# A White Noise Approach to Stochastic Currents of Brownian Motion

Martin Grothaus
Department of Mathematics, University of Kaiserslautern,
67653 Kaiserslautern, Germany
Email<EMAIL_ADDRESS>Herry Pribawanto Suryawan
Department of Mathematics, Sanata Dharma University
55281 Yogyakarta, Indonesia
Email<EMAIL_ADDRESS>José Luís da Silva
CIMA, University of Madeira, Campus da Penteada,
9020-105 Funchal, Portugal
Email<EMAIL_ADDRESS>
###### Abstract

In this paper we study stochastic currents of Brownian motion $\xi(x)$,
$x\in\mathbb{R}^{d}$, by using white noise analysis. For
$x\in\mathbb{R}^{d}\backslash\\{0\\}$ and for $x=0\in\mathbb{R}$ we prove that
the stochastic current $\xi(x)$ is a Hida distribution. Moreover for
$x=0\in\mathbb{R}^{d}$ with $d>1$ we show that the stochastic current is not a
Hida distribution.

Keywords: Stochastic currents, extended Skorokhod integral, white noise
analysis.

##  1 Introduction

The concept of current is fundamental in geometric measure theory. The
simplest version of current is given by the functional

$\varphi\mapsto\int_{0}^{T}(\varphi(\gamma(t)),\gamma^{\prime}(t))_{\mathbb{R}^{d}}\,\mathrm{d}t,\quad
0<T<\infty,$

in a space of vector fields
$\varphi:\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d}$ and $\gamma$ is a
rectifiable curve in $\mathbb{R}^{d}$. Informally, this functional may be
represented via its integral kernel

$\zeta(x)=\int_{0}^{T}\delta(x-\gamma(t))\gamma^{\prime}(t)\,\mathrm{d}t,$

where $\delta$ is the Dirac delta distribution on $\mathbb{R}^{d}$. The
interested reader may find comprehensive account on the subject in the books
[Fed96, Mor16].

The stochastic analog of the current $\zeta(x)$ rises if we replace the
deterministic curve $\gamma(t)$, $t\in[0,T]$, by the trajectory of a
stochastic process $X(t)$, $t\in[0,T]$, in $\mathbb{R}^{d}$. In this way, we
obtain the following functional

$\xi(x):=\int_{0}^{T}\delta(x-X(t))\,\mathrm{d}X(t).$ (1.1)

The stochastic integral (1.1) has to be properly defined. Now we consider a
$d$-dimensional Brownian motion $B(t)$, $t\in[0,T]$, and the main object of
our study is

$\xi(x)=\int_{0}^{T}\delta(x-B(t))\,\mathrm{d}B(t).$ (1.2)

In this work the stochastic integral (1.2) is interpreted as an extension of
the Skorokhod integral developed in [HKPS93]. It coincides with the extension
given by the adjoint of the Malliavin gradient. There have been some other
approaches to study stochastic current, such as Malliavin calculus and
stochastic integrals via regularization, see [FGGT05, FGR09, FT10, Guo14],
among others.

An initial study of the stochastic current (1.2) using white noise theory was
done in [GT13]. The authors showed that $\xi(x)$ in (1.2) is well defined as a
Hida distribution for all $x\in\mathbb{R}^{d}$ and all dimensions
$d\in\mathbb{N}$. However the proof of Theorem 3.3 in [GT13] is not carefully
written which lead the authors to an inaccurate conclusion. In fact, for
$x=0\in\mathbb{R}^{d}$, $d>1$, we show that $\xi(0)$ is not a Hida
distribution. This is confirmed by first orders of the chaos expansion we
obtained. Moreover, we got the impression that the authors were not checking
integrability of the integrand in (1.1). Hence, they cannot apply Corollary
2.5 below. We in turn could check the assumptions of Corollary 2.5, below, for
all nonzero $x\in\mathbb{R}^{d}$, $d\in\mathbb{N}$, and for
$x=0\in\mathbb{R}$. The aim of this paper is to fill this gap and obtain
kernels of first orders of the chaos expansion of $\xi(x)$.

The organization of the paper is as follows. Section 2 provides some
background of white noise analysis. In Section 3 we prove the main results of
this paper on the existence of the Brownian currents.

##  2 Gaussian White Noise Analysis

In this section we summarize pertinent results from white noise analysis used
throughout this work, and refer to [HKPS93, KLP+96, Kuo96] and references
therein for a detailed presentation.

###  2.1 White Noise Space

We start with the Gel’fand triple

$S_{d}\subset L_{d}^{2}\subset S^{\prime}_{d},$

where $S_{d}:=S(\mathbb{R},\mathbb{R}^{d})$, $d\in\mathbb{N}$, is the space of
vector valued Schwartz test functions, $S^{\prime}_{d}$ is its topological
dual and the central Hilbert space
$L_{d}^{2}:=L^{2}(\mathbb{R},\mathbb{R}^{d})$ of square integrable vector
valued measurable functions. For any $f\in L_{d}^{2}$ given by
$f=(f_{1},\ldots,f_{d})$ its norm is

$|f|^{2}=\sum_{i=1}^{d}\int_{\mathbb{R}}|f_{i}(x)|^{2}\,\mathrm{d}x.$

Let $\mathcal{B}$ be the $\sigma$-algebra of cylinder sets on
$S^{\prime}_{d}$. Since $S_{d}$ equipped with its standard topology is a
nuclear space, by Minlos’ theorem there is a unique probability measure
$\mu_{d}$ on $(S^{\prime}_{d},\mathcal{B})$ with the characteristic function
given by

$C(\varphi):=\int_{S^{\prime}_{d}}e^{i\langle
w,\varphi\rangle}\,\mathrm{d}\mu_{d}(w)=\exp\left(-\frac{1}{2}|\varphi|^{2}\right),\quad\varphi\in
S_{d}.$

Hence, we have constructed the white noise probability space
$(S^{\prime}_{d},\mathcal{B},\mu_{d})$. In the complex Hilbert space
$L^{2}(\mu_{d}):=L^{2}(S^{\prime}_{d},\mathcal{B},\mu_{d};\mathbb{C})$ a
$d$-dimensional Brownian motion is given by

$B(t,w)=(\langle w_{1},\eta_{t}\rangle,\ldots,\langle
w_{d},\eta_{t}\rangle),\;w=(w_{1},\ldots,w_{d})\in
S^{\prime}_{d},\quad\eta_{t}:=1\\!\\!1_{[0,t)},\;t\geq 0.$

In other words, $(B(t))_{t\geq 0}$ consists of $d$ independent copies of
$1$-dimensional Brownian motions. For all $F\in L^{2}(\mu_{d})$ one has the
Wiener-ItÃŽ-Segal chaos decomposition

$F(w)=\sum_{n=0}^{\infty}\langle:w^{\otimes n}:,F_{n}\rangle,\quad
F_{n}\in(L_{d,\mathbb{C}}^{2})^{\hat{\otimes}n},$

where $:w^{\otimes n}:\in(S^{\prime}_{d,\mathbb{C}})^{\hat{\otimes}n}$ denotes
the $n$-th order Wick power of $w\in S^{\prime}_{d,\mathbb{C}}$ and
$\langle\cdot,\cdot\rangle$ denotes the dual pairing on
$(S^{\prime}_{d,\mathbb{C}})^{\otimes n}\times(S{}_{d,\mathbb{C}})^{\otimes
n}$ which is a bilinear extension of $(\cdot,\bar{\cdot})$, where
$(\cdot,\cdot)$ is the inner product on $(L_{d,\mathbb{C}}^{2})^{\otimes n}$
in the sense of a Gel’fand triple. Here $V_{\mathbb{C}}$ denotes the
complexification of the real vector space $V$ and $\hat{\otimes}n$ denotes the
$n$-th power symmetric tensor product. Note that $\langle:\cdot^{\otimes
n}:,\cdot\rangle$, $n\in\mathbb{N}_{0}$, in the second variable extends to
$(L_{d,\mathbb{C}}^{2})^{\hat{\otimes}n}$ in the sense of an $L^{2}(\mu_{d})$
limit.

###  2.2 Hida Distributions and Characterization

By the standard construction with the Hilbert space $L^{2}(\mu_{d})$ as
central space, we obtain the Gel’fand triple of Hida test functions and Hida
distributions.

$(S_{d})\subset L^{2}(\mu_{d})\subset(S_{d})^{\prime}.$

We denote the dual pairing between elements of $(S_{d})^{\prime}$ and
$(S_{d})$ by $\langle\\!\langle\cdot,\cdot\rangle\\!\rangle$. For $F\in
L^{2}(\mu_{d})$ and $\varphi\in(S_{d})$, with kernel functions $F_{n}$ and
$\varphi_{n}$, respectively, the dual pairing yields

$\langle\\!\langle F,\varphi\rangle\\!\rangle=\sum_{n=0}^{\infty}n!\langle
F_{n},\varphi_{n}\rangle.$

This relation extends the chaos expansion to $\Phi\in(S_{d})^{\prime}$ with
distribution valued kernels
$\Phi_{n}\in(S^{\prime}_{d,\mathbb{C}})^{\hat{\otimes}n}$ such that

$\langle\\!\langle\Phi,\varphi\rangle\\!\rangle=\sum_{n=0}^{\infty}n!\langle\Phi_{n},\varphi_{n}\rangle,$

for every generalized test function $\varphi\in(S_{d})$ with kernels
$\varphi_{n}\in(S_{d,\mathbb{C}})^{\hat{\otimes}n}$, $n\in\mathbb{N}_{0}$.

Instead of repeating the detailed construction of these spaces we present a
characterization in terms of the $S$-transform.

######  Definition 2.1.

Let $\varphi\in S_{d}$ be given. We define the Wick exponential by

$e_{\mu_{d}}(\cdot,\varphi):=\frac{e^{\langle\cdot,\varphi\rangle}}{\mathbb{E}(e^{\langle\cdot,\varphi\rangle})}=C(\varphi)e^{\langle\cdot,\varphi\rangle}=\sum_{n=0}^{\infty}\frac{1}{n!}\langle:.^{\otimes
n}:,\varphi^{\otimes n}\rangle\in(S_{d})$

and the $S$-transform of $\Phi\in(S_{d})^{\prime}$ by

$S\Phi(\varphi):=\langle\\!\langle\Phi,e_{\mu_{d}}(\cdot,\varphi)\rangle\\!\rangle.$

######  Example 2.2.

For $d\in\mathbb{N}$ the $S$-transform of $d$-dimensional white noise
$(W(t))_{t\geq 0}$ is given by $SW(t)(\varphi)=\varphi(t)$, for all
$\varphi\in S_{d}$, $t\geq 0$, see [HKPS93]. Here $(W(t))_{t\geq 0}$ is the
derivative of $(B(t))_{t\geq 0}$ as a Hida space valued process. That is, each
of its components takes values in $(S_{d})^{\prime}$.

######  Definition 2.3 ($U$-functional).

A function $F:S_{d}\longrightarrow\mathbb{C}$ is called a $U$-functional if:

  1. 1.

For every $\varphi_{1},\varphi_{2}\in S_{d}$ the mapping
$\mathbb{R}\ni\lambda\mapsto F(\lambda\varphi_{1}+\varphi_{2})\in\mathbb{C}$
has an entire extension to $z\in\mathbb{C}$.

  2. 2.

There are constants $0<C_{1},C_{2}<\infty$ such that

$\left|F(z\varphi)\right|\leq
C_{1}\exp\big{(}C_{2}|z|^{2}\|\varphi\|^{2}\big{)},\quad\forall
z\in\mathbb{C},\varphi\in S_{d}$

for some continuous norm $\|\cdot\|$ on $S_{d}$.

We are now ready to state the aforementioned characterization result.

######  Theorem 2.4 (cf. [KLP+96], [PS91]).

The $S$-transform defines a bijection between the space $(S_{d})^{\prime}$ and
the space of $U$-functionals. In other words, $\Phi\in(S_{d})^{\prime}$ if and
only if $S\Phi:S_{d}\to\mathbb{{C}}$ is a $U$-functional.

Based on Theorem 2.4 a deeper analysis of the space $(S_{d})^{\prime}$ can be
developed. The following corollary concerns the Bochner integration of
functions with values in $(S_{d})^{\prime}$ (for more details and proofs see
e.g. [HKPS93], [KLP+96], [PS91] for the case $d=1$).

######  Corollary 2.5.

Let $(\Omega,\mathcal{F},m)$ be a measure space and
$\lambda\mapsto\Phi_{\lambda}$ be a mapping from $\Omega$ to
$(S_{d})^{\prime}$. We assume that the $S$-transform of $\Phi_{\lambda}$
fulfills the following two properties:

  1. 1.

The mapping $\lambda\mapsto S\Phi_{\lambda}(\varphi)$ is measurable for every
$\varphi\in S_{d}$.

  2. 2.

The $U$-functional $S\Phi_{\lambda}$ satisfies

$|S\Phi_{\lambda}(z\varphi)|\leq
C_{1}(\lambda)\exp\left(C_{2}(\lambda)|z|^{2}\|\varphi\|^{2}\right),\quad
z\in\mathbb{C},\varphi\in S_{d},$

for some continuous norm $\|\cdot\|$ on $S_{d}$ and for some $C_{1}\in
L^{1}(\Omega,m)$, $C_{2}\in L^{\infty}(\Omega,m)$.

Then

$\int_{\Omega}\Phi_{\lambda}\,\mathrm{d}m(\lambda)\in(S_{d})^{\prime}$

and

$S\left(\int_{\Omega}\Phi_{\lambda}\,\mathrm{d}m(\lambda)\right)(\varphi)=\int_{\Omega}S\Phi_{\lambda}(\varphi)\,\mathrm{d}m(\lambda),\quad\varphi\in
S_{d}.$

Moreover, the integral exists as a Bochner integral in some Hilbert subspace
of $(S_{d})^{\prime}.$

######  Example 2.6 (Donsker’s delta function).

As a classical example of a Hida distribution we have the Donsker delta
function. More precisely, the following Bochner integral is a well defined
element in $(S_{d})^{\prime}$

$\delta(x-B(t))=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{i(\lambda,x-B(t))_{\mathbb{R}^{d}}}\,\mathrm{d}\lambda,\quad
x\in\mathbb{R}^{d}.$

The $S$-transform of $\delta(x-B(t))$ for any $z\in\mathbb{C}$ and $\varphi\in
S_{d}$ is given by

$S\delta(x-B(t))(z\varphi)=\frac{1}{(2\pi
t)^{d/2}}\exp\left(-\frac{1}{2t}\sum_{j=1}^{d}(x_{j}-\langle
z\varphi_{j},\eta_{t}\rangle)^{2}\right).$ (2.1)

It is well known that the Wick product is a well defined operation in Gaussian
analysis, see for example [KLS96], [HOUZ10] and [KSWY98].

######  Definition 2.7.

For any $\Phi,\Psi\in(S_{d})^{\prime}$ the Wick product $\Phi\Diamond\Psi$ is
defined by

$S(\Phi\Diamond\Psi)=S\Phi\cdot S\Psi.$ (2.2)

Since the space of $U$-functional s is an algebra, by Theorem 2.4 there exists
an element $\Phi\Diamond\Psi\in(S_{d})^{\prime}$ such that (2.2) holds.

##  3 Stochastic Currents of Brownian Motion

In this section we investigate in the framework of white noise analysis the
following functional

$\varphi\mapsto\int_{0}^{T}(\varphi(B(t)),\mathrm{d}B(t))_{\mathbb{R}^{d}},$
(3.1)

on a given space of vector fields
$\varphi:\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d}$. The functional (3.1)
can be represented via its integral kernel

$\xi(x):=\int_{0}^{T}\delta(x-B(t))\,\mathrm{d}B(t),\quad x\in\mathbb{R}^{d}.$

We interpret the stochastic integral as an extended Skorokhod integral

$\displaystyle\int_{0}^{T}\delta(x-B(t))\,\mathrm{d}B(t)$
$\displaystyle:=\left(\int_{0}^{T}\delta(x-B(t))\Diamond
W_{1}(t)\,\mathrm{d}t,\ldots,\int_{0}^{T}\delta(x-B(t))\Diamond
W_{d}(t)\,\mathrm{d}t\right)$ $\displaystyle=:(\xi_{1}(x),\ldots,\xi_{d}(x)),$

where $W=(W_{1},\ldots,W_{d})$ is the white noise process as in Example 2.2.
If the integrand is a square integrable function then this stochastic integral
coincides with the Skorokhod integral. In this interpretation, we call
$\xi(x)$ stochastic currents of Brownian motion.

Below we show that $\xi(x)$, $x\in\mathbb{R}^{d}\backslash\\{0\\}$ is a well
defined functional in $(S_{d})^{\prime}$. From now on, $C$ is a real constant
whose value is immaterial and may change from line to line.

######  Theorem 3.1.

For $x\in\mathbb{R}^{d}\backslash\\{0\\}$, $0<T<\infty$, the Bochner integral

$\xi_{i}(x)=\int_{0}^{T}\delta(x-B(t))\Diamond W_{i}(t)\,\mathrm{d}t$ (3.2)

is a Hida distribution and its $S$-transform at $\varphi\in S_{d}$ is given by

$S\left(\int_{0}^{T}\delta(x-B(t))\Diamond
W_{i}(t)\,\mathrm{d}t\right)(\varphi)=\frac{1}{(2\pi)^{{}^{d/2}}}\int_{0}^{T}\frac{1}{t^{d/2}}e^{-\frac{|x-\langle\eta_{t},\varphi\rangle|_{\mathbb{R}^{d}}^{2}}{2t}}\varphi_{i}(t)\,\mathrm{d}t.$
(3.3)

###### Proof.

First we compute the $S$-transform of the integrand

$(0,T]\ni t\mapsto\Phi_{i}(t):=\delta(x-B(t))\Diamond W_{i}(t).$

Using Definition 2.7, Example 2.6, and Example 2.2 for any $\varphi\in S_{d}$
we have

$\displaystyle t\mapsto S\Phi_{i}(t)(\varphi)$
$\displaystyle=S\big{(}\delta(x-B(t))\big{)}(\varphi)SW_{i}(t)(\varphi)$
$\displaystyle=\frac{1}{(2\pi
t)^{d/2}}\exp\left(-\frac{1}{2t}|x-\langle\eta_{t},\varphi\rangle|_{\mathbb{R}^{d}}^{2}\right)\varphi_{i}(t),$

which is Borel measurable on $(0,T]$. Furthermore, for any $z\in\mathbb{C}$,
$t\in(0,T]$ and all $\varphi\in S_{d}$ we obtain

$\displaystyle\big{|}S\Phi_{i}(t)(z\varphi)\big{|}$ $\displaystyle\leq$
$\displaystyle\big{|}\frac{1}{\left(2\pi
t\right)^{d/2}}\exp\left(-\frac{1}{2t}\big{|}x-\langle\eta_{t},z\varphi\rangle\big{|}_{\mathbb{R}^{d}}^{2}\right)z\varphi_{i}(t)\big{|}$
$\displaystyle\leq$ $\displaystyle\frac{1}{\left(2\pi
t\right)^{d/2}}\exp\left(-\frac{1}{2t}|x|^{2}\right)\exp\left(\frac{1}{t}|x|\big{|}\langle\eta_{t},z\varphi\rangle\big{|}\right)\exp\left(\frac{1}{2t}|z|^{2}\big{|}\langle\eta_{t},\varphi\rangle\big{|}^{2}\right)\big{|}z\varphi_{i}(t)\big{|}$
$\displaystyle\leq$ $\displaystyle\frac{1}{\left(2\pi
t\right)^{d/2}}\exp\left(-\frac{1}{2t}|x|^{2}\right)\exp\left(|x||z|\big{|}\varphi\big{|}_{\infty}\right)\exp\left(\frac{1}{2}|z|^{2}\big{|}\varphi\big{|}^{2}\right)\exp\left(|z|\big{|}\varphi\big{|}_{\infty}\right)$
$\displaystyle\leq$ $\displaystyle\frac{C}{\left(2\pi
t\right)^{d/2}}\exp\left(-\frac{1}{2t}|x|^{2}\right)\exp\left(\frac{1}{2}|x|^{2}\right)\exp\left(\frac{1}{2}|z|^{2}\big{|}\varphi\big{|}^{2}\right)\exp\left(\frac{1}{2}|z|^{2}\big{|}\varphi\big{|}_{\infty}^{2}\right)$
$\displaystyle\leq$ $\displaystyle\frac{C}{\left(2\pi
t\right)^{d/2}}\exp\left(-\frac{1}{2t}|x|^{2}\right)\exp\left(\frac{1}{2}|x|^{2}\right)\exp\left(\frac{1}{2}|z|^{2}\|\varphi\|^{2}\right),$

where $\|\cdot\|$ is the continuous norm on $S_{d}$ defined by
$\|\varphi\|:=\sqrt{\big{|}\varphi\big{|}^{2}+\big{|}\varphi\big{|}_{\infty}^{2}}$.
The first factor $\frac{C}{\left(2\pi
t\right)^{d/2}}\exp\left(-\frac{1}{2t}|x|^{2}\right)$ is integrable with
respect to the Lebesgue measure $\mathrm{d}t$ on $[0,T]$. To be more precise,
using the formula

$\int_{u}^{\infty}y^{\nu-1}e^{-\mu
y}\,\mathrm{d}y=\mu^{-\nu}\Gamma\left(\nu,\mu u\right),\quad
u>0,\mathrm{Re}(\mu)>0,$

where $\Gamma\left(\cdot,\cdot\right)$ is the complementary incomplete gamma
function, one can show that

$\int_{0}^{T}\frac{1}{t^{d/2}}\exp\left(-\frac{1}{2t}|x|^{2}\right)\mathrm{d}t=2^{d/2-1}|x|^{2-d}\Gamma\left(\frac{d}{2}-1,\frac{|x|^{2}}{2T}\right).$

As the second factor
$\exp\left(\frac{1}{2}|x|^{2}\right)\exp\left(\frac{1}{2}|z|^{2}\|\varphi\|^{2}\right)$
is independent of $t\in(0,T]$, the result now follows from Corollary 2.5. ∎

######  Corollary 3.2.

For $x=0$ and $d=1$ the stochastic current $\xi(0)$ is a Hida distribution,
that is, the Bochner integral

$\xi(0)=\int_{0}^{T}\delta(B(t))\Diamond W(t)\,\mathrm{d}t$

is a Hida distribution. Moreover its $S$-transform at $\varphi\in S_{1}$ is
given by

$S\left(\int_{0}^{T}\delta(B(t))\Diamond
W(t)\,\mathrm{d}t\right)(\varphi)=\frac{1}{\sqrt{2\pi}}\int_{0}^{T}\frac{1}{\sqrt{t}}e^{-\frac{\langle\eta_{t},\varphi\rangle^{2}}{2t}}\varphi(t)\,\mathrm{d}t.$

###### Proof.

By adapting the proof of Theorem 3.1 we obtain for any $z\in\mathbb{C}$,
$t\in(0,T]$ and all $\varphi\in S_{1}$

$|S\Phi(t)(z\varphi)|\leq\frac{C}{\left(2\pi
t\right)^{1/2}}\exp\left(\frac{1}{2}|z|^{2}\|\varphi\|^{2}\right).$

Since the function $(0,T]\ni t\mapsto t^{-1/2}$ is integrable with respect to
the Lebesgue measure, Corollary 2.5 implies the statement of the corollary. ∎

######  Remark 3.3.

We would like to comment on the chaos expansion of the stochastic current of
Brownian motion. To this end we identify the space $L_{d}^{2}$ with the
Hilbert space $L^{2}(m):=L^{2}(E,\mathcal{B},m)$, where
$E:=\mathbb{R}\times\\{1,\ldots,d\\}$, $\mathcal{B}$ is the product
$\sigma$-algebra on $E$ of the Borel $\sigma$-algebra on $\mathbb{R}$ and the
power set of $\\{1,\ldots,d\\}$ and $m=\mathrm{d}x\otimes\Sigma$ is the
product measure of the Lebesgue measure on $\mathbb{R}$ and the counting
measure on $\\{1,\ldots,d\\}$. That is, for all $f,g\in L^{2}(m)$ we have

$(f,g)_{L^{2}(m)}=\int_{E}f(x,i)g(x,i)\,\mathrm{d}m(x,i)=\sum_{i=1}^{d}\int_{\mathbb{R}}f(x,i)g(x,i)\,\mathrm{d}x.$

The $n$-th order chaos of a Hida distribution can be computed by the $n$-th
order derivative of its $S$-transform at the origin. More precisely, for
$\Psi\in(S_{d})^{\prime}$ and $\varphi\in(S_{d})$ consider the function

$\mathbb{R}\ni s\mapsto U(s):=S\Psi(s\varphi)\in\mathbb{C}.$

Then the $n$-th order chaos $\Psi^{(n)}$ of $\Psi$ applied to
$\varphi^{\otimes n}\in S_{d,\mathbb{C}}^{\hat{\otimes}n}$ is given by

$\langle\Psi^{(n)},\varphi^{\otimes
n}\rangle=\frac{1}{n!}\frac{\mathrm{d}^{n}}{\mathrm{d}s^{n}}U(s)\big{|}_{s=0},$

see [Oba94, Lemma 3.3.5]. In our situation we have

$S\Phi_{i}(s\varphi)=\frac{1}{(2\pi
t)^{d/2}}\exp\left(-\frac{1}{2t}|x-\langle\eta_{t},s\varphi\rangle|_{\mathbb{R}^{d}}^{2}\right)s\varphi_{i}(t),\quad
i\in\\{1,\ldots,d\\}.$

Hence, for the stochastic currents of Brownian motion the first chaos are
given by

$\displaystyle\xi^{(0)}(x)$ $\displaystyle=(0,\ldots,0).$
$\displaystyle\xi_{i}^{(1)}(x)$
$\displaystyle=\left(\underbrace{0,\ldots,0}_{i-1},\frac{1}{(2\pi)^{d/2}}\int_{0}^{T}\frac{1}{t^{d/2}}e^{-\frac{|x|^{2}}{2t}}\delta_{t}\,\mathrm{d}t,0,\ldots,0\right).$
$\displaystyle\left(\xi_{i}^{(2)}(x)\right)_{j,k}$
$\displaystyle=-\frac{1}{4(2\pi)^{d/2}}\int_{0}^{T}\frac{1}{t^{d/2+1}}e^{-\frac{|x|^{2}}{2t}}\left(\mathrm{Id}_{ki}x_{j}\eta_{t}\otimes\delta_{t}+\mathrm{Id}_{ji}x_{k}\delta_{t}\otimes\eta_{t}\right)\mathrm{d}t,$

where $\mathrm{Id}$ denotes the identity matrix on $\mathbb{R}^{d}$ and
$\delta_{t}$ denotes the Dirac distribution at $t>0$. Note that for $x=0$ and
$d>1$ the first chaos $\xi_{i}^{(1)}(0)$ is divergent, hence in this case
$\xi(0)$ cannot be a Hida distribution. In all the other cases the integrals
are well defined as Bochner integrals in a suitable Hilbert subspace of
($S^{\prime}_{d})^{\hat{\otimes}n}$, $n=0,1,2.$ This follows from the
estimates for integrability we derived in the proof of Theorem 3.1. Indeed the
estimates derived to apply Corollary 2.5 imply that the integrands $\Phi_{i}$,
$1\leq i\leq d$, are Bochner integrable in some Hilbert subspace $H_{-}$ of
$(S^{\prime}_{d})$ equipped with a norm $\|\cdot\|_{-}$, see proof of [KLP+96,
Thm. 17]. More precisely, there one shows that $\|\Phi_{i}\|_{-}$, $1\leq
i\leq d$, is integrable. That implies Bochner integrability of the kernels of
$n$-th order in the generalized chaos decomposition in a suitable Hilbert
subspace of $S^{\prime}(\mathbb{R}^{n})$.

##  4 Conclusion and Outlook

In this paper we give a mathematical rigorously meaning to the stochastic
current $\xi(x)$, $x\in\mathbb{R}^{d}\setminus\\{0\\}$ and $\xi(0)$,
$0\in\mathbb{R}$, of Brownian motion in the framework of white noise analysis.
On the other hand, for $x=0\in\mathbb{R}^{d}$, $d>1$, we showed that $\xi$(0)
is not a Hida distribution. The first orders of the chaos expansion leave open
whether the $\xi(x)$, $x\in\mathbb{R}^{d}\backslash\\{0\\}$, are regular
generalized functions or even square integrable. That is, it is not obvious
whether $\xi^{(n)}(x)\in(L_{d,\mathbb{C}}^{2})^{\hat{\otimes}n}$ or not for
$x\in\mathbb{R}^{d}\backslash\\{0\\}$ and $n\in\mathbb{N}$. In a future paper
we plan to extend these results to a larger class of stochastic processes,
e.g., fractional Brownian motion and grey Brownian motion.

### Acknowledgments

This work was partially supported by a grant from the Niels Hendrik Abel Board
and by the Center for Research in Mathematics and Applications (CIMA) related
with the Statistics, Stochastic Processes and Applications (SSPA) group,
through the grant UIDB/MAT/04674/2020 of FCT-Fundação para a Ciência e a
Tecnologia, Portugal. We gratefully acknowledge the financial support by the
DFG through the project GR 1809/14-1.

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Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

PMAI@IJCAI22: International IJCAI Workshop on Process Management in the AI
era, July 23, 2022, Vienna, Austria

[orcid=0000-0002-0624-4431<EMAIL_ADDRESS>] [1]

<EMAIL_ADDRESS>]

[1]

[1]Corresponding author.

# Towards Automated Process Planning and Mining

Peter Fettke  Alexander Rombach  German Research Center for Artificial
Intelligence (DFKI), Saarbruecken, Germany  Saarland University, Saarbruecken,
Germany

(2022)

# Mining process data for process improvements

Peter Fettke  Alexander Rombach  German Research Center for Artificial
Intelligence (DFKI), Saarbruecken, Germany  Saarland University, Saarbruecken,
Germany

(2022)

# (Discussion/Short Paper)

Peter Fettke  Alexander Rombach  German Research Center for Artificial
Intelligence (DFKI), Saarbruecken, Germany  Saarland University, Saarbruecken,
Germany

(2022)

###### Abstract

AI Planning, Machine Learning and Process Mining have so far developed into
separate research fields. At the same time, many interesting concepts and
insights have been gained at the intersection of these areas in recent years.
For example, the behavior of future processes is now comprehensively predicted
with the aid of Machine Learning. For the practical application of these
findings, however, it is also necessary not only to know the expected course,
but also to give recommendations and hints for the achievement of goals, i.e.
to carry out comprehensive process planning. At the same time, an adequate
integration of the aforementioned research fields is still lacking. In this
article, we present a research project in which researchers from the AI and
BPM field work jointly together. Therefore, we discuss the overall research
problem, the relevant fields of research and our overall research framework to
automatically derive process models from executional process data, derive
subsequent planning problems and conduct automated planning in order to
adaptively plan and execute business processes using real-time forecasts.

###### keywords:

Automated Planning Process Mining Machine Learning Business Process Management

##  1 Introduction

Business Process Management (BPM) plays a vital role for organizations. The
consistent design, implementation, execution and monitoring of business
processes promises efficiency increases and cost reduction. In today’s
organizations, BPM follows the paradigm of continuous process improvement.
After a process is designed, implemented and executed, the process execution
data is retrospectively analyzed with methods such as Process Mining in order
to optimize process design in a cyclic manner. In this framework however,
process agility (e.g. process variants) has to be defined at design time in
order to achieve full process automation, which is typically not feasible in
real-world scenarios. Approaches using Process Mining and Machine Learning
(ML) to overcome this problem exist, however stakeholders are left to their
own on how to use these insights [1]. At the same time, existing research
regarding applications of AI planning in BPM focuses on the design phase and
lacks planning of dynamic process executions. Such methods also require an
extensive manual definition of the planning problem beforehand. To add on
that, conventional Process Mining techniques are not necessarily suitable as a
foundation for subsequent automated planning, since discovered process models
purposely filter out information and/or simplify process details for the sake
of readability.

Figure 1: Research gap and overall research framework

Our work is positioned within this research gap at the intersection of AI
Planning, ML and Process Mining, as depicted in Figure 1. We propose
developing new methods to achieve online automated planning leading to an
agile execution of business processes. In that regard, we aim at answering the
following research question: How is an automated planning of processes
achieved using automatically derived process models and automatically
constructed planning models?

##  2 Overall research framework

We propose a procedure model divided into five phases leading from executional
process data to executable planning models, as depicted in the right part of
Figure 1. The general starting point is an ERP system providing process
execution data in shape of relational data and execution logs. The second
stage deals with extracting a process execution log covering the occurring
cases and events. Based on that, the next important step is to automatically
derive a process model. The key aspect hereby is that the resulting model
entails enough process details to allow for AI planning, e.g. not only the
general process flow, but also implicit domain knowledge, involved resources
or potential resource allocation constraints. Such elements can then be used
to automatically construct a sophisticated planning domain in subsequent
phases. In this regard we also propose taking a broader view and starting to
think about mining entire system models which describe the architecture of
(hierarchical) systems including their static as well as dynamic elements
based on distributed runs observable from event logs [2]. Stage four covers
the automated extraction of a state space model from the previously discovered
model, which will then finally be (ideally automatically) transformed into a
process planning model to conduct AI planning in stage five. A plethora of
different techniques can be integrated during this phase. One possibility is
using ML-based methods like DeepQ learning, as it will be portrayed in section
4. These automated conversions once again illustrate the before mentioned
necessity of comprehensive process models, conventional Process Mining
techniques usually cannot provide. The planning model is used to obtain an
optimal execution of the conceptual process model. Resulting process plans are
executed and thus generate new executional process data, leading to an
iterative pipeline that ultimately allows to conduct automated process
planning.

##  3 Application cases

Figure 2: Abstract view of an exemplary application case

We designed different application cases to develop and evaluate our approach,
most prominently a classic flow-shop scheduling problem shown in Figure 2
modeled using Heraklit [2].

In this scenario, a set of different products have to be produced on a
production line consisting of different machine stations and workers to
operate them, where each product has to cross all stations. Figure 2 shows an
exemplary setup consisting of 10 stations and 10 workers. The scenario
includes several variable factors, e.g. the process execution time of one
production step depends on the underlying product, station as well as the
operating worker with a distinct experience level. The general steps to
perform at each station can differ, adding another dimension to the planning
problem. On top of that, the process includes walking times from one station
to another whenever necessary, which in turn is dependent on the layout of the
production line, i.e. how the individual stations are distributed.

##  4 Preliminary results

Regarding phases two and three, multiple modeling frameworks can be
considered. While XES is a well-known standard for event logs, we came to the
conclusion that such representation might not suffice due to not providing
enough information about involved resources necessary for planning domains.
Heraklit on the other hand is a modeling language based on symbolic Petri Nets
which we regard as a good foundation for subsequent planning procedures. It
also allows us to model entire system models as described in section 2
including dependencies between individual subsystems and agents. For more
details about Heraklit, we refer to [2].

Several advances have been made regarding the latter parts of the framework,
e.g. “domain-specific Constraint Patterns” in the context of Constraint
Programming [3]. These patterns describe reoccurring problems and their
solutions and can be combined in order to simplify the construction of
planning problems. An example of such pattern is the flow-shop. It is also
possible to automatically identify occurrences of these patterns in event logs
and automatically assemble parameterized constraint models. This assembly is
ultimately based on a pattern repository, which therefore requires manual yet
reusable one-time definitions of these patterns.

We also propose a novel GPU-assisted approach for on-the-fly generation of
training samples for Deep-Q learning [4], a powerful solution for the main
challenge of providing training data for such techniques, which outperforms
conventional approaches both in terms of run-time and memory/storage
consumption. It uses the domain description extracted from a process model in
order to iteratively instantiate randomized states and compute corresponding
Q-matrices. As a Proof of Concept, the input models are preliminarily provided
in PNML file formats. In the future we plan on integrating richer
representations like heraklit in order to increase the amount of usable domain
knowledge, which is necessary in more complex use cases.

Based on this approach, we developed Deep-Q-Agents methods for optimization.
To achieve convincing results using such techniques, it is necessary to set up
domain specific configurations, which in turn requires domain knowledge. This
however also requires a human assessment of these configurations. To overcome
this problem, we suggest an automated exploration of the underlying search
space as a heuristic to find potentially optimal configurations.

##  5 Conclusion and outlook

Several research initiatives in the fields of AI Planning and Process Mining
in the context of BPM have been made so far. However, there are no satisfying
results when it comes to planning and agile executions of business processes
in an automated fashion. To this end, we propose an iterative procedure model
build upon executional process data to automatically derive process models and
automatically infer solvable planning problems to plan and optimize business
processes. Up until now, several concepts have been implemented and
advancements addressing individual parts of the procedure model have been
made. The next main step is to adequately combine these individual concepts
and put the proposed "round-trip" into practice.

###### Acknowledgements.

This work is part of the research project APPaM (Grant 01IW20006), which is
partly funded by the Federal Ministry of Education and Research (BMBF).

## References

  * Neu et al. [2022] D. Neu, J. Lahann, P. Fettke,  A systematic literature review on state-of-the-art deep learning methods for process prediction,  Artificial Intelligence Review 55 (2022) 801–827. doi:10.1007/s10462-021-09960-8.
  * Fettke and Reisig [2022] P. Fettke, W. Reisig, Systems Mining with Heraklit: The Next Step, 2022\. doi:10.48550/ARXIV.2202.01289.
  * Saller and Koehler [2021] S. Saller, J. Koehler,  Easy, adaptable and high-quality Modelling with domain-specific Constraint Patterns,  Twentieth International Workshop on Constraint Modelling and Reformulation (2021).
  * Köster et al. [2021] M. Köster, J. Groß, A. Krüger,  MACSQ: Massively Accelerated DeepQ Learning on GPUs Using On-the-Fly State Construction,  in: Parallel and Distributed Computing, Applications and Technologies: 22nd International Conference, PDCAT 2021, Springer-Verlag, Berlin, Heidelberg, 2021, p. 383–395. doi:10.1007/978-3-030-96772-7_35.


# HEP Software Foundation Community White Paper Working Group – Conditions
Data

Marko Bracko 3 Marco Clemencic 4,a Dave Dykstra 5 Andrea Formica 4 Giacomo
Govi 6,b Michel Jouvin 7,b David Lange 3,8,c Paul Laycock 9 and Lynn Wood
University of Maribor, Ljubljana, SloveniaJožef Stefan Institute, Ljubljana,
SloveniaCERN, Geneva, SwitzerlandFermi National Accelerator Laboratory,
Batavia, IL, USADSM/IRFU (Institut de Recherches sur les Lois Fondamentales de
l’Univers), CEA Saclay (Commissariat à l’Énergie Atomique), Gif-sur-Yvette,
FranceLAL, Université Paris-Sud and CNRS/IN2P3, Orsay, FrancePrinceton
University, Princeton, NJ, USAPhysics Department, Brookhaven National
Laboratory, Upton, NY, USAPacific Northwest National Laboratory, Richland,
Washington, USA

###### Abstract

To produce the best physics results, high energy physics experiments require
access to calibration and other non-event data during event data processing.
These conditions data are typically stored in databases that provide
versioning functionality, allowing physicists to make improvements while
simultaneously guaranteeing the reproducibility of their results. With the
increased complexity of modern experiments, and the evolution of computing
models that demand large scale access to conditions data, the solutions for
managing this access have evolved over time. In this white paper we give an
overview of the conditions data access problem, present convergence on a
common solution and present some considerations for the future.

|  | HSF-CWP-2017-03
---|---|---
|  | January 16, 2019

aaaSupported by the US-DOE, DE-AC02-07CH11359bbbCommunity White Paper
Editorial Board MembercccCorresponding Author

##  1 Introduction

Access to conditions data is critical to producing the best physics results
from HEP experiments. It is somewhat surprising then that few people can
accurately define what conditions data means. The challenges for conditions
data access are many, notably the requirement to provide simultaneous read
access to conditions data for distributed computing resources at kHz rates.
This is a highly non-trivial problem that has brought difficulties, especially
to the biggest experiments with the largest distributed computing resources;
it is easy to get wrong. Equally those shared difficulties provoked
collaboration on shared solutions, particularly between ATLAS and CMS
Viegas:2008zz (1, 2, 3, 4), and this white paper benefits greatly from that
work.

The purpose of this white paper then is to answer the following questions:

  1. 1.

What is conditions data ?

  2. 2.

What are the major use cases and challenges for conditions data access ?

  3. 3.

What does best practice for conditions data access management look like ?

  4. 4.

What challenges are posed by conditions data access in the future ?

The reader should note that this paper focuses primarily on conditions data
access and therefore not all aspects of conditions data management are
addressed111Conditions data creation workflows and management tools,
authentication and authorization are all examples of topics that deserve
further attention..

The paper is organised as follows: section 2 answers the first two questions,
thus defining the scope of the work presented here. Section 3 describes best
practice in conditions data access management by means of a detailed
prototype. The requirements for the future success of HEP in terms of managing
conditions data access are discussed in section 4, attempting to answer the
final question in the list above, before drawing conclusions in the final
section.

##  2 Conditions data in HEP experiments

Although conditions data is a commonly used term in most HEP experiments, its
definition remains ambiguous for many non-experts. Such ambiguity is not
helpful and has caused considerable problems in the past, some examples of
which will be given later. Conditions data is distinct from the particle
collision event data collected from the experimental detectors, hereafter
simply referred to as event data. However, conditions data is not all non-
event data but rather a particular subset.

From the detector expert perspective, conditions data can be broadly defined
as the non-event data required by event data-processing software to correctly
reconstruct the raw detector event data. In the case of simulated event data,
the scope extends to the correct simulation, digitisation and reconstruction
of the data. The most obvious examples of conditions data are calibration and
alignment constants, but this is not a complete picture. In general, from the
detector perspective, conditions data is the subset of non-event data required
to maintain, operate and optimise detectors. This in turn can be broken down
into the following categories:

  1. a.

Detector and readout configuration parameters.

  2. b.

Detector Control System (DCS) data, typically monitoring values of, e.g.
voltage and current that are provided directly by hardware.

  3. c.

Higher-level detector and system monitoring information, typically provided by
custom software with the purpose of evaluating detector performance.

  4. d.

Detector calibration and alignment data.

Conditions data in the detector context thus largely consists of (d), together
with the subset of (a) and (b) that are required for event data processing.
Other non-event data may also be required for event data processing, for
example particle physics accelerator parameters. Thus, in practice, any non-
event data from any source that are required for event data processing can be
considered as conditions data.

The use cases for conditions data are described in section 2.1. In general,
conditions data vary with time but with a granularity much coarser than a
particle collision event, typically having an interval of validity ranging
from one year to one data acquisition run222The minimum granularity is of
course arbitrary, but care should be taken to keep the conditions data volume
reasonable, further discussed in section 2.2..

###  2.1 Workflows and use cases

The original use cases for conditions data were in offline software, but with
the advent of software-based high level triggers this is no longer the only
use case. Broadly speaking, the major conditions data access use cases are:

  1. 1)

Online event data processing of raw detector event data

  2. 2)

Offline event data processing of

    1. (a)

raw detector event data

    2. (b)

fully reconstructed event data

    3. (c)

simulated event data

  3. 3)

Higher-level analysis of processed event data

To a large extent this paper deals with the first two use cases while analysis
is covered briefly in section 2.1.1. Online event data processing typically
has infrequent updates of conditions, with stability and predictability
preferred over optimal performance. The exceptions are those conditions that
are critical to the performance of trigger algorithms, e.g. the position of
the beam-spot in the case of ATLAS and CMS. Online event data processing runs
on dedicated resources with a dedicated server for conditions data access.

Offline event data processing in general starts with the reconstruction of raw
detector event data, while for simulated data it requires simulation,
digitisation and reconstruction of the generated events. In modern
experiments, there is usually subsequent data reduction of the output of fully
reconstructed event data to produce smaller samples that are better targeted
for physics analysis. Offline event data processing is typically repeated
several times following improvements in software and/or conditions. It
generally involves distributed event data processing on the grid, making it
the most challenging use case for conditions data access, requiring conditions
data caching to run at scale. HPC and similar off-grid resources are also
viable offline event data processing locations and have their own special
requirements, in particular a lack of external connectivity.

####  2.1.1 Conditions for analysis

Versions of conditions data for analysis have often been managed in ad hoc
ways, if at all, with hard-coding of conditions data into analysis software
releases being considered normal333The situation is improving for running
experiments, albeit with different solutions than those used for offline event
data processing.. Belle II Ritter:2018jxh (5, 6) has adopted a different
approach where conditions data for analysis will be treated in the same way as
the rest of their conditions data. They will use a conditions data access
management system similar to the one presented here and, given the growing
interest in the analysis community, feedback on the suitability of these
solutions is expected in the near future.

###  2.2 Data volumes, read and write rates

Typically, write-rates for conditions data must support of the order of 1 Hz
to ensure good support for several independent systems writing conditions data
every minute, with the majority of conditions data being updated much less
frequently than this. On the other hand, read-rates up to several kHz must be
supported for distributed computing workflows, where thousands of jobs needing
the same conditions data may start at the same time. Conditions data are
typically written once and read frequently.

Conditions data tend to scale in a controlled way during the lifetime of an
experiment, typically producing data volumes of gigabytes to terabytes. It is
worth noting however that for ATLAS, where the same database instances were
used for conditions data as well as the DCS and trigger data, the DCS and
trigger data dominated the offline and online database instances,
respectively. Raw DCS data, which typically have very high granularity, are
not generally needed for event data processing. Smoothed DCS data which have
been treated to provide a granularity appropriate for event data processing
are used instead. High granularity raw DCS data are vital for understanding
problems with detectors, but mixing these two very different use cases in the
same database schema led to several difficulties. Firstly it was difficult to
optimise the database tables, and secondly conditions data access for offline
event data processing required overly complicated database clients and
infrastructure to cope with offline conditions data access at scale. It is
therefore strongly recommended by this working group to factorise conditions
data access from other use cases, even where these are related.

##  3 Conditions Database Archetype

The archetypal solution to a conditions database management system (CDMS) is
shown below. The conditions data payloads are stored in a master database and
are accessed using a client-server design through a REST interface. All of the
experiments consulted gave feedback that achieving a high degree of separation
between client and server was very desirable. Due to the read-rate
requirements, caching is extremely important and good experience was seen when
using web-proxy caches, e.g. the Squid cache shown here. Some key design
principles are detailed in the rest of this section.

Figure 1: The Conditions Database Management System archetype.

###  3.1 Payload technology

Experiments will inevitably choose their favourite payload technology. This
working group recommends placing most emphasis on homogeneity and long-term
maintenance when making this choice. Inhomogeneity and home-grown solutions
all place additional burdens on projects that typically lack the resources to
support this after the initial build and commissioning phase of an experiment.
CMS Guida:2015gvw (2) has very good experience of removing choice and only
supporting boost-serialised C++ objects, with all classes belonging to one
package in their software framework. Such a strategy lends itself more readily
to long-term maintenance and minimises hurdles to data preservation. It is
noted that there are payload formats used routinely in industry which would
lend themselves more to higher-level functionality without the need of the
software framework but, while this is attractive, the choice of format tends
to be driven by software framework developers.

One very important consideration for payload technologies is the issue of
schema evolution. This is further exacerbated in the (frequent) case that the
experiment uses the same framework for the online and offline event data
processing use cases. Conditions payloads are determined offline and therefore
their versioning follows the offline release which will usually evolve more
rapidly than the online release. This implies a forward compatibility
constraint such that the online software can correctly process conditions
payloads generated by the offline software which may use a later version. In
the case of CMS, as boost does not support forward-compatibility in its schema
evolution, conditions payloads for a particular set of event data must be
written by a version of the software framework that is at most as old as the
version used by the online software when recording those event data.

###  3.2 Database back-end

One of the key features of the design is that it is agnostic to particular
choices of database back-end. This was also one of the key features of COOL
Valassi:2008zka (7, 8), but due to the lack of caching in COOL, the queries
themselves had to become more complicated. Thus on ATLAS, which uses an Oracle
back-end, a significant amount of Oracle DBA effort was required to tune the
queries and make them performant. It is therefore important to realise that
real flexibility with respect to choices in database back-ends only comes when
the system as a whole is simplified.

###  3.3 Client-side requirements

The client layer should be as simple as possible and should be as agnostic of
the rest of the architecture as it is possible to be in order to improve
maintainability. The database insertion tools in particular benefit from
adopting a simple, e.g. REST, interface. The client layer needs to take care
of payload deserialisation, as the remaining architectural components will
deal with serialised objects. Experience also shows that clients should be
able to manage multiple proxies and servers to provide robustness against
server failures.

###  3.4 Caching

CMS and ATLAS absolutely require an intermediate layer, Frontier
Dykstra:2011zz (9, 10), between the client and server to provide caching
capabilities. Considering offline event data processing using distributed
computing resources, where thousands of jobs start at the same time and
require the same conditions data, this is a clear requirement and one that can
be well met using web-proxy technologies. An alternative solution would be to
use a distributed file-system with good caching capabilities, and the LHCb
ClemencicCHEP2018 (11) and ALICE ALICE-CDB (12) experiments have gained
experience using CVMFS 1742-6596-331-4-042003 (13). The simplicity of this
solution makes it very attractive and it has thus been adopted as a strategy
by the NA62 LaycockCHEP2018 (14) experiment. This in turn suggests that, as a
design requirement, it should be possible to represent a conditions database
on a file-system. The primary challenge here is to make the file-system
mapping use the CVMFS caching layers efficiently. Several experiments also
have experience using SQLite replicas, including for HPC usage. These are
attractive for workflows where the exact subset of conditions is known in
advance, e.g. some simulated data workflows, but in general a performant
caching layer is preferable thanks to its flexibility.

###  3.5 Data Model

The data model for conditions data access management is an area where the
experiments have almost universally agreed on a design, shown in figure 2.

Figure 2: The data model for conditions data access management (see text).

A global tag is the top-level configuration of all conditions data. For a
given payload type444A payload type can be e.g. the muon detector alignment.
and a given interval of validity555A period of time for which a conditions
data payload is considered valid., a global tag will resolve to one, and only
one, conditions data payload. The Global Tag resolves to a particular payload
type Tag via the Global Tag Map table. A payload type Tag consists of many
non-overlapping intervals of validity or entries in the IOV table. Finally,
each entry in the IOV table maps to a payload via its unique hash key in the
Payload table. A relational database is a good choice for this design.

The data model design has several key features. Firstly, conditions data
payloads are uniquely identified by a hash which is the sole reference to any
given conditions data payload. The payload data has been separated from the
data access management metadata and can both in principle and in practice be
placed in a separate storage system. This could also be important for data
preservation, as the entire metadata component will occupy a trivial data
volume and could exist in, e.g. an SQLite file, while the payload storage
could be handled separately. Secondly, IOVs are resolved independently of
payloads and are also cacheable. Efficient caching is a key design requirement
for any conditions database access system that must support high rate data
access.

###  3.6 A git-based approach

For workflows which are completely offline and asynchronous with respect to
data-taking, a subset of the problems discussed previously, LHCb has adopted a
different approach. Using git as the versioning system, conditions data
payloads are placed in a directory structure, one directory for each payload
type. A file is used to map timestamps to payload files, and a simple format
is used to allow a level of indirection to improve performance. The file
format allows a timestamp to point either directly to a payload file or to a
directory, thus allowing partitioning of the lookup. Versioning is taken care
of by creating a git tag, which is equivalent to a global tag, while the
conditions data payloads are stored on CVMFS.

##  4 Future and roadmap

The conditions data access management model described here will be tested by
Belle II in 2019 and by CMS and ATLAS in LHC Run 3. Assuming they work as
expected, this will meet the conditions data access performance demands of HEP
for the coming decade. Even at the HL-LHC HL-LHC (15), conditions data volumes
are not expected to exceed a Terabyte per year and the rate of requests,
determined by the computing resources of the experiments, are expected to peak
at tens of kHz. Based on experience with similar conditions data access
management systems, particularly the Run 2 CMS implementation, the outlook is
positive.

Nevertheless there are concerns. The most important issue for experiments will
be maintenance and operation in the face of evolving hardware and
infrastructure and a dwindling number of conditions data management experts.
These concerns provide even more motivation to consolidate the conditions data
access model to a simple and modular design, such as that presented here.
Going further, the logical next step of the collaborative, cross-experiment
work started by members of this working group would ultimately result in an
experiment-agnostic HEP conditions data service. The challenge for the
community is realising that ambition.

##  5 Conclusion

Conditions data management is an important component of HEP software and one
that has relatively few experts. While conditions data volumes are easily
accommodated by several database technologies, conditions data access use
cases can be demanding. In particular, it is critical that solutions support
access rates at the level of tens of kHz.

Several experiments have converged on a common design for conditions data
access management. A key feature is a high degree of separation between a
relatively simple client and the server, a design that is well-suited to REST
interfaces. Loosely-coupled industry standard components provide much of the
robust software stack. An intermediate layer with caching capability is
required to support kHz rates of read requests, and web-proxies have performed
very well. Combined with a relational data model that isolates payloads from
metadata, the design can support a wide variety of HEP workflows both now and
in the coming decade. The main challenges the community faces will be finding
resources for the maintenance and operation of this solution in the face of
evolving hardware and infrastructure. Having achieved a common design, the
solution to this problem may be the logical next step, a common HEP conditions
data service.

## References

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