Probabilistic integrals: mathematical aspects
Dr. Sonia Mazzucchi accepted the invitation on 10 May 2010
Prof. Sergio Albeverio accepted the invitation on 10 May 2010
Integrals over spaces of paths or more generally of fields has 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 measures) are special cases. In probability the concept of flat integral 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 integrals, will be presented, while the theory and the applications of path integrals of Feynman's type will be presented in Path integrals: mathematical aspects.
Measure 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 \(\sigma-\) addittive measure is a non trivial task. In fact a Lebesgue-type measure cannot be defined on infinite dimensional Hilbert spaces. Indeed the assumption of the existence of a \(\sigma\)-additive measure \(\mu\) which is invariant under rotations and translations and assigns a positive finite measure to all bounded open sets, leads to a contraddiction. In fact by taking an orthonormal system \(\{e_i\}_{i\in\N}\) in an infinite dimensional Hilbert space ${\mathcal H}$ 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 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. This argument can be generalized to the case where the Hilbert space $\mathcal H$ is replaced by an infinite dimensional Banach space $\mathcal B$.
Wiener measure
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 provides a mathematical model for the physical Brownian motion studied by A. Eistein e M. Smoluchowski between 1905 and 1909. The construction of Wiener measure presented here emphasizes the relation between the rigorous work by Wiener and the heuristic concept of path integral, applied for instance in the Feynman formulation of quantum mechanics. Let $0=t _0<t_1<t_2<\cdot\cdot\cdot<t_n\equiv t$. The Wiener measure $P$ is defined in such a way that the probability $P(X _{t_1}\in B_1,\dots ,X _{t_n}\in B_n)$ that the process crosse at time $t_i$ the Borel subsets $B _i\subseteq\R^d$ is given by: \begin{equation}\tag{1}\int_{B_n}\cdots\int_{B_1}\prod_{j=0}^{n-1}\rho(\Delta t_j,\Delta x_j)dx_1\ldots dx_n,\end{equation} where $\Delta a_i\equiv a_{i+1}-a_i$, $a_0\equiv 0$ and $\rho$ is given by $$\rho(t,x)=\frac{e{-\frac{\vert x\vert ^2}{2 t}}}{(2\pi t)^{\frac{d}{2}}}$$ It follows that \begin{equation}\tag{2}\prod_{j=0}^{n-1}\rho(\Delta t_j,\Delta x_j)=\frac{ e^{-\frac{1}{2}\sum_{j=0}^{n-1}\frac{\vert\Delta x_j\vert^2}{\Delta t_j}} }{2\pi \Delta t_{n-1}\ldots \Delta t_{0} }. \end{equation} By introducing a "poligonal path" $\gamma :[0,t]\to \R^d$, such that $\gamma (t_j)=x_j$ and $\gamma (\tau)$ for $\tau \in [t_j,t_{j+1}]$ coincides with the constant velocity path connecting $x_j$ with $x _{j+1}$: $$\gamma (s)=\sum_{j=0}^{N-1}\chi_{[t_j, t_{j+1}]}(s)\Big( x_j+\frac{x_{j+1}-x_j}{t_{j+1}-t_j}(s-t_j)\Big),\qquad s\in [0,t], $$ where $\chi_{[t_j, t_{j+1}]}$is the characteristic function of the interval $[t_i, t_{i+1}]$. Let us denote with $\mathcal P$ the set of all poligonal paths. Let \begin{equation}\tag{3} S^\circ_t(\gamma)\equiv\frac{1}{2}\int _0^t \vert\dot\gamma (s)\vert ^2ds \end{equation} be the action evaluated along the path $\gamma$: \begin{equation}\tag{4} S^\circ_t(\gamma)\equiv\frac{1}{2}\sum_{j=0}^{n-1} \Big| \frac{\Delta x _j}{\Delta t _j}\Big|^2 \Delta t _j.\end{equation} Let us denote with \begin{equation}\tag{5} D\gamma\equiv Z^{-1}\prod_{t\in\{t_1,\ldots, t_n \}}d\gamma (t), \end{equation} with \begin{equation}\tag{6} Z\equiv (2\pi \Delta t_{n-1}\ldots \Delta t_0)^{\frac{d}{2}}, \end{equation} and $d\gamma (t)=dx_j$ for $t=t_j$.Equation (1) can be written as \begin{equation}\tag{7} \int _\Delta e^{-S_t^\circ (\gamma)}Delta \end{equation} with $$\Delta=\{ \gamma\in {\mathcal P} \; | \; \gamma (t_1)\in B _1,\ldots , \gamma (t_n)\in B _n\}.$$ In such a way Wiener measure can be written as \begin{equation}\tag{8} P(X _{t_1}\in B_1, \ldots, X _{t_n}\in B_n)=\int _\Delta e^{-S_t^\circ (\gamma)}D\gamma. \end{equation} The right hand side can be interpreted as a "path integral" on the space of poligonal paths wuith respect to the measure $e^{-S_t^\circ (\gamma)}D\gamma$. $D\gamma$ can be regarded as a "flat measure" on $\R ^{nd}$ normalized in such a way that $\int_{[0,1]^{nd}}D\gamma =(2\pi \Delta t_{n-1}\ldots \Delta t_0)^{-\frac{d}{2}}$). $D\gamma$ is multiplied by the factor $e^{-S_t^\circ (\gamma)}$, (that can be regarded as a Gibbs factor, in the sense of statistical mechanics, as $S_t^\circ$ is an action functional). Clearly $e^{-S_t^\circ (\gamma)}\Delta\gamma$ is the standard Gaussian measure $d\mu _{{\mathcal H}_n}(\gamma )$ associated to the Hilbert space ${\mathcal H}_n\equiv({\mathbb R}^{nd},\langle\cdot,\cdot\rangle)$, where $\langle\gamma,\gamma'\rangle=\int _0^t\dot \gamma(s)\cdot\dot\gamma'(s)ds$. The Fourier transform $\hat\mu_{{\mathcal H}_n}$ of the measure $\mu _{{\mathcal H}_n}$, defined by $$\hat\mu_{{\mathcal H}_n}(\alpha)=\int e^{i\langle\alpha,\gamma\rangle }d\mu_{{\mathcal H}_n}(\gamma), \qquad\forall\alpha\in {\mathcal H}_n,$$ is equal to $$\hat\mu_{{\mathcal H}_n}(\alpha)=e^{-\frac{1}{2}\vert \alpha\vert^2_{{\mathcal H}_n}}.$$ ${\mathcal H}_n$ converges naturally when $n\to\infty$ to the Cameron-Martin space ${\mathcal H}\equiv\{ \gamma\in C_{(0)} ([0,t];{\mathbb R}^d)\; |\; \vert\gamma\vert_{\mathcal H}<\infty\}$, that is to the Hilbert space of absolutely continuous paths $\gamma :[0,t]\to\R^d$ such that $\gamma (0)=0$ and $ \vert\gamma\vert _{\mathcal H}^2\equiv\langle\gamma,\gamma\rangle\equiv\int_0^t\vert\dot\gamma(s)\vert ^2ds$. One has the convergence of the Fourier transforms: $$\hat\mu_{{\mathcal H}_n}(\alpha _n)\rightarrow\hat\mu_{\mathcal H}(\alpha )\equiv e^{-\frac{1}{2}\vert \alpha\vert^2_{\mathcal H}},\qquad n\to\infty,$$ for any $\alpha\in {\mathcal H}$, where $\alpha_n$ is the projection of $\alpha$ on ${\mathcal H}_n$.\par It is possible to prove \cite{Mi,Gro1} that there exists a finite additive measure on ${\mathcal H}$, its Fourier transform is $\hat\mu_{\mathcal H}$ and its $\sigma$-additive extension on the Banach space $E=C_{(0)}([0,t];{\mathbb R}^d)$ is the Wiener measure on $E$.
Applications
Other Gaussian path integrals
powers of the Laplacian. fields
Path integrals associated with diffusion processes
Path integrals associated with infinitely divisible stochastic processes or infinitely divisible measures
Path integrals associated with stochastic (P)DE's
Stochastic quantization
Asymptotics of path integrals
Laplace method
Extensions
discrete spaces, fractals
Applications
Physics Hydrodynamics, Biology, Economics
