Probabilistic integrals: mathematical aspects

Post-publication activity

Integrals over spaces of paths or more generally of fields have been introduced as heuristic tools in several areas of physics and mathematics. Mathematically, they should be intended as extensions of finite dimensional integrals suitable to cover the applications the heuristic path integrals were originally thought for.

Eponymous naming conventions are: functional integrals, infinite dimensional integrals, field integrals. Feynman path integrals (or functionals), and Wiener path integrals (or integrals with respect to Wiener-type measures) are special cases. In probability the concept of flat integral(or integrals with respect to a flat measure) also occurs. A particular realization of Gaussian path integrals is given by "white noise functionals".

In the present article, the mathematical theory of path integrals of probabilistic type, as Wiener path integrals, will be presented, while the theory and the applications of path integrals of Feynman's type is presented in another article of Scholarpedia: Path integral: mathematical aspects.

In fact, a general setting is presented, having in mind mainly stochastic processes taking values in finite dimensional spaces and their applications. The case of the infinite dimensional processes related to random fields and stochastic partial differential equations will be discussed under another heading.

1 Stochastic processes and probability measures on spaces of paths
2 The historical development of probabilistic path integrals
3 Measures on infinite dimensional spaces
4 Kolmogorov theorem
5 Construction of Markov stochastic processes via Markov kernels
6 Stochastic processes with stationary and independent increments: Lévy processes
7 The Wiener measure and the Wiener process
8 Applications of the Wiener measure
- 8.1 The Feynman-Kac formula
- 8.2 The solution of the Dirichlet problem for the Laplace equation
9 Other stochastic processes
10 Fourier transform of measures. Bochner-Minlos theorem
11 Gaussian measures and Gaussian integrals
12 Asymptotics of path integrals
13 Applications
14 References
15 Further reading

Stochastic processes and probability measures on spaces of paths

We can look at the theory of path integrals, i.e. of integrals on function spaces, of probabilistic type from two different, but equivalent, points of view.

On the one hand, for the readers who are more familiar with infinite dimensional analysis, one should introduce a function space $\Gamma$, endowed with a $\sigma$-algebra ${\cal A} (\Gamma)$ and, given a $\sigma$-additive positive measure $\mu$ with $\mu (\Gamma)=1$, define the integral of a bounded measurable function $f:\Gamma\to{\mathbb C} $ as the (absolutely convergent) Lebesgue integral $\int_\Gamma f(\gamma) d\mu(\gamma)$.

On the other hand, for the readers who are more familiar with a probabilistic language, one could introduce a stochastic process $X=(X_t)_{t\in I}$, i.e., a family of random variables (i.e. measurable maps) $X_t:\Omega\to {\mathbb R}$ indexed by the element $t$ of an interval $I\subset {\mathbb R}$, defined on a common probability space $(\Omega, {\cal A},{\mathbb P})$, where $\Omega $ is an nonempty set, ${\cal A}$ is a $\sigma-$algebra of subsets of $\Omega$ and ${\mathbb P}$ is a probability measure on ${\cal A}$. If, more generally, the random variables $X_t$ have range in a measurable space $(E, {\cal A}')$ instead of ${\mathbb R}$, then the stochastic process is said to have state space $E$. As examples we can take $E={\mathbb R}^d$ or $E$ a finite dimensional manifold or $E$ an infinite dimensional space with an appropriately chosen $\sigma$-field ${\cal A}'$ (usually the Borel $\sigma$-field).

As anticipated, the two approaches are deeply related. Given a stochastic process $\{X_t\}_{t\in I}$ on $(\Omega, {\cal A},P)$ with (a "suitably regular") state space $E$, it is possible to construct a probability measure $\mu$ on the set $\Gamma=E^I$ of all functions $\gamma:I\to E$, called the ‘‘sample paths’’ of the process, equipped with the $\sigma$-algebra ${\cal A}(\Gamma)$ generated by the sets of the form \begin{equation}\tag{1}B(t_1,...,t_n;B_1,...,B_n):=\{\gamma \in \Gamma : \gamma (t_1)\in B_1, ...,\gamma (t_n)\in B_n\}, \quad \hbox{for some }\, t_1, t_2,...,t_n\in I, \,B_1,B_2,...,B_n\in {\cal A}'. \end{equation} The measure $\mu$ is defined on the sets of the form (1) as $$\mu (B(t_1,...,t_n;B_1,...,B_n)):=P(\{\omega\in \Omega : X_{t_{1}}(\omega )\in B_1,...,X_{t_{n}}(\omega )\in B_n).$$

Conversely, any probability measure $\mu $ on $(\Gamma\equiv E^I, {\cal A}(\Gamma))$ gives rise to a stochastic process $X_t$, $t\in I$, on the probability space $(\Omega, {\cal A}, P)=(\Gamma, {\cal A}(\Gamma), \mu)$, where $\gamma\in \Gamma$, $ t \in I$ and $X_t(\gamma):=\gamma (t)$.

A natural generalization of the concept of stochastic process is the random field, i.e., a family of random variables $\{X_y\}_{y\in Y}$ indexed by the elements of a set $Y$, which replaces the time interval $I\subset\R$.

The historical development of probabilistic path integrals

The advent, at the beginning of last century, of Borel and Lebesgue's development of measure and integration theory for functions on finite dimensional spaces opened the way, on the one hand, to abstract integration theory and, on the other hand, to measure and integration theory on infinite dimensional spaces, which just started to be systematically studied at that time (the impulses coming here from other topics, like calculus of variations). N. Wiener in the early 20s made the crucial discovery of a natural measure on the space of continuous paths, connecting it with the description of physical Brownian motion studied (via heuristic limits from symmetric random walks) by Einstein and Smoluchowski about 15 years before. Precursors of such limits of random walks are to be found both in the statistical astronomical work by Thiele (in the 1870's) and in Bachelier's studies in finance (1900). Kolmogorov (1933) in the development of the axiomatic foundations of probability theory and the theory of stochastic processes proved his fundamental theorem on the construction of measures on infinite dimensional spaces. This construction is in terms of limits of projective systems of probabilities on product spaces. This, complemented by a support theorem of Kolmogorov and Chentsov, yields the Wiener measure as a particular case, as well as the case of product measures on general spaces. Kolmogorov's theorem has also become the basis for vast extensions, let us mention in particular the one provided by Minlos theorem, which covers the cases of distributional spaces (which are natural in many applications, e.g., in physics and engineering). L. Gross' theory of abstract Wiener spaces provides another important extension, to Hilbert and Banach spaces, close to the spirit of Wiener's space construction and thus also suitable for the development of infinite dimensional stochastic analysis. Among the most studied classes of measures on infinite dimensional spaces, let us mention Gaussian measures, connected via the Wiener-Itô-Segal isomorphism with Fock spaces (of great relevance in quantum field theory). Also of importance are other measures associated with stochastic processes and corresponding to projective systems constructed from Markov kernels (see the section below and the ones on diffusion processes and Lévy processes). These were developed starting from around 1940 (in the work by Bernstein, Kakutani, Itô, Skorohod, Doob).

Measures on infinite dimensional spaces

From a mathematical point of view, the implementation of an integration theory of Lebesgue type on a space of paths, i.e., on an infinite dimensional space, in terms of a $\sigma-$ additive measure is a non trivial task. Contrary to the case of a finite-dimensional Euclidean space, it is impossible to construct a nontrivial Lebesgue-type measure on an infinite dimensional Hilbert space ${\mathcal H}$, i.e., a regular Borel measure which is invariant under rotations or translations. Indeed, the assumption of the existence of a $\sigma$-additive measure $\mu$ with Euclidean invariance properties, which assigns a positive finite measure to all bounded open sets, leads to a contraddiction. In fact, by taking an orthonormal system $\{e_i\}_{i\in\N}$ and by considering the open balls $B_i=\{ x\in {\mathcal H}, \Vert x-e_i\Vert<1/2\}$, one has that they are pairwise disjoint and their union is contained, e.g., in the open ball $B(0,2)=\{ x\in {\mathcal H}, \Vert x\Vert<2\}.$ By the Euclidean invariance of the Lebesgue-type measure $\mu$ one can deduce that $\mu (B_i)=a$, $0<a<+\infty$, for all $i\in{\mathbb N}$. By the $\sigma$-additivity one has \[ \mu(B(0,2))\geq\mu(\cup_iB_i)=\sum _i\mu (B_i)=+\infty, \] but, on the other hand, $\mu(B(0,1))$ should be finite as $B(0,2)$ is bounded.

It is interesting to point out that this argument forbids also the existence of a standard Gaussian measure on any infinite-dimensional Hilbert space, because of its rotational invariance. The study of this problem led to the development of the theory of abstract Wiener spaces (see Gross (1967), Kuo (1975)).

As a matter of fact the argument above can be generalized to the case where the Hilbert space $\mathcal H$ is replaced by an infinite-dimensional topological vector space $X$. Let us denote by $X^*$ the algebraical dual of $X$ and by $R$ a linear subspace of $X^*$. Let ${\cal B}_R$ be the smallest $\sigma$-algebra on $X$ in which any $\xi\in R$ is measurable. The couple $(X,{\cal B}_R)$ turns out to be a measurable additive group, in the sense that the map $T:X\times X\to X$ defined as $T(x,y):=x-y$ is measurable from $(X\times X, {\cal B}_R\times {\cal B}_R) $ to $(X,{\cal B}_R)$. By applying the theory of invariant (Haar) measure on topological groups (see Yamasaki (1985)), we obtain that there cannot exist a nontrivial invariant measure (under translations) on $(X,{\cal B}_R)$.

The first non trivial example of a Gaussian probability measure on the Banach space $C[0,t]$ of continuous functions on the interval [0,t] ("paths") was provided by Norber Wiener in the years 1920-1923. Wiener constructed a stochastic process $X\equiv (X _t)_{t\geq 0}$, which is nowadays called "Wiener process" or "mathematical Brownian motion" as it presents a mathematical model for the physical Brownian motion studied by A. Einstein e M. Smoluchowski between 1905 and 1909. This also gives a rigorous realization of the process introduced by L. Bachelier (1900) to describe prices evolution in finance.

Kolmogorov theorem

The main tool for the construction of probability measures on infinite dimensional spaces starting from their finite dimensional approximations is the celebrated Kolmogorov existence theorem. It was originally stated and proved by Kolmogorov in the case of measures on the space $\Omega={\mathbb R}^{I}$ of real valued functions (called paths) defined on an interval $I$ of the real line and later generalized by S. Bochner to projective limit spaces. In the following we shall present a version which is sufficiently general for our purposes.

Let $(E, {\cal A}')$ be a measure space. We shall assume $E$ to be a Polish space, i.e., a topological space with a countable basis and topology derived from a complete metric, while $ {\cal A}'$ will be the Borel $\sigma-$algebra ${\cal B}(E)$ on $E$. In the most common applications, $E$ will be $\R^d$ or a $d-$ dimensional Riemannian manifold $M$, or a separable real Hilbert space.

Let ${\cal F } (I)$ be the set of all finite subsets of the interval $I\subset {\mathbb R}$, endowed with the partial order relation $\leq$ defined by $J\leq K $ if $J\subseteq K$. The set ${\cal F } (I)$ has the structure of a directed set, in the sense that for any $J,K\in{\cal F } (I)$ there is an $H\in {\cal F } (I)$ such that $J\leq H$ and $K\leq H$. Given a $J\in {\cal F } (I)$, with $J=\{t_1,t_2,...,t_n\}$, $0\leq t_1<t_2...<t_n$, $t_i\in I$ $i=1,..., n$, let us consider the set $E^J$ of all maps from $J$ to $E$. An element of $E^J$ is an $n-$tuple $(\gamma(t_1),\gamma(t_2),...,\gamma(t_n) )\in E^n$, $n$ being the cardinality of $J$ (clearly, $E^J$ is naturally isomorphic to $E^n$). Let us consider on $E^J$ the product topology and the Borel $\sigma-$ algebra ${\cal B}(E^J)$. Furthermore, let us consider the projection $\Pi_J:E^I\to E^J$ which assigns to each path $\gamma\in E^I$ its restriction to the set $J$, namely, if $J=\{t_1,t_2,...,t_n\}\subset I$, then $$\Pi_J:E^I\ni \gamma\mapsto (\gamma(t_1),\gamma(t_2),...,\gamma(t_n) )\in E^{J}.$$ Let us focus on the cylinder sets, i.e., the subsets of $\Omega=E^I$ of the form $\Pi_J^{-1}(B_J)$ for some $J\in {\cal F } (I)$ and some Borel set $B_J\in {\cal B}(E^J)$. Let ${\cal C}$ denote the set of all cylinder sets, and let ${\cal A}$ be $\sigma$-algebra generated by ${\cal C}$.

Given a measure $\mu$ on $(\Omega, {\cal A})$, for any $J\in {\cal F } (I)$ one defines a measure $\mu_J$ on $(E^J, {\cal B}({E}^J))$ as $\mu_J:=\Pi_J(\mu)$, i.e., $$\mu_J(B_J):=\mu(\Pi_J^{-1}(B_J)), \qquad B_J\in {\cal B}({E}^J).$$ Given two elements $J,K\in {\cal F} (I)$, with $J\leq K$, let $\Pi_J^K:E^K\to E^J$ be the projection map, which is continuous and hence Borel measurable. By construction, the measures $\mu_J$ on $(E^J, {\cal B}(E^J))$ and $\mu_K$ on $(E^K, {\cal B}(E^K))$ are related by the equation $\mu_J=\Pi_J^K(\mu_K)$, that means \begin{equation}\tag{2}\mu_J(B_J):=\mu_K((\Pi_J^K)^{-1}(B_J)), \qquad B_J\in {\cal B}(E^J),\end{equation} as one can verify by means of the equation $\Pi_J=\Pi^K_J\circ\Pi_K$. A family of measures $\{\mu_J\}_{J\in {\cal F}(I)}$ satisfying the compatibility condition (2) is called a projective family of measures.

In the case of probability measures, the converse is also true. Indeed, according to the Kolmogorov existence theorem, given a family of probability measures $\{\mu_J\}_{J\in {\cal F}(I)}$ satisfying the compatibility condition (2), there exists a unique probability measure $\mu$ on $({E}^{I},{\cal A})$ such that for any $J\in {\cal F}(I) $ $\mu_J=\Pi_J(\mu)$. In other words, it is possible to construct a measure on the (infinite dimensional) space of paths $\Omega ={E}^{I}$ by means of its finite dimensional approximations. The measure $\mu$ described by the Kolmogorov theorem is called the projective limit of the projective family $\{\mu_J\}$.

A cylinder function is a function $f:{E}^I\to {\mathbb C}$ of the form $f(\gamma)=F (\gamma(t_1),\gamma(t_2),...,\gamma(t_n) )$, for some $t_1,...,t_n\in I$ and a Borel function $F:E^n\to {\mathbb C}$. Clearly, a cylinder function is measurable with respect to the $\sigma$-algebra ${\cal A}$ generated by the cylinder sets. Given a measure $\mu$ on $E^I$ constructed as the projective limit of a projective family $\{\mu_J\}$, the (path) integral of a cylinder function $f$ with respect to the measure $\mu$ can be computed as $$\int_{{E}^I}f(\gamma)d\mu (\gamma)=\int_{{E}^n}F(x_1,...,x_n)d\mu_J (x_1,...,x_n), \qquad J=\{t_1,...,t_n\}.$$

By adopting a probabilistic language and introducing the stochastic process $X=(x_t)_{t\in I}$, defined by $X_t\equiv \gamma(t)$, on $\Omega\equiv E^I$, the Kolmogorov theorem states that the process is completely determined by its finite dimensional distributions, namely by the quantities $$P(X_{t_1}\in B_1, ...,X_{t_n}\in B_n\})=\mu_J(B_1\times ...\times B_n), \qquad J=\{t_1,...,t_n\}, B_i\in {\cal B}(E), i=1,...,n.$$ The expected value of a measurable function $f:\Omega \to {\mathbb C}$ is defined to be the integral ${\mathbb E}[f(X)]=\int_{{E}^I}f(\gamma)d\mu (\gamma)$.

A different type of extension theorem which does not need any topological assumption on the state space $E$ has been proved by Ionescu-Tulcea (1949).

Construction of Markov stochastic processes via Markov kernels

Kolmogorov theorem provides a tool for the construction of probability measures on the space of paths $E^I$, once a projective family of measures $\{\mu_J\}_{J\in {\cal F}(I)}$ is given. We are in a position to show an important method for the construction of a projective family $\{\mu_J\}_{J\in {\cal F}(I)}$, which, incidentally, anticipates the connection between probability theory (in particular Markov processes) and parabolic equations associated to second order elliptic operators (namely, the theory of Markov semigroups).

Thinking of a probability measure on $E^I$ as a description of the possible trajectories of a point particle moving in the topological space $E$, an important object is the transition probability $P_{t_{j}, t_{j+1}}(x,B)$ that the particle starting at time $t_j$ a the point $x\in E$, reaches at time $t_{j+1} $ the set $B\in {\cal B}(E)$. If this probability depends only on $t_{j}, t_{j+1},x$ and $B$, but not on the trajectory up to time $t_j$, namely if the particle has no "memory", the Markov property is satisfied. Assuming that, given an intermediate time instant $t_k$ with $t_{j}<t_k< t_{j+1}$, the random motion during the interval $[t_j,t_k]$ is independent of the motion during $[t_k,t_{j+1}]$, the following relation, called Chapman-Kolmogorov equation, holds \[ \tag{3} P_{t_j, t_{j+1}}(x,B)=\int_E P_{t_j, t_{k}}(x,dx')P_{t_{k}, t_{j+1}}(x', B) , \qquad \forall x\in E, B\in {\cal B}(E), t_{j}<t_k< t_{j+1}\] A Markov kernel is a map $P:E\times {\cal B}(E)\to [0,1]$ satisfying the following conditions:

the map $x \in E\mapsto P(x,B)$ is measurable for each $B\in {\cal B}(E)$,
the map $B\in {\cal B}(E) \mapsto P(x,B)$ is a probability measure on $ {\cal B}(E)$ for each $x\in E$

A family $\{P_{t,s}\}$ of Markov kernels indexed by $(s,t)\in I^2$ such that $s\leq t$, satisfying the condition (3) is called Markov transition function.

Given a Markov transition function $\{P_{t,s}\}$, $s\leq t$, and a probability measure $\nu_0$ on $(E,{\cal B}(E))$ (which plays the role of an initial distribution), it is possible to build up a projective family of probability measures $\{\mu_J\}_{J\in {\cal F}(I)}$ as \begin{equation}\tag{4} \mu_J(B_J)=\int_{E\times B_J}P_{t_n,t_{n-1}}(x_{n-1},dx_n)\dots P_{t_2,t_1}(x_1,dx_2) P_{t_1,t_0}(x_0,dx_1)d\nu(x_0) \end{equation} where $J=\{t_1,t_2,...,t_n\}$, with $t_0<t_1<t_2<...<t_n<+\infty$, and $B_J\in {\cal B}({\mathbb R}^n)$. The compatibility condition (2) is a direct consequence of Eq. (3). Due to the Kolmogorov existence theorem, there exists a unique probability measure $\mu$ on the space of paths $E^I $ such that $$\mu(\{\gamma \in E^I : (\gamma(t_1),\gamma(t_2),...,\gamma(t_n))\in B_J)=\mu_J(B_J)$$ The path integral of a cylinder function $f:E^I\to {\mathbb C}$ of the form $f(\gamma)=F(\gamma(t_1),\gamma(t_2),...,\gamma(t_n))$ for some $F\in {\cal B}(E^n) $ is defined to be $$\int _{E^I}f(\gamma)d\mu (\gamma) =\int _{E^{n+1}}F(x_1, ..., x_n)P_{t_n,t_{n-1}}(x_{n-1},dx_n)\dots P_{t_2,t_1}(x_1,dx_2) P_{t_1,t_0}(x_0,dx_1)d\nu(x_0).$$

The associated stochastic process $X=(X_t)_{t\in I}$ on $(E^I, {\cal A},\mu)$ is called a Markov process. If the initial distribution $\nu_0$ is the Dirac point measure at $x\in E$, then the process is said to start at $x$. The expectation with respect to the associated probability measure is denoted by ${\mathbb E}^x$. The kernels $\{P_{t,s}\}$, $s\leq t$, are the transition probabilities of $X_t$, i.e., $$P_{t,s}(x,B)=P(X_t\in B \,|\, X_s=x)={\mathbb E}^x[{\bf 1}_{X_t^{-1}(B)}].$$

Stochastic processes with stationary and independent increments: Lévy processes

If $I=[0,+\infty)$ and for any $t,s\in I$, $s\leq t$, and $x\in E$ the transition probabilities $ P_{t,s}(x,\, \cdot \,)$ depend only on the difference $t-s$ then the stationarity property is satisfied and $P_{t, s}(x,\, \cdot \,)\equiv P_{t-s}(x,\, \cdot \,)$, where $P_t(x,\, \cdot \,)):=P_{t,0}(x, \, \cdot \,)$ by definition. From a rigorous mathematical point of view, the set of transition probabilities is represented by a semigroup of Markov kernels, i.e., a family of Markov kernels $\{P_t\}_{t\in \R}$ satisfying the semigroup property $P_{t+s}(x,B)=\int_E P_t(x,dx')P_s(x', B)$ for all $t,s\in \R_+$, $x\in E$, $B\in {\cal B}(E)$.

Given a semigroup of Markov kernels, the Kolmogorov theorem provides a unique probability measure $\mu $ on the space of paths $E^I $ such that for all $J=\{t_1,t_2,...,t_n\}$, with $0<t_1<t_2<...<t_n<+\infty$, and $B_J\in {\cal B}(E^n)$, \begin{equation}\tag{5}\mu(\{\gamma \in E^I : (\gamma(t_1),\gamma(t_2),...,\gamma(t_n))\in B_J)=\int_{ B_J}P_{t_n-t_{n-1}}(x_{n-1},dx_n)\dots P_{t_2-t_1}(x_1,dx_2) P_{t_1}(0,dx_1).\end{equation} The path integral of a cylinder function $f:E^I\to {\mathbb C}$ of the form $f(\gamma)=F(\gamma(t_1),\gamma(t_2),...,\gamma(t_n))$, for some $F\in {\cal B}({\mathbb R}^n) $ and $\{t_1,t_2,...,t_n\}\subset I$ with $0<t_1<t_2<...<t_n$, is defined to be \begin{equation}\tag{6}\int _{E^I}f(\gamma)d\mu (\gamma) =\int _{E^{n}}F(x_1, ..., x_n)P_{t_n-t_{n-1}}(x_{n-1},dx_n)\dots P_{t_2-t_1}(x_1,dx_2) P_{t_1}(0,dx_1).\end{equation}

The stochastic process $X=\{X_t\}_{t\in I}$ on $\Omega =E^I$ naturally associated to the measure $\mu$ by setting $X_t(\gamma):=\gamma (t)$, $\gamma \in \Omega$, $t\in I$, has, by construction, independent and stationary increments on the probability space $(\Omega, {\cal A},\mu)$. Indeed, for any $s,t\in I$, the random variables $X_t-X_s$ and $X_{t-s}$ have the same distributions. Moreover, by the explicit form of the finite dimensional distribution of the process (see Eq (5)), for any $t_1,...,t_n\in I$ with $t_1<t_2<...<t_n$, the $n-1$ random variables $X_{t_2}-X_{t_1}, X_{t_3}-X_{t_2}, ...,X_{t_n}-X_{t_{n-1}}$ are independent.

If, furthermore, the measures $P_t$ converge weakly as $t\downarrow 0$ to the Dirac $\delta $ measure, i.e., for all continuous bounded functions $f:E\to {\mathbb R}$ one has $\lim_{t\to 0} \int _E f(y)P_t(x,dy)=f(x)$, then it is possible to prove (see e.g. Applebaum (2009)) that the process $X_t$ has the following continuity property: \begin{equation}\tag{7}\lim_{t\to s}P(|X_t-X_s|>a)=0, \qquad \forall a>0, t,s \in I. \end{equation} Such a stochastic process is called a Lévy process.

An elegant characterization of Lévy processes on ${\mathbb R}^d$ is provided by the Lévy-Khinchine formula, which describes the general form of the characteristic function of the process, namely of the map $x\mapsto {\mathbb E}[e^{ix \cdot X_t}]$, $x\in {\mathbb R}^d$. Indeed, there exists a vector $b\in {\mathbb R}^d$, a positive-definite symmetric $d\times d$ matrix $A$ and a Borel measure $\nu_L$ on ${\mathbb R}^d\setminus 0$ satisfying $\int_{{\mathbb R}^d\setminus 0}{\mathrm min}(1, |y|^2)\nu_L(dy)<\infty$, such that for any $t\geq 0$ and $x\in {\mathbb R}^d$, $${\mathbb E}[e^{ix X_t}]=e^{t\eta (x)},$$ where $$\eta(x)=i b\cdot x -\frac{1}{2}xAx+\int_{ {\mathbb R}^d\setminus 0 }[e^{ix\cdot y }-1-ix\cdot y{\bf 1}_{B(0,1)}(y)]\nu_L(dy).$$ Here $B(0,1)=\{y\in {\mathbb R}^d \colon |y|<1\}$.

The Wiener measure and the Wiener process

Let us focus on the semigroup of Markov kernels on ${\mathbb R}^d$ defined by $P_t(x,B)=\int_B(2\pi t)^{-d/2}e^{-\frac{|x-y|^2}{2t}}dy$, $t>0$ for $ B\in {\cal B}({\mathbb R}^d)$ and $P_0(x,dy)=\delta_x(dy)$. The resulting probability measure $\mu $ on the space of paths ${\bf q} :[0,T]\to {\mathbb R}^d$ is called the Wiener measure and the associated process is denoted by $W_t$ and called the Wiener process (or mathematical Brownian motion), as it provides a mathematical model for the physical Brownian motion studied by A. Einstein e M. Smoluchowski (see, e.g. Albeverio (1997)). The density $p_t$ of $P_t$, $t>0$, with respect to Lebesgue measure, i.e. $p_t(y)=(2\pi t)^{-d/2}e^{-\frac{|x-y|^2}{2t}}$, $y\in {\mathbb R}^d$, is called heat kernel (on ${\mathbb R}^d$).

The measure of a cylinder set of paths ${\bf q}:[0,T]\to {\mathbb R}^d$ (or equivalently the probability $P(W _{t_1}\in B_1,\dots ,W _{t_n}\in B_n)$ that the process is at time $t_i$ in the Borel subset $B _i\in {\cal B}\R^d$), $i=1, ...,n$, $0<t_1<...<t_n$, namely Eq. (5), becomes \begin{equation}\tag{8} \mu (\{ {\bf q}(t_1)\in B_1,..., {\bf q}(t_n)\in B_n\})= \int_{B_n}\cdots\int_{B_1} \left((2\pi)^n (t_{n}-t_{n-1})\ldots (t_{1}-t_0) \right)^{-d/2}e^{-\frac{1}{2}\sum_{j=0}^{n-1}\frac{\vert x_{i+1}-x_i\vert^2}{t_{i+1}-t_i}} dx_1\ldots dx_n, \end{equation} where $t_0\equiv 0$, $x_0\equiv x$. By introducing (using the notation of Path integral) "piecewise linear paths" ${\bf q}_c :[0,T]\to \R^d$, such that ${\bf q}_c (t_j)=x_j$ and ${\bf q}_c (\tau)$ for $\tau \in [t_j,t_{j+1}]$ coincides with the constant velocity path connecting $x_j$ with $x _{j+1}$: $${\bf q}_c (\tau)=\sum_{j=0}^{N-1}{\bf 1}_{[t_j, t_{j+1}]}(\tau)\Big( x_j+\frac{x_{j+1}-x_j}{t_{j+1}-t_j}(\tau-t_j)\Big),\qquad \tau\in [0,t],$$ the exponent on the right hand side of (8) is equal to the classical action functional $S_0({\bf q})\equiv\frac{1}{2}\int _0^T \vert\dot {\bf q} (\tau)\vert ^2d\tau $ evaluated along the path ${\bf q}_c$, i.e., $S_0({\bf q}_c)\equiv\frac{1}{2}\sum_{j=0}^{n-1} \Big| \frac{x_{j+1}- x _j}{t_{j+1}- t _j}\Big|^2 (t_{j+1}- t _j)$. By setting $Z\equiv ((2\pi)^n (t_n-t_{n-1})\ldots (t_1-t_0))^{\frac{d}{2}}$, and $d {\bf q}=dx_1...dx_n$, Eq. (8) can be heuristically written as $Z^{-1}\int e^{-S_0({\bf q})}d {\bf q}$, obtaining the intuitive formula (7) in Path integral. It has the interpretation of a "path integral" on the space of polygonal paths with respect to the measure $Z^{-1}e^{-S_0 ({\bf q})}d {\bf q}$. The symbol $d {\bf q}$ can be regarded as a "flat measure" on $\R ^{nd}$, $Z$ a normalization constant and the term $e^{-S_0 ({\bf q})}$ as a Gibbs factor, in the sense of statistical mechanics, as $S_0$ is an action functional). In fact, the Wiener measure defined originally on the whole path space $({\mathbb R}^d)^{[0,+\infty)}$ is supported on the space $C([0,+\infty))$ of continuous paths. Moreover, the Wiener measure is supported, for each $\theta <1/2$, on the space of $\theta$-Hölder continuous paths. This is because for any $p\geq 2$ there exists a $c_p>0$ such that $${\mathbb E}\left[|W_t-W_s|^p\right]\leq c_p|t-s|^{p/2}, \quad t,s\geq 0.$$ Within an abstract formulation, the Kolmogorov-Chensov theorem establishes that if a process $X=(X_t)_{t\in [0, +\infty)}$ defined on a probability space $(\Omega ,{\cal F},{\mathbb P})$ satisfies \begin{equation}\tag{9}{\mathbb E}[|X_t-X_s|^\alpha]\leq c|t-s|^{1+\beta}\end{equation} for some $\alpha>0$, $\beta>0$ and $c>0$, then there exists another process $\tilde X=(\tilde X_t)_{t\in [0, +\infty)}$ such that $X_t=\tilde X_t$ ${\mathbb P}$-almost surely for all $t\in [0,+\infty)$, and every trajectory of the process $\tilde X$ is $\theta-$Hölder continuous or every $\theta \in (0, \beta/ \alpha)$.

Furthermore, in the case of the Wiener process $W_t$, it is possible to prove that the sample paths are almost surely nowhere differentiable. For a detailed description of the Wiener process see, e.g., Revuz and Yor (1999).

Applications of the Wiener measure

In the following we present some suggestive probabilistic representations of solutions of partial differential equations. For an overview of the scope of functional integration tecniques, in particular when applied to quantum or statistical physics, see, e.g. Simon (2005), Kac (1980) and Path integral.

The Feynman-Kac formula

The density of the Markov kernel $p_t(x,y)=(2\pi t)^{-d/2}e^{-\frac{|x-y|^2}{2t}}$ , $t>0$, which leads to the construction of the Wiener measure is, in fact, also the fundamental solution of the heat equation, in the sense that the associated initial value problem \begin{align} \frac{\partial}{\partial t}u(t,x)&=\frac{1}{2}\Delta_x u(t,x), \tag{10}\\ u(0,x)&=u_0(x), \quad x\in {\mathbb R}^d, t\in [0,+\infty), \end{align} has, for $u_0:{\mathbb R}^d\to {\mathbb R}$ a Borel bounded function, a classical solution of the form $$u(t,x)=\int_{{\mathbb R}^d}u_0(y)p_t(x,y)dy=\int_{{\mathbb R}^d}u_0(y)\frac{e^{-\frac{|x-y|^2}{2t}}}{(2\pi t)^{d/2}}dy ={\mathbb E}^x[u_0(W_t)]$$ As pointed out in 1949 by M. Kac (who was actually inspired by a lecture by R. Feynman at Cornell University), a probabilistic representation can also be proved for the solution of the perturbed initial value problem \begin{align} \frac{\partial}{\partial t}u(t,x)&=\frac{1}{2}\Delta_x u(t,x)-V(x)u(t,x), \tag{11}\\ u(0,x)&=u_0(x), \quad x\in {\mathbb R}^d, t\in [0,+\infty), \end{align} where $V:{\mathbb R}^d\to {\mathbb R}$ is continuous and bounded from above (these conditions can be relaxed, see, e.g., Simon (2005) for further details). The following Wiener integral representation for the solution of the heat equation with potential (11) $$u(t,x)={\mathbb E}^x[u_0(W_t)e^{-\int_0^t V(W_\tau)d\tau}]$$ is known as Feynman-Kac formula.

The solution of the Dirichlet problem for the Laplace equation

Let $D\subset {\mathbb R}^d$ be an open connected (nonempty) set, $x\in D$ and let $\tau_D:\Omega \to {\mathbb R}^+$ be the first hitting time of the complement $D^c$ of $D$ by the Wiener process starting at $x$, that is $$\tau_D=\inf \{t>0 |W(t)\in D^c\},$$ where $\inf \emptyset =+\infty$. The random variable $\tau _D$ is also called the first exit time from $\bar D$. We shall assume that any point $x$ belonging to the boundary $\partial D$ of $D$ is regular, in the sense that $P^x(\tau_D=0)=1$ for $x\in \partial D$, where $P^x$ denotes the distribution of the Wiener process $W$ starting at $x$. It turns out (see, e.g., Dynkin and Yushkevich (1969)) that in the case $d=2$ a sufficient condition for the regularity of a point $x\in \partial D$ is to be the vertex of a triangle $T\subset D^c$ (in particular, if $D$ is delimited by smooth curves, every point of $\partial D$ is regular), while in the case $d=3$ a point $x\in \partial D$ which is the vertex of some tetrahedron $T\subset D^c$ is regular.

Let us consider the boundary value problem associated to the Laplace equation \begin{align} \Delta u(x)=0 , \qquad & x\in D,\tag{12}\\ u(x)=f(x), \qquad & x\in \partial D, \end{align} where $f:\partial D\to {\mathbb R}$ is a bounded continuous function. Under the stated assumptions (see Chung (1982), Dynkin and Yushkevich (1969)) the function $u:\bar D\to \R$ defined by $$u(x):={\mathbb E}^x[f(W(\tau_D))]$$ is a classical solution of (12), i.e., it is of $C^2$-class in $D$, continuous on $\bar D$ satisfying $\Delta u=0$ on $D$ and coinciding with $f$ on $\partial D$.

Moreover, if $D\subset {\mathbb R}^d$ is an open connected bounded set and every point on $\partial D$ is regular, then the function $m_D:\bar D\to \R$ defined by the mean exit time $m_D(x):={\mathbb E}^x[\tau_D]$, is the unique classical solution of the Poisson equation $\Delta m=-2$ which is continuous on $\bar D$ and such that $m_=0$ on $\partial D$.

Other stochastic processes

Brownian motion on manifolds

Let $(M,g)$ be a connected closed Riemannian manifold and let $\Delta _{LB}$ be the Laplace-Beltrami operator on $M$. Let us consider the heat equation on $M$, namely \begin{align} \frac{\partial}{\partial t}u(t,x)&=\frac{1}{2}\Delta_{LB}u(t,x),\tag{13}\\ u(0,x)&=u_0(x),\quad x\in M, \, t\in [0,+\infty) \end{align} and let $p_t(x,y)$ be its fundamental solution, also called heat kernel, such that the solution of the initial value problem (13) is given by $u(t,x)=\int _Mp(x,y)u_0(y)dy$, where $dy$ is the Riemannian volume measure on $M$. Let us focus on the measure $\mu $ on $M^{[0,+\infty)}$ associated via Eq(5) to the semigroup of Markov kernels defined by $P_t(x,B):=\int_B p_t(x,y)dy$, $P_0=\delta_x$, $x\in M$. The measure $\mu$ is called Wiener measure on $M$ and the associated stochastic process $W=(W_t)$ is the Brownian motion on $M$ (starting at $x\in M$). In fact, as in the case where $M={\mathbb R}^d$, the application of the Kolmogorov-Chentsov theorem allows to restrict $\mu$ to the space $C([0,+\infty),M)$ of continuous paths on $M$. More precisely, for any $\theta \in (0,1/2)$ there exists a modification $\tilde =(\tilde W_t)_{t\geq 0}$ of $W=(W_t)_{t\geq 0}$ with $\theta$-Hölder continuous paths. By the very construction of $\mu$, the Feynman-Kac formula follows directly, namely for $V:M\to {\mathbb R}$, continuous and bounded, the solution of the heat equation on $(M,g)$ with potential $V$ \begin{align} \frac{\partial}{\partial t}u(t,x)&=\frac{1}{2}\Delta_{LB}u(t,x)-V(x)u(t,x),\tag{14}\\ u(0,x)&=u_0(x), \end{align} is given by $$u(t,x)={\mathbb E}^x\left[u_0(W(t))e^{-\int_0^tV(W_s)ds}\right]$$

The Ornstein-Uhlenbeck process

For a parameter $\alpha >0$ and a point $x\in {\mathbb R}$, let us consider the semigroup of Markov kernels on $({\mathbb R}, {\cal B}({\mathbb R}) )$ defined by:

$P_0(x, \mathrm{d}y) = \delta_x(\mathrm{d}y)$ for $t=0$ and $y\in {\mathbb R}$,
for $t>0$, the kernel $P_t(y,B)$ is defined by $P_t(y,B)=\int_B\left(2\pi(1-e^{-2\alpha t})\right)^{-1/2}\exp\left(\frac{(z-e^{-\alpha t}y)^2}{2(1-e^{-2\alpha t})}\right)dz$, where $B\in {\cal B}({\mathbb R})$ is a Borel set in $\mathbb R$.

The process $X^x=(X_t^x)_{t\geq 0}$ on $E=\mathbb{R}$ obtained from the Kolmogorov construction with $\nu=\delta_x$ is called Ornstein-Uhlenbeck process. It is Gaussian with mean $e^{-\alpha t}x$ and covariance $c_{\alpha}(s,t):=E(X_t^x X_s^x) = \frac{\exp(-\alpha \left| s-t \right|)}{2\alpha}$, $s,t \geq 0$. Note that $c_{\alpha}(t,s)=\tilde c_\alpha (|t-s|)$, where $\tilde c_\alpha (t)=\frac{\exp(-\alpha |t|)}{2\alpha}$ and the map $\tilde c_\alpha :{\mathbb R}\to {\mathbb R}$ is the fundamental solution (in the distribution sense) of $\left(-\frac{\mathrm{d}^2}{\mathrm{d}\tau^2} + \alpha\right)\tilde c_\alpha =\delta_0 $.

The standard Gaussian measure $N(0,1)$ is an invariant probability measure for the process $X=(X_t)_{t\geq 0}$. In particular, if the process is started from a $\nu_0$-distributed initial position instead of $x$, with $\nu_0= N(0,1)$, then the process is stationary for all times $t \in \mathbb{R}^+$. Furthermore it is Gaussian, with mean zero and covariance kernel $c_{\alpha}(s,t)$, $s,t\in\R$.

$X_t^x$ satisfies the stochastic differential equation (Langevin equation for the Ornstein-Uhlenbeck velocity process) $\mathrm{d}X_t^x = -\alpha X_t^x + \sqrt{2\alpha}\mathrm{d}B_t$, with $X_0^x = x$ and $B_t$ a standard Brownian motion on $\mathbb{R}$.

$\alpha-$stable processes

Let $\alpha \in (0,2)$. It can be proved that for any $c>0$, $t\geq 0$ a unique probability measure $\nu_t$ exists on $({\mathbb R},{\cal B}({\mathbb R})$ whose characteristic function $\hat \nu _t:{\mathbb R}\to{\mathbb C}$, defined by $\hat \nu _t(x)=\int e^{ixy} \nu _t(dy)$, satisfies $$\hat \nu _t(x)=e^{-ct|x|^\alpha}, \qquad x\in \R.$$ Moreover, for any $t,s>0$, the composition property $\nu_{t+s}=\nu_t *\nu_s$ holds (where $*$ denotes the convolution of measures). This implies that the family of kernels $\{P_t\}_{t\geq 0}$ defined by $P_t(x,B):=\nu_t(B-x)$, $ x \in {\mathbb R}$, $B\in {\cal B}({\mathbb R})$, is a Markov semigroup of kernels. The associated process $X=(X_t)_{t\geq 0}$ is Lévy and it is said to be $\alpha$-stable.

If $\alpha =2$ then $(X_t)_{t\geq 0}$ is the Wiener process. If $\alpha =c=1$ then $X_t$ is called the Cauchy process, in this case $\nu_t(y)=\frac{1}{\pi }\frac{t}{t^2+x^2}$, $t>0$, $x\in {\mathbb R}$.

Fourier transform of measures. Bochner-Minlos theorem

An important tool for the construction of probability measures on vector spaces or, more generally, on locally compact abelian groups, is harmonic analysis.

Let $X$ be a real vector space and $X^*$ its (algebraic) dual. Let ${\cal B} (X^*)$ be the smallest $\sigma$-algebra on $X^*$ in which the map $\xi\mapsto \xi (x)=\langle \xi , x \rangle$ is measurable for any $x\in X$. Given a positive measure $\mu$ on $(X^*, {\cal B} (X^*)$, let us define its characteristic function as the map $\hat \mu:X\to {\mathbb C}$ $$\hat \mu(x)=\int_{X^a}e^{i\xi (x)}d\mu (\xi), \qquad x\in X.$$ It is rather simple to prove that it is a positive definite function, namely it satisfies for any $n\in {\mathbb N}$, $x_1,...x_n\in X$ and $\alpha_1,...,\alpha_n\in {\mathbb C}$: $$\sum_{j,k=1}^n\alpha _j\bar \alpha_k \hat \mu (x_j-x_k)\geq 0.$$ If $X={\mathbb R}^d$ (which can be identified with its dual), it is simple to verify that $\hat \mu $ is continuous with respect to the Euclidean topology. The classical Bochner theorem states that these two properties identify univocally the characteristic functions, indeed any continuous and positive definite function on ${\mathbb R}^d$ is the characteristic function of a measure $\mu $ on ${\mathbb R}^d$.

In the case of an infinite dimensional vector space $X$, an application of Kolmogorov theorem allows us to prove that any positive-definite map $f:X\to {\mathbb C}$ which is continuous on any finite dimensional subspace $Y\subset X$ (with respect to the Euclidean topology on $Y$) is the characteristic of a measure $\mu $ on $(X^*, {\cal B} (X^*)$.

If $X$ is endowed with a topology $\tau$ making it a topological vector space, in general it is not true that a positive-definite and continuous map $f:X\to {\mathbb C}$ is the characteristic function of a measure on the topological dual $X'$ of $X$. In fact, the validity of this result depends on particular conditions either on the topology $\tau$ or on the function $f$.

When $X$ is a real separable Hilbert space $(H, \langle \, ,\, \rangle)$ (which can be identified via Riesz theorem with its topological dual), according to a result of Minlos and Sazonov, a positive-definite function $f:H\to {\mathbb R}$ is the Fourier transform of a Borel measure on $H$ iff there exists a Hilbert-Schmidt operator $T:H\to H$ on $H$ such that $f$ is continuous with respect to the norm $\| \, \cdot \,\|$ defined by $\|x\|:=\langle Tx,Tx \rangle$.

Let us consider now the case of a nuclear space $X$, i.e. a locally convex topological vector space whose topology is defined by a family $\{ \| \, \|_\alpha\}$ of Hilbert seminorms, i.e., induced by some inner product $\langle \, , \, \rangle _\alpha$ such that $ \| x \|_\alpha^2 =\langle x,x \rangle _\alpha$, and such that for any $\alpha$ there exists an $\alpha'$ with $\| \, \|_\alpha$ Hilbert-Schmidt with respect to $\| \, \|_{\alpha'}$. The last condition means that there exists an $M_{\alpha,\alpha'}>0$ such that $\sum _n\|x_n\|_\alpha ^2\leq M^2_{\alpha,\alpha'} $ for any orthonormal basis $\{x_n\}$ in $\langle \,\cdot\, , \,\cdot \, \rangle _{\alpha'}$. The generalization of Bochner theorem to this case asserts that any continuous positive definite function on a nuclear space $X$ is the Fourier transform of a measure on $X'$. An important example of nuclear space is the Schwartz space $S({\mathbb R}^d)$ of smooth functions on $\mathbb{R}^d$ for which the derivatives of all orders are rapidly decreasing, whose dual is the space $S'({\mathbb R}^d)$ of Schwartz distributions.

Gaussian measures and Gaussian integrals

Bochner theorem allows the characterization of measures $\mu$ on ${\mathbb R}^d$ or, more generally, on topological vector spaces $X$ via their Fourier transform $\hat \mu$. From this point of view, a Borel measure $\mu_G$ on ${\mathbb R}^d$ is Gaussian if its characteristic function is of the form $\hat \mu_G(x)=e^{i\langle x,a\rangle}e^{i\langle x,Qx\rangle}$, $x\in {\mathbb R}^d$, for some vector $a\in {\mathbb R}^d$ and some nonnegative symmetric $d\times d$ matrix $Q$. The vector $a$ is called the mean while the matrix $Q$ is called the covariance operator of $\mu_G$. They are related to the first two moments of the measure in the following way: $$a=\int xd\mu_G(x), \qquad \langle x, Q y\rangle =\int \langle x, z-a\rangle \langle y, z-a\rangle d\mu_G(z), \quad x,y \in {\mathbb R}^d$$ In the case where $Q=0$, $\mu_G$ is the $\delta$ point measure at $a\in {\mathbb R}^d$.

If ${\mathbb R}^d$ is replaced by a real infinite dimensional Hilbert space $H$, we say that a Borel measure on $H$ is Gaussian iff for any $x\in H$ the law of the random variable $\langle x, \, \cdot \,\rangle$ is Gaussian. By applying the Bochner-Milnos-Sazonov theorem (see above), it turns out that a measure $\mu_G$ on $H$ is Gaussian iff its Fourier transform is of the form $$\hat \mu_G(x)=e^{i\langle x,a\rangle}e^{i\langle x,Qx\rangle}$$ for some vector $a\in H$ and some non negative symmetric trace-class operator $Q:H\to H$. From this result, one can infer that on an infinite dimensional Hilbert space there cannot exists a Gaussian measure having the identity as covariance operator, as the latter is not trace class.

In the case we consider the dual $X'$ of a nuclear space $X$, endowed by the $\sigma $-algebra generated by the cylider sets of the form $\{\xi \in X' : (\xi (x_1),..., \xi (x_n))\in B\}$ for some $x_1, ...,x_n\in X$ and some Borel set $B\subset {\mathbb R}^n$, a measure $\mu_G$ is said to be Gaussian if for each finite dimensional subspace $Y\subset X$ the restriction of $\mu_G$ to the $Y$-cylinder sets is Gaussian. In this case, it arises that for any $\xi \in X'$ and any contimuous nonnegative linear map $Q:X\to X'$ there exists a unique Gaussian measure $\mu_G $ on $X'$ such that $\hat \mu_G(x) =e^{i\langle xi , x\rangle }e^{i\langle Q x , x\rangle }$, $\langle \, , \,\rangle $ denoting the dual pairing between $X'$ and $X$.

Examples of Gaussian processes

Let us consider the Hilbert space $H=L^2([0,T])$ , $T>0$, and the integral operator $Q:H\to H$ with kernel $q(s,t)$. In all the examples below it is possible to prove that $Q$ is bounded, symmetric, positive and trace-class and the function $x\mapsto e^{-\frac{1}{2}\langle x,Q x\rangle}$ is the Fourier transform of a unique Gaussian measure on $L^2([0,T])$ with mean $a=0$ and covariance operator $Q$.

In the case $q(s,t)=s\land t$ the measure $\mu_G$ is a realization of the Wiener measure.

If, more generally, one considers the kernel $q(s,t):=\frac{1}{2}\left(|t|^{2h}+|s|^{2h}+|t-s|^{2h}\right)$, with $h\in (0,1)$, again the operator $Q$ is positive and trace class and the associated Gaussian measure is the distribution of the fractional Brownian motion. In particular if $h=1/2$, we get again the Brownian motion. If $h\neq 1/2$, then the increments of the process are not independent. In particular, if $h>1/2$ they are positively correlated, while if $h<1/2$ they are negatively correlated.

If instead of the choice $q(s,t) = s \wedge t$ for the Wiener measure (to the standard Brownian motion process $B_t$) we make the choice

$$q(s,t) = \begin{cases} s(t-1), 0 \le s \le t \\ t(1-s), t \le s \le 1, \end{cases}$$ then we still have a positive trace-class operator $Q$ in $L^2[0,1]$. The inverse of $Q$ in $L^2[0,1]$ is the operator $(Q^{-1}f)(t) = f''(t)$ for all $f\in D(Q^{-1}) = H^{1,2}(0,1) \cap H^{1,2}_0(0,1)$, see e.g. Da Prato (2006). The corresponding Gaussian measure is called the Brownian bridge process. A process $\beta=(\beta(t))_{t\in [0,1]}$ defined by $\beta(t) = B(t) - tB(1)$, $t \in [0,1]$, has the probability distribution of the Brownian bridge process.

If we make the choice

$$q(t,s)=\frac{1}{2k^2(1-\exp(-a\delta))}\left(\exp(-\delta|t-s|)+ \exp(-\delta(a-|s-t|))\right)$$ where $k>0$ and $\delta>0$ are parameters, the resulting process is the so-called Hoegh-Krohn process, also known as periodic Ornstein-Uhlenbeck process (see Albeverio et al. (2009) or Brzezniak and van Neerven(2001) ).

Examples of Gaussian integrals

The computation of concrete Gaussian integrals permits to study the interplay with certain problems in differential and integral equations. A classical computation relates the expectation of a particular functional of the 1-dimensional Wiener process $W_t$ ( i.e. a particular integral with respect to Wiener measure) with the eigenvalues of the Sturm-Liuville problem in $L^2([0,1])$: $$ f''(t)+\lambda p(t) f(t) =0, \qquad t\in [0,1],$$ with the boundary values $f(0)=f'(1)=0$, where $p$ is a non-negative continuous function on $[0,1]$. The relation is $${\mathbb E}[e^{i\alpha \int_0^1p(t)W^2_tdt}]=\prod _{j=1}^\infty (1-2\alpha \lambda_j)^{-1/2},\qquad \alpha \in {\mathbb R}.$$ For other examples of explicit computations with respect to Gaussian measures see, e.g., Simon (2005), Albeverio and Mazzucchi (2015), Albeverio, Kondratiev, Kozitsky, Röckner (2009). A collection of concrete computations of integrals involving Brownian motion and related processes appears in Borodin and Salminen (2002). For computations involving other types of infinite dimensional measures (associated, e.g., to processes with jumps) see, e.g., Barndorff-Nielsen, Mikosh, Resnick (2001), Duquesne, Barndorff-Nielsen, Bertoin (2010), Cont and Tankov (2004) and Mandrekar and Rüdiger (2015).

Equivalence and orthogonality of Gaussian measures

Important work for applications involves controlling certain transformations on the infinite-dimensional space where the measures have their support. For example, given two Gaussian measures $\mu,\nu$ on an Hilbert space $H$ with mean $a\in H$ resp. $0\in H$ and the same covariance operator $Q$, they are absolutely continuous with respect to each other if and only if the vector $a$ is in the range of $\sqrt Q$ in $H$. In this case, the corresponding Radon-Nikodym derivative (the density of $\nu$ with respect to $\mu$) reads $$\frac{d\nu}{d\mu}(x)=e^{-\frac{1}{2}\|(\sqrt Q)^{-1}(a)\|^2}e^{\langle (\sqrt Q)^{-1}(a),(\sqrt Q)^{-1}(x)\rangle},$$ for all $x\in H$ (with $\| \; \|$, resp. $\langle \; , \; \rangle $, being the norm, resp. the scalar product, in $H$), see, e.g., Kuo (1975), Th. 3.1. On the other hand, if $a=0$ but $\mu$ and $\nu$ have different covariance operators $Q_\mu$ resp. $Q_\nu$, then if $\mu$ is absolutely continuous with respect to $\nu$, then the operators $Q_\mu$ and $Q_\nu$ have to be related by $$Q_\nu=\sqrt Q_\mu T \sqrt Q_\mu,$$ with $T$ a positive bounded invertible linear operator such that $I-T$ is an Hilbert-Schmidt operator on $H$. In fact, this is a special case of a theorem by Feldman and Hajek (see, e.g., Kuo (1975)).

Asymptotics of path integrals

The classical Laplace method concerns the asymptotic expansion in (fractional) powers of a small parameter $\epsilon>0$ for integrals of the form $I(\epsilon):=\int_{\R^n}e^{-\frac{1}{\epsilon}S(x)}g(x)dx$, with $g,S:\R^n\to \R$ smooth and such that the integral exists, in Lebesgue sense. In the case the phase function $S$ has a single absolute minimum say at $x=x_c\in\R^n$ which is non-degenerate (so that the determinant $D(x)$ of the Hessian matrix $\left( \frac{\partial ^2 S}{\partial x_i \partial x_j}\right)_{i,j=1....n}$ is strictly positive) the expansion in ascending powers of $\epsilon$ takes the form $$I(\epsilon)=\sum_{j=0}^Na_j(x_c)\epsilon^j+R_N(\epsilon, x_c),$$ where the coefficients $a_j(x_c)$ can be computed from $S,g$ and their derivatives. The leading term given by $$a_0(x_c)=(2\pi\epsilon)^{n/2}D(x_c)^{-1/2}e^{-\frac{1}{\epsilon}S(x_c)}g(x_c)$$ and the remainder $R_N(\epsilon, x_c)$ satisfies the estimate $|R_N(\epsilon,x_c)|\leq C_N(x_c)\epsilon^{N+1}$, with $C_N(x_c)$ independent of $\epsilon$. The case of a finite number of local non-degenerate minima can be reduced (by a smooth partition of unity) to a sum of terms of this form, corresponding to expansions around each minimum. In the case of infinitely many local non degenerate minima the same holds, provided the sum over the contributions of the single minima converges. In the case of degeneracy of a minimum, the form of the expansion around that minimum depends essentially on the form of degeneracy, in particular the leading term, instead of a power $\epsilon ^{n/2}$, has a power depending on the degeneracy. As a matter of fact the expansion generally involves powers and terms which are logarithmic in $\epsilon$, see, e.g. Combet (1982). The extension of these methods to the infinite dimensional case, where $\R^n$ is replaced by an infinite dimensional Hilbert space has also been worked out. In this case a (smooth, not necessarily Gaussian) reference measure has been used, see Albeverio and Steblovskaya (1999). Other types of dependence on the parameters have been studied by other methods, in relation to the heat equation with potentials or stochastic differential equations. In this connection, expectations of the form $$\int_H \psi (\sqrt \epsilon Y) e^{-\frac{1}{\epsilon}F(\sqrt \epsilon Y)}P(dY)$$ over a separable Hilbert space $H$ arise, where $P$ is a Gaussian probability measure. Work in the case where $P$ is Wiener measure goes back to Donsker's school, with contributions from Schilder, Pincus, Varadhan, see, e.g., Ellis and Rosen (1982), Simon (2005).

Applications

Application of probabilistic path integrals are numerous and spread out over many different areas, from mathematics and natural sciences to engineering and technical sciences, as well as to socio-economical sciences. First of all, let us mention some applications within mathematics (to areas other than stochastics), starting from analysis on Eucliedean spaces or manifolds. We already saw that solutions of some parabolic PDEs, like the heat equation, allow for representations in terms of Wiener integrals. Extensions of such representations either to other linear systems of PDEs or to nonlinear PDEs exist, including, e.g., representations of hydrodynamical equations by infinite dimensional integrals, see. e.g., Burdzy (2014) and Stroock (2012). Such representations have also been applied to many problems of differential geometry and related areas, where heat kernels (of semigroups associated to second order elliptic or hypoelliptic operators of geometric relevance) play an important role. In particular, it is relevant to mention formulas connecting the trace of the heat semigroup on a Riemannian manifold $M$ (i.e. a sum involving eigenvalues of the Laplace-Beltrami operator on $M$) with a sum of lengths of periodic geodesics (counting multiplicity). Since in turn the heat semigroup can be expressed by a Wiener integral on the manifold $M$, one gets an interpretation of such formulas in terms of infinite dimensional probabilistic integrals. These extend to much more general context of certain second order differential operators on Riemannian manifolds, where the closed geodesics are replaced by periodic orbits of an underlying classical Hamiltonian system. In this case the formulas are to be understood as asymptotic expansions with respect to a suitable parameter, the expansion being obtained by a Laplace method applied to the infinite dimensional Wiener type integral yielding the probabilistic representation for the semigroup. Such generalized trace formulas can be looked upon as rigorous mathematical implementation of ideas of semiclassical qiuantization (Gutzwiller’s trace formulas), which have found important applications in the study of the relations between chaotic classical and chaotic quantum systems, see, e.g. Schuss (2010) and Ikeda and Matsumoto (1999). For related applications to homotopy theory see, e.g., Sunada (1992). Probabilistic infinite dimensional integrals are also a powerful tool for providing upper and lower bounds for heat kernels, in the case of manifolds with singularities resp. degeneracies (see, e.g., Ledoux (2013) and Burdzy (2014)). Related small time, resp. large time, expansions have been obtained by representations in terms of such integrals, see, e.g., Uemura (1987) and Albeverio and Arede (1985). Particularly striking are those connected to index theory and topological invariants, see Bismut (1984) and Albeverio, Hahn and Sengupta (2004) .

Even though path integration techniques play a central role in present days physics (see, e.g., the Scholarpedia article Path integral by Zinn-Justin), many of the path integrals involved are not yet brought into a mathematical rigorous form. Nevertheless, the applications of those which are under mathematical control are already numerous and spread over many areas of physics. Various early applications are mentioned in the collections of essays by Wax (1954) (including areas like astrophysics and signal analysis). Kac (1980) discusses applications both in classical (potential theory) and quantum (spectral) problems. Simon (2005) presents important results reached in the study of bound states problem (number and locations of eigenvalues of Schoedinger operators). Other applications concern the stability of matter, lower estimates for multi particle Hamiltonians (see Lieb and Seiringer (2010) ). Recently Bose-Einstein condensation phenomena have been discussed in terms of probabilistic integrals in De vecchi and Ugolini (2014) . Other areas of physics where there are good applications of rigorous probabilistic integrals are equilibrium and non-equilibrium statistical mechanics and kinetic theory, both classical (see, e.g., Kac (1980)) and quantum, see, e.g., Jona-Lasinio (1976), Presutti (2009) and Albeverio et al (2009). For applications in solid state physics see, e.g. the discussion on polaron model in Schulman (1981). For work in hydrodynamics see, e.g., Albeverio, Flandoli and Sinai (2002) . In quantum field theory, probabilistic integrals have played a decisive role in constructing non trivial low dimensional models, see, e.g., Glimm and Jaffe (1987) and Simon (1974).

In the modeling of long polymer chains in chemistry and physics, methods of infinite dimensional integration have found important applications. An interesting example is Edwards’ model describing polymers in ${\mathbb R}^d$ by means of a Gibbs-type measure with respect to a Wiener measure. The Gibbs factor makes unlikely self-intersections of the Brownian path modeling the long molecular chains constituting the polymer. Rigorous results exist for $d=1,2,3$, see, e.g., the references in Streit et al (2015). Lett us also mention that polymer related models and corresponding probabilistic methods have found applications to DNA modeling, see, e.g. Cotta-Ramusino and Maddocks (2010) and Bellomo and Pulvirenti (2000). Probabilistic path integrals also have applications in various areas of biology (e.g. epidemiology, immunology, genetic, population dynamics), see, e.g., Ricciardi (1977) and, for some newer developments, Bovier el al (2015) . Neurobiology has also been a favorite area of application of such integrals, from simple axon model to neuron networks, see, e.g., Tuckwell (1989).

In economics, the use of infinite dimensional integrals is implicit in pioneering work by Bachelier (see , e.g., the article by Schachermayer in Albeverio, Schachermayer and Talagrand (2003) ), which has lead to an intense activity in probabilistic methods in mathematical finance producing, e.g., the famous Black-Scholes formula for the price of a European call option (expressed by a Wiener integral involving a functional of a brownian motion process). This is just a prototype of more general models and computations, see, e.g., Øksendal (2003). Also in macroeconomical modeling some use of probabilistic integrals is present, e.g. in Mallianis and Broch (1982) . For some use of modeling by probabilistic integrals in sciences of society see, e.g. Weidlich (2000).

Probabilistic integrals are applied also in ecology and climate research, following the pioneering work by Haselmann, see, e.g., Imkeller (2002). Engineering applications include stochastic control theory and robotics, signal transmission, filtering and civil engineerings, see, e.g., Blankenship et al (2000) and Kree and Soize (1986).

Let us stress that what we have mentioned does not in any way expect to be exhaustive, both in terms of topics and references, but should be rather considered as a first orientation in a fascinating rapidly expanding area of research.

References

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Probabilistic integrals: mathematical aspects

Contents

Stochastic processes and probability measures on spaces of paths

The historical development of probabilistic path integrals

Measures on infinite dimensional spaces

Kolmogorov theorem

Construction of Markov stochastic processes via Markov kernels

Stochastic processes with stationary and independent increments: Lévy processes

The Wiener measure and the Wiener process

Applications of the Wiener measure

The Feynman-Kac formula

The solution of the Dirichlet problem for the Laplace equation

Other stochastic processes

Brownian motion on manifolds

The Ornstein-Uhlenbeck process

$\alpha-$stable processes

Fourier transform of measures. Bochner-Minlos theorem

Gaussian measures and Gaussian integrals

Examples of Gaussian processes

Examples of Gaussian integrals

Equivalence and orthogonality of Gaussian measures

Asymptotics of path integrals

Applications

References

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