\int_{0}^{1}P(X_{s})\alpha_{s}(X).dB_{s}\,,$

for some $\alpha(X)\in L^{2}_{a}(dt\times d\mu,{\rm I\\!R}^{d})$, adapted to
the filtration $({\mathcal{F}}_{t}(X),t\in[0,1])$. Moreover, if
$F(X)\in{\mathcal{S}}(X)$, then from the Gaussian case, we have

(4.18) $F(X)=E[F(X)]+\int_{0}^{1}E[D_{s}F(X)|{\mathcal{B}}_{s}]dB_{s}\,,$

where $({\mathcal{B}}_{s},s\in[0,1])$ is the filtration of the Brownian
motion. It follows from Theorem 3, taking the conditional expectations of both
sides of the equality (4.18), that

$F(X)=E[F(X)]+\int_{0}^{1}P(X_{s})E[D_{s}F(X)|{\mathcal{F}}_{s}(X)]dB_{s}\,.$

Hence in this case
$P(X_{s})\alpha_{s}(X)=P(X_{s})E[D_{s}F(X)|{\mathcal{F}}_{s}(X)]=P(X_{s})E[E[D_{s}F(X)|{\mathcal{F}}_{1}(X)|{\mathcal{F}}_{s}(X)]=P(X_{s})E[\hat{D}_{s}F(X)|{\mathcal{F}}_{s}(X)]$
$ds\times d\mu$-a.s. Choose now a sequence $(F_{n}(X),n\geq
1)\subset{\mathcal{S}}(X)$ which approximates $F(X)$ in ${\rm I\\!M}_{2,1}$.
Then $(\hat{\nabla}F_{n}(X),n\geq 1)$ converges to $\hat{\nabla}F(X)$ in
$L^{2}(\mu,H)$ (in other words $(\hat{\nabla}F_{n})$ converges to
$\hat{\nabla}F$ in $L^{2}(\nu,H)$), consequently

$\lim_{n}E\int_{0}^{1}|P(X_{s})E[\hat{D}_{s}F_{n}(X)-\hat{D}_{s}F(X)|{\mathcal{F}}_{s}(X)]|^{2}ds=0\,,$

hence the result follows for any $F(X)\in{\rm I\\!M}_{2,1}$.

######  Corollary 6.

Assume that $F=F(X)\in{\rm I\\!M}_{2,1}$ such that $\hat{\nabla}F=0$
$\nu$-a.s. (or $\hat{\nabla}F(X)=0$ $\mu$-a.s), then $F$ is $\nu$-a.s. (or
$F(X)$ is $\mu$-a.s.) a constant.

Proof: The proof follows directly from Theorem 9.

We obtain the Poincaré inequality from Theorem 9, whose proof is trivial since
the conditional expectation in the representation formula (4.18) operates as a
contraction:

######  Theorem 10.

For any $F(X)\in{\rm I\\!M}_{2,1}$, we have

$E[|F(X)-E[F(X)]|^{2}]\leq E[|{\mathbb{P}}(X)\hat{\nabla}F(X)|_{H}^{2}]\,.$

######  Theorem 11.

For any $F\in{\rm I\\!M}_{2,1}$, it holds true that

(4.19) $E[F^{2}(X)\log F^{2}(X)]\leq E[F^{2}(X)]\log
E[F^{2}(X)]+2E[|{\mathbb{P}}(X)\hat{\nabla}F(X)|_{H}^{2}]\,.$

Proof: It suffices to prove for any $F(X)\in{\rm I\\!M}_{2,1}$ which is
strictly positive and lower bounded by a positive constant. Taking
$f(X)=F^{2}(X)/E[F^{2}(X)]$, it suffices to show that

$E[f(X)\log
f(X)]\leq\frac{1}{2}\>E\left[\frac{1}{f(X)}|{\mathbb{P}}(X)\hat{\nabla}f(X)|_{H}^{2}\right]\,.$

Using Theorem 9, we can represent $f(X)$ as

$\displaystyle f(X)$ $\displaystyle=$ $\displaystyle
1+\int_{0}^{1}P(X_{s})E[\hat{D}_{s}(f(X))|{\mathcal{F}}_{s}(X)]\cdot dB_{s}$
$\displaystyle=$ $\displaystyle
1+\int_{0}^{1}\frac{P(X_{s})E[\hat{D}_{s}(f(X))|{\mathcal{F}}_{s}(X)]}{E[f(X)|{\mathcal{F}}_{s}(X)]}E[f(X)|{\mathcal{F}}_{s}(X)]\cdot
dB_{s}\,.$

Hence $f(X)$ can be represented as an exponential martingale’s final value:

$f(X)=\exp\left[\int_{0}^{1}\frac{P(X_{s})E[\hat{D}_{s}(f(X))|{\mathcal{F}}_{s}(X)]}{f_{s}(X)}\cdot
dB_{s}-\frac{1}{2}\>\int_{0}^{1}\left|\frac{P(X_{s})E[\hat{D}_{s}(f(X))|{\mathcal{F}}_{s}(X)]}{f_{s}(X)}\right|^{2}ds\right]\,,$

where $f_{s}(X)=E[f(X)|{\mathcal{F}}_{s}(X)]$. Let $\gamma$ be the probability
measure on the path space defined by $d\gamma=f(X)d\mu$, then the Girsanov
Theorem and Bayes’ formula entail that the relative entropy of $\gamma$ w.r.t.
$\mu$, denoted by $H(\gamma|\mu)$ is equal to

$\displaystyle H(\gamma|\mu)$ $\displaystyle=$ $\displaystyle\int\log
f(X)d\gamma$ $\displaystyle=$
$\displaystyle\frac{1}{2}\>E_{\gamma}\int_{0}^{1}\left|\frac{P(X_{s})E[\hat{D}_{s}(f(X))|{\mathcal{F}}_{s}(X)]}{f_{s}(X)}\right|^{2}ds$
$\displaystyle=$
$\displaystyle\frac{1}{2}\>E_{\gamma}\int_{0}^{1}\left|\frac{E[P(X_{s})\hat{D}_{s}(f(X))|{\mathcal{F}}_{s}(X)]}{f_{s}(X)}\right|^{2}ds$
$\displaystyle=$
$\displaystyle\frac{1}{2}\>E_{\gamma}\int_{0}^{1}\left|E_{\gamma}[P(X_{s})\hat{D}_{s}(\log
f(X))|{\mathcal{F}}_{s}(X)]\right|^{2}ds$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\>E_{\gamma}\int_{0}^{1}\left|P(X_{s})\hat{D}_{s}(\log
f(X))|\right|^{2}ds$ $\displaystyle=$
$\displaystyle\frac{1}{2}\>E\left[\frac{1}{f(X)}|{\mathbb{P}}(X)\hat{\nabla}f(X)|_{H}^{2}\right]\,.$

Remark:  Maybe it is pedagogically interesting to note this inequality in
terms of the measure induced by the the diffusion process on the path space,
namely $\nu$:

(4.20) $\int f^{2}\log f^{2}d\nu-\left(\int f^{2}d\nu\right)\log\int
f^{2}d\nu\leq 2\int|{\mathbb{P}}\hat{\nabla}f|_{H}^{2}d\nu\,$

for any $f\in{\rm I\\!M}_{2,1}$.

##  5\. Examples and Further Applications

We shall give applications of the analysis that we have developped to some
specific cases beginning from the simplest to more complicated situations

###  5.1. Classical Situation

Let $(X_{t},t\geq 0)$ be the solution of the equation

$X_{t}(x)=x+\int_{0}^{t}b(X_{r}(x))dr+\Sigma B_{t}\,,$

where $B=(B_{t},t\geq 0)$ is an $d$-dimensional Brownian motion, $b:{\rm
I\\!R}^{m}\to{\rm I\\!R}^{m}$ is a Lipschitz-continuous vector field and
$\Sigma\in L({\rm I\\!R}^{m},{\rm I\\!R}^{d})$. Regularizing $b$ with a
mollifier, we obtain a smooth and Lipschitz vector field $b_{\varepsilon}$,
let $X^{\varepsilon}$ be the solution of SDE

$X^{\varepsilon}_{t}(x)=x+\int_{0}^{t}b_{\varepsilon}(X^{\varepsilon}_{r}(x))dr+\Sigma
B_{t}\,.$

For any $h\in H$, where $H=H^{1}({\rm I\\!R}_{+},{\rm I\\!R}^{d})$ is the
Cameron-Martin space of $B$, we have

$\nabla_{h}X^{\varepsilon}_{t}=\int_{0}^{t}\partial
b_{\varepsilon}(X_{r})\nabla_{h}X^{\varepsilon}_{r}dr+\Sigma h(t)$

and the Gronwall lemma implies that, for any $T>0$,

$\sup_{t\leq T}\|\nabla X^{\varepsilon}_{t}\|_{H\otimes{\rm
I\\!R}^{m}}\leq\|\Sigma\|e^{TK}\,,$

where $K$ is the Lipschitz constant of $b$ and $\|\Sigma\|$ is the operator
norm of $\Sigma$. We can pass to the limit as $\varepsilon\to 0$ to obtain
also

$\sup_{t\leq T}\|\nabla X_{t}\|_{H\otimes{\rm
I\\!R}^{m}}\leq\|\Sigma\|e^{TK}\,.$

The following theorem is now a straight application of Theorem 11

######  Theorem 12.

Let $T>0$ be arbitrary and denote by $\nu^{T}$ the law of the process
$(X_{t},t\in[0,T])$. Then, for any smooth, cylindrical function on
$W_{T}=C([0,T],{\rm I\\!R}^{d})$, we have

$\int_{W_{T}}F^{2}(x)\log\frac{F^{2}(x)}{\nu^{T}(f^{2})}\nu^{T}(dx)\leq
2\|\Sigma\|e^{TK}\int_{W_{T}}|\nabla F(x)|^{2}\nu^{T}(dx)\,,$

in particular, for any $t\in(0,T]$, denoting by $\nu_{t}$ the law of the
random variable $X_{t}$, we have

$\int_{{\rm
I\\!R}^{m}}f^{2}(y)\log\frac{f^{2}(y)}{\nu_{t}(f^{2})}\nu_{t}(dy)\leq
2\|\Sigma\|e^{TK}\int_{{\rm I\\!R}^{m}}|\partial f(y)|^{2}\nu_{t}(dy)\,,$

for any smooth function $f:{\rm I\\!R}^{m}\to{\rm I\\!R}$.

##  6\. Log-Sobolev for Dyson’s Brownian Motion

In the example above, the drift coefficient has nice properties and the log-
Sobolev inequality is almost trivial. Let us look at a more singular case,
namely, let $(X_{t}=(X^{1}_{t},\ldots,X_{t}^{d}),\,t\geq 0)$ be the Dyson
Brownian motion, i.e., the strong solution of the following stochastic
differential equation:

(6.21) $dX_{t}^{i}=dB_{t}^{i}+\gamma\sum_{1\leq j\neq i\leq
d}\frac{1}{X_{t}^{i}-X^{j}_{t}}dt\,,\,i=1,\ldots,d;\,\,\gamma>0,$

with $X_{0}=x_{0}$ such that $x_{1}<\ldots<x_{d}$ We refer the reader to
references [3, 4, 5] for the existence of the strong solutions, where they use
the theory of dissipative mappings in their construction and to [9] for the
origins of the equation. For notational convenience, we shall write above
equation as

(6.22) $dX_{t}=\gamma m(X_{t})dt+dB_{t}\,,X_{0}=x_{0}\in D(m),$

where $m$ is the mapping from a subset of ${\rm I\\!R}^{d}$, noted as $D(m)$
into ${\rm I\\!R}^{d}$ whose components are given in the equation (6.21). In
fact $m$ is a set-valued, dissipative map. It is maximal and this implies that
its values reduce to one-point subsets of ${\rm I\\!R}^{d}$ (cf. [26]). As a
typical example for such a function, one can take the gradient of a concave
function on ${\rm I\\!R}^{d}$. Equation 6.21 corresponds to the case
$m=\partial f$, where $f(x)=\sum_{i<j}\log(x_{j}-x_{i})$ if
$x_{1}<\ldots<x_{d}$ and $f(x)=-\infty$ otherwise. Let us announce first an
important technical result:

######  Lemma 2.

Let $X=(X_{t},t\geq 0)$ be the unique strong solution of SDE 6.22, then, for
any $T>0$, $h,k\in H([0,T],{\rm I\\!R}^{d})$ we have

$\sup_{s\leq t}\left|X_{s}(w+h)-X_{s}(w+k)\right|\leq
2\sqrt{t}\gamma|h-k|_{H([0,t],{\rm I\\!R}^{d})}$

P-almost surely. Consequently, for any $t>0$, $X_{t}\in\cap_{p}{\rm
I\\!D}_{p,1}({\rm I\\!R}^{d})$, in fact $|\nabla X_{t}|_{{\rm
I\\!R}^{d}\otimes H([0,t],{\rm I\\!R}^{d})}\in L^{\infty}(P)$ with the bound
$2t\gamma$, where $H([0,t],{\rm I\\!R}^{d})$ (respectively $H([0,T],{\rm
I\\!R}^{d})$ denotes the Cameron-Martin space for ${\rm I\\!R}^{d}$-valued
functions on $[0,t]$ (respectively $[0,T]$).

Proof: We will suppose that $\gamma=1$ and that $(B_{t})$ is the linear
Brownian motion, which amounts to working on the canonical Wiener space (for
practical reasons). From the equation 6.22, it comes

$X_{t}(w+h)=B_{t}(w)+h(t)+\int_{0}^{t}m(X_{s}(w+h))ds\,,$

hence

$X_{t}(w+h)-X_{t}(w)=h(t)+\int_{0}^{t}[m(X_{s}(w+h))-m(X_{s}(w))]ds\,.$

Let $y_{t}^{h}(w)=X_{t}(w+h)-X_{t}(w)$, then $(y_{t}^{h},\,t\in[0,T])$
satisfies

$y_{t}^{h}(w)=h(t)+\int_{0}^{t}[m(y^{h}_{s}(w)+X_{s}(w))-m(X_{s}(w))]ds\,,$

hence

(6.23)
$y_{t}^{h}(w)-y_{t}^{k}(w)=h(t)-k(t)+\int_{0}^{t}[m(y_{s}^{h}(w)+X_{s}(w))-m(y_{s}^{k}(w)+X_{s}(w))]ds\,.$

It follows from equation 6.23 and from the dissipativity of $m$ that

$\displaystyle|y_{t}^{h}-y_{t}^{k}|^{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\frac{d}{ds}|y_{s}^{h}-y_{s}^{k}|^{2}ds$
$\displaystyle=$ $\displaystyle
2\int_{0}^{t}(y_{s}^{h}-y_{s}^{k},\dot{h}(s)-\dot{k}(s))_{{\rm
I\\!R}^{d}}+2\int_{0}^{t}(y_{s}^{h}-y_{s}^{k},m(y^{h}_{s}+X_{s})-m(y^{k}_{s}+X_{s}))ds$
$\displaystyle\leq$ $\displaystyle 2\sup_{s\leq
t}|y_{s}^{h}-y_{s}^{k}|\int_{0}^{t}|\dot{h}(s)-\dot{k}(s)|ds$
$\displaystyle\leq$ $\displaystyle 2\sqrt{t}\sup_{s\leq
t}|y_{s}^{h}-y_{s}^{k}|\left(\int_{0}^{t}|\dot{h}(s)-\dot{k}(s)|^{2}ds\right)^{1/2}$

Since the last line of above inequality is increasing w.r.t. $t\in{\rm
I\\!R}_{+}$, the claim follows. The fact that $X_{t}\in\cap_{p>1}{\rm
I\\!D}_{p,1}$ follows from [20], Lemma 2 at p.72.

An immediate consequence of Lemma 2 is Fernique’s bound:

######  Corollary 7.

For any $t>0$, there exists some $\varepsilon=\varepsilon(\gamma,t)$, such
that

$E\left[\exp\left(\varepsilon\,\sup_{s\leq t}|X_{s}|^{2}\right)\right]<\infty$

Remark:  Let $T>0$ be any fixed number and let us denote by $\nu$ the law of
the process restricted to the time horizon $[0,T]$, starting from $x_{0}$.
Evidently $\nu$ is the unique solution of the martingale problem (cf. [19])
associated to the operator $K$ defined as

$Kf(x)=\sum_{i=1}^{d}\left[\frac{1}{2}\frac{\partial^{2}f}{\partial
x_{i}^{2}}(x)+\gamma m_{i}(x)\frac{\partial f}{\partial x_{i}}(x)\right]\,.$

It is immediate to see that under $\nu$, the coordinate map $x\to x(t)$ of
$C_{x_{0}}([0,T],{\rm I\\!R}^{d})$ can be represented as

$dx_{t}=\gamma m(x_{t})dt+d\beta_{t}\,,$

where $(\beta_{t},t\in[0,T])$ is a $\nu$-Brownian motion.

The following is now clear:

######  Proposition 4.

The covariant derivative defined on the paths of Dyson process restricted to
$[0,T]$, i.e., for $F(X)=f(X_{t_{1}},\ldots,X_{t_{n}})$,

$\hat{\nabla}_{h}F(X)=E[\nabla_{h}F(X)|{\mathcal{F}}_{T}(X)]\,,$

where $h\in H([0,T],{\rm I\\!R}^{d})$ and
$\hat{\nabla}F(X)=\sum_{i}\hat{\nabla}_{h_{i}}h_{i}$, such that $(h_{i})$ is a
basis in $H([0,T],{\rm I\\!R}^{d})$, are closable operators on
$L^{p}({\mathcal{F}}_{T}(X))$ for any $p>1$. Consequently, we have the
following representation theorem:

$F(X)=E[F(X)]+\int_{0}^{T}E[D_{s}F(X)|{\mathcal{F}}_{s}(X)]\cdot dB_{s}\,.$

From Proposition 4 we obtain as explained in the general case, the logarithmic
Sobolev inequality:

######  Proposition 5.

For any ${\mathcal{F}}_{T}(X)$ measurable, cylindrical function $F(X)$, we
have

$E[F^{2}(X)\log(F^{2}(X)/E[F^{2}(X)])]\leq
2\gamma\sqrt{T}E[|\hat{\nabla}F(X)|_{H([0,T],{\rm I\\!R}^{d})}^{2}]\,.$

##  7\. Intertwining Relations and Logarithmic Sobolev Inequality

Let us come back to the general situation studied in the paper: if $f$ be a
smooth function function on ${\rm I\\!R}^{n}$ and if $(Q_{t},t\in[0,1])$ is
the semi-group associated to the Markov process induced by
$(X_{t},t\in[0,1])$, then $(Q_{1-t}f(X_{t}(x)),t\in[0,1])$ is an
$({\mathcal{F}}_{t}(X))$-martingale and, due to the Itô formula, it can be
represented as

(7.24)
$Q_{1-t}f(X_{t}(x))=Q_{1}f(x)+\int_{0}^{t}\partial_{x}Q_{1-s}f(X_{s}(x))\cdot\sigma(X_{s}(x))dB_{s}\,.$

From Theorem 9, we obtain a second representation:

(7.25)
$Q_{1-t}f(X_{t}(x))=Q_{1}f(x)+\int_{0}^{t}P(X_{s}(x))E[\hat{D}_{s}f(X_{1}(x))|{\mathcal{F}}_{s}(X)]\cdot
dB_{s},$

where $\hat{D}_{s}f(X_{1}(x))=E[D_{s}f(X_{1}(x))|{\mathcal{F}}_{1}(X)]$. Ito
Isometry principle implies that

$\partial_{x}Q_{1-s}f(X_{s}(x))\cdot\sigma(X_{s}(x))=P(X_{s}(x))E[\hat{D}_{s}f(X_{1}(x))|{\mathcal{F}}_{s}(X)]$

$ds\times dP$-almost surely, where

$\hat{D}_{t}f(X_{1})=E[D_{t}f(X_{1})|{\mathcal{F}}_{1}(X)]\,.$

It will be tempting to compare in detail these two representations: As

$\hat{D}_{t}f(X_{1})=(\hat{D}_{t}X_{1})^{\star}\partial f(X_{1})\in{\rm
I\\!R}^{d}\,,$

we obtain

######  Theorem 13.

For any smooth function $f$ on ${\rm I\\!R}^{n}$, we have the following
commutation-intertwining relation:

(7.26) $\sigma(X_{t})^{\star}\partial
Q_{1-t}f(X_{t})=P(X_{t})E\left[(\hat{D}_{t}X_{1})^{\star}\partial
f(X_{1})|{\mathcal{F}}_{t}(X)\right]\,.$

Corollary 5 gives that

$(\hat{D}_{t}X_{1})^{\star}=\sigma^{\star}(X_{t})K_{t}^{\star}J_{1}^{\star}\,,$

where $J_{t}=\hat{\partial}X_{t}(x)$ and $K_{t}=J_{t}^{-1}$. Consequently, we
get

$\displaystyle E[\hat{D}_{t}f(X_{1})|{\mathcal{F}}_{t}(X)]$ $\displaystyle=$
$\displaystyle
E[(\hat{D}_{t}X_{1})^{\star}\hat{\partial}f(X_{1})|{\mathcal{F}}_{t}(X)]$
$\displaystyle=$ $\displaystyle
E[\sigma^{\star}(X_{t})K_{t}^{\star}J_{1}^{\star}\hat{\partial}f(X_{1})|{\mathcal{F}}_{t}(X)]$
$\displaystyle=$
$\displaystyle\sigma^{\star}(X_{t})K_{t}^{\star}E[J_{1}^{\star}\hat{\partial}f(X_{1})|{\mathcal{F}}_{t}(X)]\,.$

To calculate the conditional expectation of the last line, we use the flow
property of the diffusion process: For $s<t\leq 1$, let $X^{s}_{t}(x,\omega)$
denote the solution of the SDE defining the diffusion process at the instant
$t$, starting from the point $x\in{\rm I\\!R}^{n}$ at the instant $s$. The
flow property implies that
$X_{1}(x,\omega)=X^{t}_{1}(X_{t}(x,\omega),\tilde{\omega})$, where
$\tilde{\omega}$ denotes a Brownian path independent of $\\{B_{\tau}:\tau\leq
t\\}$ Taking the derivative of both sides, from the chain rule, it comes

$\partial_{x}X_{1}(x,\omega)=\partial
X^{t}_{1}(X_{t}(x,\omega),\tilde{\omega})\partial X_{t}(x,\omega)$

almost surely. Hence

$\displaystyle E[J_{1}^{\star}\partial f(X_{1})|{\mathcal{F}}_{t}(X)]$
$\displaystyle=$ $\displaystyle E[\partial X_{1}^{\star}\partial
f(X_{1})|{\mathcal{F}}_{t}(X)]$ $\displaystyle=$ $\displaystyle E[(\partial
X^{t}_{1}(X_{t},\tilde{\omega})\partial X_{t}(x))^{\star}\partial
f(X^{t}_{1}\circ X_{t})|{\mathcal{F}}_{t}(X)]$ $\displaystyle=$ $\displaystyle
E\left[\partial X_{t}^{\star}E[(\partial
X^{t}_{1})^{\star}\circ(X_{t},\tilde{\omega})\partial f(X^{t}_{1}\circ
X_{t})|{\mathcal{B}}_{t}]|{\mathcal{F}}_{t}(X)\right]\,,$

where $({\mathcal{B}}_{t},t\geq 0)$ represents the filtration of the
underlying Brownian motion. Due to the independence and the stationarity of
the Brownian increments, the inner conditional expectation can be written as:

$\displaystyle E[(\partial
X^{t}_{1})^{\star}\circ(X_{t},\tilde{\omega})\partial f(X^{t}_{1}\circ
X_{t})|{\mathcal{B}}_{t}]$ $\displaystyle=$ $\displaystyle E[(\partial
X^{t}_{1})^{\star}(\xi)\partial f(X^{t}_{1}(\xi))]|_{\xi=X_{t}(x)}$
$\displaystyle=$ $\displaystyle E[(\partial X_{1-t})^{\star}(\xi)\partial
f(X_{1-t}\xi))]|_{\xi=X_{t}(x)}$

and in particular, it is ${\mathcal{F}}_{t}(X)$-measurable. Let us define the
semigroup $(M_{t},t\geq 0)$ on the vector-valued functions as

$M_{t}\alpha(\xi)=E[(\partial X_{t}(\xi))^{\star}\alpha(X_{t}(\xi))]\,,$

$\xi\in{\rm I\\!R}^{n}$. Using this semigroup we end up with the identity

$\displaystyle E[J_{1}^{\star}\partial f(X_{1})|{\mathcal{F}}_{t}(X)]$
$\displaystyle=$ $\displaystyle E[\partial
X_{t}^{\star}|{\mathcal{F}}_{t}(X)]\,M_{1-t}\partial f(X_{t}(x))$
$\displaystyle=$ $\displaystyle\hat{\partial}X^{\star}_{t}(x)M_{1-t}\partial
f(X_{t}(x))\,.$

Finally we find that

(7.28)
$E[\hat{D}_{t}f(X_{1}(x))|{\mathcal{F}}_{t}(X)]=\sigma(X_{t}(x))^{\star}M_{1-t}\partial
f(X_{t}(x))$

almost surely. Although the expression (7.28) is what we can obtain by
differentiation under the expectation, it has an important byproduct that we
write separately as a lemma:

######  Lemma 3.

For any smooth function $f$ on ${\rm I\\!R}^{n}$, the process
$(\hat{\partial}X^{\star}_{t}(x)M_{1-t}\partial f(X_{t}(x)),\,t\in[0,1])$ is
an ${\mathcal{F}}_{\cdot}(X(x))$-martingale.

The following result may be called as a modified Poincaré inequality:

######  Theorem 14.

Let $Z$ be the functional defined as

$Z(x)=\int_{0}^{1}|\sigma^{\star}(X_{t}(x))(\hat{\partial}X_{t}(x)^{\star})^{-1}|^{2}dt\,.$

We then have, for any $f\in C_{b}^{2}({\rm I\\!R}^{n})$, the following
inequality:

(7.29) $E[|f(X_{1}(x))-E[f(X_{1}(x))]|^{2}]\leq
E\left[Z\left(\hat{\partial}X_{1}(x)\cdot\hat{\partial}X_{1}^{\star}(x)\partial
f(X_{1}(x)),\partial f(X_{1}(x))\right)_{{\rm I\\!R}^{n}}\right]\,.$

Remark:  Let operator-valued function $\Gamma(x,y)$ be defined as

(7.30)
$\Gamma(x,y)=E[Z\hat{\partial}X_{1}(x)\cdot\hat{\partial}X_{1}^{\star}(x)|X_{1}(x)=y]\,,$

then we can read the same inequality in ${\rm I\\!R}^{n}$ as

$\int_{{\rm I\\!R}^{n}}\left|f(y)-\int_{{\rm
I\\!R}^{n}}f(z)q_{1}(x,z)dz\right|^{2}q_{1}(x,y)dy\leq\int_{{\rm
I\\!R}^{n}}\left(\Gamma(x,y)\partial f(y),\partial f(y)\right)_{{\rm
I\\!R}^{n}}q_{1}(x,y)dy\,,$

where $y\to q_{1}(x,y)$ is the density of the law of $X_{1}(x)$.

Proof: From the martingale representation theorem, from the relation (7.28)
and from the Itô isometry, we obtain

$\displaystyle E[|f(X_{1}(x))-E[f(X_{1}(x))]|^{2}]$ $\displaystyle=$
$\displaystyle E\int_{0}^{1}|\sigma^{\star}(X_{t}(x))M_{1-t}\partial
f(X_{t}(x))|^{2}dt$ $\displaystyle=$ $\displaystyle
E\int_{0}^{1}|\sigma^{\star}(X_{t}(x))(\hat{\partial}X^{\star}_{t})^{-1}(\hat{\partial}X^{\star}_{t})M_{1-t}\partial
f(X_{t}(x))|^{2}dt$ $\displaystyle\leq$ $\displaystyle
E\int_{0}^{1}|\sigma^{\star}(X_{t}(x))(\hat{\partial}X^{\star}_{t})^{-1}|^{2}|(\hat{\partial}X^{\star}_{t})M_{1-t}\partial
f(X_{t}(x))|^{2}dt$ $\displaystyle\leq$ $\displaystyle
E\int_{0}^{1}|\sigma^{\star}(X_{t}(x))(\hat{\partial}X^{\star}_{t})^{-1}|^{2}|(\hat{\partial}X^{\star}_{1})\partial
f(X_{1}(x))|^{2}dt\,,$

where the inequality at the last line follows from Lemma 3.

Same observation gives also an extension of the logarithmic Sobolev inequality
for the measure $q_{1}(x,y)dy$ which is the law of the random variable
$\omega\to X_{1}(\omega,x)$:

######  Theorem 15.

Let $f\in C^{2}_{b}({\rm I\\!R}^{n})$, then, we have

(7.31) $E\left[f^{2}(X_{1})\log\frac{f^{2}(X_{1})}{E[f^{2}(X_{1})]}\right]\leq
2\int_{{\rm I\\!R}^{n}}\left(\Gamma(x,y)\partial f(y),\partial
f(y)\right)_{{\rm I\\!R}^{n}}q_{1}(x,y)dy\,,$

where $\Gamma$ is defined with (7.30).

Proof: We proceed as in the proof of Theorem 11 with $f\in C_{b}^{2}({\rm
I\\!R}^{n})$ which is strictly positive. Moreover we can suppose that
$E[f(X_{1}(x))]=1$. In this case the claimed inequality takes the form

$E\left[f(X_{1})\log f(X_{1})\right]\leq\int_{{\rm
I\\!R}^{n}}(\Gamma(x,y)\partial f(y),\partial f(y))_{{\rm
I\\!R}^{n}}\frac{1}{f(y)}q_{1}(x,y)dy\,.$

Let $d\gamma=f(X_{1}(x))dP$, similar calculations of entropy $H(\gamma|P)$ as
in the proof of Theorem 11 imply that

$\displaystyle H(\gamma|P)$ $\displaystyle=$ $\displaystyle\int\log
f(X_{1}(x))d\gamma$ $\displaystyle=$
$\displaystyle\frac{1}{2}\>E_{\gamma}\int_{0}^{1}\left|\frac{P(X_{s})E[\hat{D}_{s}f(X_{1})|{\mathcal{F}}_{s}(X)]}{Q_{1-s}f(X_{s})}\right|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\>E_{\gamma}\int_{0}^{1}\left|\frac{P(X_{s})\sigma^{\star}(X_{s}(x))M_{1-s}\partial
f(X_{s}(x))}{Q_{1-s}f(X_{s})}\right|^{2}$ $\displaystyle\leq$ $\displaystyle
E_{\gamma}\int_{0}^{1}\left|\frac{\sigma^{\star}(X_{s})(\hat{\partial}X_{s}^{\star})^{-1}\hat{\partial}X_{s}^{\star}M_{1-s}\partial
f(X_{s})}{Q_{1-s}f(X_{s})}\right|^{2}ds$ $\displaystyle\leq$ $\displaystyle
E_{\gamma}\int_{0}^{1}|\sigma^{\star}(X_{s})(\hat{\partial}X_{s}^{\star})^{-1}|^{2}\left|\frac{\hat{\partial}X_{s}^{\star}M_{1-s}\partial
f(X_{s})}{Q_{1-s}f(X_{s})}\right|^{2}ds\,.$

Note that, from Lemma 3, the process

$(\frac{\hat{\partial}X_{s}^{\star}M_{1-s}\partial
f(X_{s})}{Q_{1-s}f(X_{s})},s\in[0,1])$

is a $\gamma$-martingale, consequently we have

$\displaystyle H(\gamma|P)$ $\displaystyle\leq$ $\displaystyle
E_{\gamma}\left[\left|\frac{\hat{\partial}X_{1}^{\star}\partial
f(X_{1})}{f(X_{1})}\right|^{2}\int_{0}^{1}|\sigma^{\star}(X_{t})(\hat{\partial}X_{s}^{\star})^{-1}|^{2}dt\right]$
$\displaystyle=$ $\displaystyle
E\left[\frac{1}{f(X_{1})}|\hat{\partial}X_{1}^{\star}\partial
f(X_{1})|^{2}Z\right]\,,$

where $Z$ is defined in the announcement of Theorem 14.

Left side of the equation (7.31) depends only on the law of the random
variable $X_{1}(x)$, but at the right hand side there is the term
$\Gamma(x,y)$ which is defined as

$\Gamma(x,y)=E[Z\hat{\partial}X_{1}(x)\cdot\hat{\partial}X_{1}^{\star}(x)|X_{1}(x)=y]\,,$

where

$Z(x)=\int_{0}^{1}|\sigma^{\star}(X_{t}(x))(\hat{\partial}X_{t}(x)^{\star})^{-1}|^{2}dt\,.$

Consequently, the right hand side of the inequality (7.31) depends rather
explicitly on $\sigma$, while the left hand side depends only on
$a=\sigma\sigma^{\star}$, hence only on the law of the process. Meanwhile,
this inequality remains valid for any $\sigma$ which is $n\times d$-matrix
valued function satisfying the relation $a=\sigma\sigma^{\star}$. In
particular the second dimension $d$ is not fixed. Let us define

$\Sigma_{a}=\cup_{d\geq 1}\\{\sigma\in C_{b}^{1}({\rm I\\!R}^{n},M(n\times
d)):\,\sigma\,\sigma^{\star}=a\\}\,,$

where $M(n\times d)$ denotes $n\times d$-matrices.

######  Corollary 8.

Let us define the functional $Q(\partial f,\partial f)$ for $f\in
C_{b}^{2}({\rm I\\!R}^{n})$ as

$\inf_{\sigma\in\Sigma_{a}}\int_{{\rm I\\!R}^{n}}(\Gamma_{\sigma}(x,y)\partial
f(y),\partial f(y))_{{\rm I\\!R}^{n}}q_{1}(x,y)dy\,.$

Let $\nu_{1}^{x}$ be the law of the random variable $\omega\to
X_{1}(x,\omega)$. Then it holds true that

(7.32) $\int_{{\rm
I\\!R}^{n}}f^{2}(y)\log\frac{f^{2}(y)}{\nu_{1}^{x}(f^{2})}\nu_{1}^{x}(dy)\leq
Q(\partial f,\partial f)\,.$

##  8\. An example: Poincaré and Log-Sobolev inequalities on the Heisenberg
Group

Let us apply the ideas explained above to prove the logarithmic Sobolev
inequality for the heat kernel on the Heisenberg group $G={\rm I\\!R}^{3}$,
where the group operation $\star$ is defined for $(x,y,z)$ and $(a,b,c)$ as

$(x,y,z)\star(a,b,c)=(x+a,y+b,z+c+\frac{1}{2}\>(xb-ya))\,.$

Let $b$ and $w$ be two real-valued, independent Brownian motions, the
diffusion process with which we are concerned is the unique solution of the
following SDE:

$x_{t}=\int_{0}^{t}\sigma(x_{s})dB_{s}\,,$

where $B$ is the two-dimensional Brownian motion $B=(b,w)$ and $\sigma$ is the
matrix given by

$\sigma(x,y,z)=\left[{\begin{array}[]{cc}1&0\\\ 0&1\\\ -y/2&x/2\\\
\end{array}}\right]$

The solution of this equation is given explicitly as
$(x_{t}=(b_{t},w_{t},\frac{1}{2}\>z_{t}(b,w)),t\in{\rm I\\!R}_{+})$, where
$(z_{t})$ is the Lévy area process:

$z_{t}=\int_{0}^{t}(b_{s}dw_{s}-w_{s}db_{s})\,.$

We shall denote by $(Q_{t},t\geq 0)$ the semi-group corresponding to the
corresponding Markov process defined by $Q_{t}f(p)=E[f(p\star x_{t})]$,
$p\in{\rm I\\!R}^{3}$. We denote by $X$ and $Y$ the left-invariant vector
fields defined as

$\displaystyle X$ $\displaystyle=$
$\displaystyle\partial_{x}-\frac{y}{2}\partial_{z}$ $\displaystyle Y$
$\displaystyle=$ $\displaystyle\partial_{y}+\frac{x}{2}\partial_{z}\,,$

and with $\hat{X}$ and $\hat{Y}$ the right invariant ones:

$\displaystyle\hat{X}$ $\displaystyle=$
$\displaystyle\partial_{x}+\frac{y}{2}\partial_{z}$ $\displaystyle\hat{Y}$
$\displaystyle=$ $\displaystyle\partial_{y}-\frac{x}{2}\partial_{z}\,.$

Note that the hypoelliptic infinitesimal generator ${\rm I\\!L}$ of $(Q_{t})$
is given by

${\rm I\\!L}=\frac{1}{2}\>(X^{2}+Y^{2})\,.$

It follows from direct calculation that $\hat{X}$ and $\hat{Y}$ commute with
${\rm I\\!L}$, hence with the semigroup $(Q_{t})$. Let now $A$ be the isometry
defined by

$-A=\left[{\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&-1\\\
\end{array}}\right]$

and let us denote by $(y_{t},t\geq 0)$ the process defined by
$y_{t}=(-b_{t},-w_{t},\frac{1}{2}\>z_{t}(b,w))$, note that $y_{t}=Ax_{t}$ and
the law of $y$ is equal to the law of $x$. Moreover, $y$ defines also a Markov
process with semi-group defined as $R_{t}f(p)=E[f(y_{t}\star p)]$ with
infinitesimal generator

$\hat{{\rm I\\!L}}=\frac{1}{2}\>(\hat{X}^{2}+\hat{Y}^{2})\,.$

Although the semi-groups $(Q_{t},t\geq 0)$ and $(R_{t},t\geq 0)$ are
different, they are equal on the paths starting from the origin:

######  Lemma 4.

For any $t\geq 0$, $f\in C_{b}({\rm I\\!R}^{3})$, we have

(8.33) $\displaystyle R_{T-t}f(y_{t})$ $\displaystyle=$ $\displaystyle
E[f(y_{T})|{\mathcal{F}}_{t}(x)]=Q_{T-t}f(y_{t})$ (8.34) $\displaystyle
Q_{T-t}f(x_{t})$ $\displaystyle=$ $\displaystyle
E[f(x_{T})|{\mathcal{F}}_{t}(x)]=R_{T-t}f(x_{t})$

for any $T\geq t$, P-almost surely. In particular, we have

(8.35) $\displaystyle\hat{{\rm I\\!L}}f(y_{t})$ $\displaystyle=$
$\displaystyle{\rm I\\!L}f(y_{t})$ (8.36) $\displaystyle\hat{{\rm
I\\!L}}f(x_{t})$ $\displaystyle=$ $\displaystyle{\rm I\\!L}f(x_{t})$

Proof: Let $\theta(b,w)$ be any bounded and ${\mathcal{F}}_{t}(x)$–measurable
function. As ${\mathcal{F}}_{t}(x)=\sigma(b_{s},w_{s},s\leq t)$. We have

$\displaystyle E[f(y_{T}(b,w))\,\theta(b,w)]$ $\displaystyle=$ $\displaystyle
E[f(y_{T}(-b,-w))\theta(-b,-w)]$ $\displaystyle=$ $\displaystyle
E\left[f(x_{T}(b,w))\theta(-b,-w)\right]$ $\displaystyle=$ $\displaystyle
E[E[f(x_{T})|{\mathcal{F}}_{t}(x)]\theta(-b,-w)]$ $\displaystyle=$
$\displaystyle E\left[Q_{T-t}f(x_{t}(b,w))\theta(-b,-w)\right]$
$\displaystyle=$ $\displaystyle E[Q_{T-t}f(x_{t}(-b,-w))\theta(b,w)]$
$\displaystyle=$ $\displaystyle E[Q_{T-t}f(y_{t}(b,w))\theta(b,w)]\,,$

by the invariance of the law of the Brownian motion under the isometries. The
second claim follows by symmetry, the equalities (8.35) and (8.36)
follow111These two relations follow also directly by elementary calculation
from the first part by differentiation under the expectation.

######  Lemma 5.

For any smooth function $f$ on ${\rm I\\!R}^{3}$ and $t,\,T\geq 0$, we have

$\displaystyle\hat{X}R_{T}f(y_{t})$ $\displaystyle=$
$\displaystyle\hat{X}Q_{T}f(y_{t})$ $\displaystyle\hat{Y}R_{T}f(y_{t})$
$\displaystyle=$ $\displaystyle\hat{Y}Q_{T}f(y_{t})$

almost surely.

Proof: As the second one is similar, let us prove the first relation. Since
both sides are measurable w.r.t. the sigma algebra generated by $y_{t}$, it
suffices to test them on the functions $k(y_{t})$, with $k$ being smooth. Let
$r_{t}$ be the density of the law of $y_{t}$ w.r.t. the Lebesgue measure of
${\rm I\\!R}^{3}$. Then we have

$\displaystyle E[\hat{X}R_{T}f(y_{t})k(y_{t})]$ $\displaystyle=$
$\displaystyle-E[R_{T}f(y_{t})(\hat{X}k(y_{t})+k(y_{t})\hat{X}\log
r_{t}(y_{t}))]$ $\displaystyle=$
$\displaystyle-E[Q_{T}f(y_{t})(\hat{X}k(y_{t})+k(y_{t})\hat{X}\log
r_{t}(y_{t}))]$ $\displaystyle=$ $\displaystyle
E[\hat{X}Q_{T}f(y_{t})k(y_{t})]\,,$

hence the proof follows.

######  Proposition 6.

For any smooth function $f$ on ${\rm I\\!R}^{3}$ and $t\geq 0$, we have

  * •

$f(y_{1})=-\int_{0}^{1}Q_{1-s}\hat{X}f(y_{s})db_{s}-\int_{0}^{1}Q_{1-s}\hat{Y}f(y_{s})dw_{s}\,,$

  * •

in particular the following bound hold true:

$E[|f(y_{1})-E[f(y_{1})]|^{2}]\leq
E\left[|\hat{X}f(y_{1})|^{2}+|\hat{Y}f(y_{1})|^{2}\right]\,.$

Proof: We will apply the Itô formula to the process
$(R_{1-t}f(y_{t}),t\in[0,1])$ which is also a martingale for smooth $f$. From
Lemma 4, $R_{1-t}f(y_{t})=Q_{1-t}f(y_{t})$ and from Lemma 5,
$\hat{X}R_{1-t}f(y_{t})=\hat{X}Q_{1-t}f(y_{t})$ and
$\hat{Y}R_{1-t}f(y_{t})=\hat{Y}Q_{1-t}f(y_{t})$. Moreover, as $\hat{X}$ and
$\hat{Y}$ commute with $Q_{s}$, we get

$\displaystyle R_{1-t}f(y_{t})$ $\displaystyle=$ $\displaystyle
Q_{1-t}f(y_{t})$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}\hat{X}Q_{1-s}f(y_{s})db_{s}-\int_{0}^{t}\hat{Y}Q_{1-s}f(y_{s})dw_{s}$
$\displaystyle=$
$\displaystyle-\int_{0}^{t}Q_{1-s}\hat{X}f(y_{s})db_{s}-\int_{0}^{t}Q_{1-s}\hat{Y}f(y_{s})dw_{s}\,.$

At first sight, in the equality (8) the integrals should be taken as
Stratonovitch integrals but, the left hand side is a continuous martingale,
consequently the only terms remaining after calculations are Itô integrals,
thanks to the relation

$\hat{{\rm I\\!L}}f(y_{t})={\rm I\\!L}f(y_{t})\,.$

To complete the proof, it suffices to remark that the integrands in the
equation (8) are martingales, hence their squares are submartingales,
consequently their expectations increase with $s\in[0,1]$:

$\displaystyle\sup_{s\leq 1}E[|Q_{1-s}\hat{X}f(y_{s})|^{2}]$
$\displaystyle\leq$ $\displaystyle E[|\hat{X}f(y_{1})|^{2}]$
$\displaystyle\sup_{s\leq 1}E[|Q_{1-s}\hat{Y}f(y_{s})|^{2}]$
$\displaystyle\leq$ $\displaystyle E[|\hat{Y}f(y_{1})|^{2}]\,.$

We are ready to prove

######  Theorem 16.

For any smooth function $f$ on ${\rm I\\!R}^{3}$, we have the following
Poincaré inequality

$E[|f(x_{1})-E[f(x_{1})]|^{2}]\leq E[|Xf(x_{1})|^{2}+|Yf(x_{1})|^{2}]\,.$

Proof: By the equality of the laws of $x$ and $y$ and using the inequality of
Proposition 6, we have

$\displaystyle E[|f(x_{1})-E[f(x_{1})]|^{2}]$ $\displaystyle=$ $\displaystyle
E[|f(y_{1})-E[f(y_{1})]|^{2}]$ $\displaystyle\leq$ $\displaystyle
E\left[|\hat{X}f(y_{1})|^{2}+|\hat{Y}f(y_{1})|^{2}\right]\,.$

Moreover, the term inside the expectation of the right hand side is the square
of the norm of the operator of Paul-André Meyer, called “opérateur carré du
champs” (cf. [7]) of the right Markov process $(y_{t}\star p,t\in{\rm
I\\!R},p\in{\rm I\\!R}^{3})$ and it can be written as

$|\hat{X}f(y_{1})|^{2}+|\hat{Y}f(y_{1})|^{2}=-\frac{1}{2}\>[\hat{{\rm
I\\!L}}f^{2}(y_{1})-2f(y_{1})\hat{{\rm I\\!L}}f(y_{1})]\,,$

and from Proposition 6, it is also equal to

$-\frac{1}{2}\>[{\rm I\\!L}f^{2}(y_{1})-2f(y_{1}){\rm
I\\!L}f(y_{1})]=|Xf(y_{1})|^{2}+|Yf(y_{1})|^{2}\,,$

and the claim follows using again the equality of the laws of the processes
$x$ and $y$.

Having developped all the necessary techniques, we can now prove the
logarithmic Sobolev inequality for the measure defined as the law of the
process at the instant $t=1$, which is regarded as a measure on the Heisenberg
group in ${\rm I\\!R}^{3}$

######  Theorem 17.

For any smooth function on ${\rm I\\!R}^{3}$, we have

(8.38) $E\left[f^{2}(x_{1})(\log f^{2}(x_{1})-\log E[f^{2}(x_{1})])\right]\leq
2(E[|Xf(x_{1})|^{2}+|Yf(x_{1})|^{2}])\,.$

Proof: We begin with the right process and define the martingale
$M_{t}=E[f(y_{1})|{\mathcal{F}}_{t}(x)]$, while supposing that $f$ is strictly
positive and $E[f(x_{1})]=E[f(y_{1})]=1$. Using the Itô formula and Lemmas 4,
5:

$\displaystyle dM_{t}$ $\displaystyle=$
$\displaystyle-\frac{\hat{X}R_{1-t}f(y_{t})}{R_{1-t}f(y_{t})}db_{t}-\frac{\hat{Y}R_{1-t}f(y_{t})}{R_{1-t}f(y_{t})}dw_{t}$
$\displaystyle=$
$\displaystyle-\frac{\hat{X}Q_{1-t}f(y_{t})}{Q_{1-t}f(y_{t})}db_{t}-\frac{\hat{Y}Q_{1-t}f(y_{t})}{Q_{1-t}f(y_{t})}dw_{t}$
$\displaystyle=$
$\displaystyle-\frac{Q_{1-t}\hat{X}f(y_{t})}{Q_{1-t}f(y_{t})}db_{t}-\frac{Q_{1-t}\hat{Y}f(y_{t})}{Q_{1-t}f(y_{t})}dw_{t}\,,$

where we have used the fact that $Q_{t}$ commutes with $\hat{X}$ and
$\hat{Y}$. Let us denote by $\nu$ the probability $d\nu=f(y_{1})dP$. Then, we
get from above

$\displaystyle E[f(y_{1})\log f(y_{1})]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\>E_{\nu}\int_{0}^{1}\left[|E_{\nu}[\hat{X}\log
f(y_{1})|{\mathcal{F}}_{t}(x)]|^{2}+|E_{\nu}[\hat{Y}\log
f(y_{1})|{\mathcal{F}}_{t}(x)]|^{2}\right]dt$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\>E_{\nu}[|\hat{X}\log f(y_{1})|^{2}+|\hat{Y}\log
f(y_{1})|^{2}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\>E\left[\frac{1}{f(y_{1})}(|\hat{X}f(y_{1})|^{2}+|\hat{Y}f(y_{1})|^{2})\right]$

Replacing $f(y_{1})$ by $f^{2}(y_{1})/E[f^{2}(y_{1})]$, we obtain from above

(8.39) $E\left[f^{2}(y_{1})\log(f^{2}(y_{1})/E[f^{2}(y_{1})])\right]\leq
2E[|\hat{X}f(y_{1})|^{2}+|\hat{Y}f(y_{1})|^{2})]\,.$

The left hand side of (8.39) is equal to

$E\left[f^{2}(x_{1})(\log f^{2}(x_{1})-\log E[f^{2}(x_{1})])\right]$

and the right hand side is equal to the expectation of the “carré du champs”
applied to $f$ at the point $x_{1}$ and this finishes the proof.

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A. S. Üstünel, Bilkent University, Math. Dept., Ankara, Turkey
<EMAIL_ADDRESS>