5
$\begingroup$

I know that the Wiener measure for the Brownian motion $\{B_t\}_{t\ge 0}$ on the probability space $(\Omega, \mathscr{F},P)$ can be defined as $\mu=P\circ B^{-1}$ acting on the sigma-algebra generated by the cylindrical sets: $$Cly:=\{x\in C([0,\infty),\Bbb{R}^d)|x(t_1)\in A_1,...,x(t_n)\in A_n\}$$ Here $A_i$ are Borel sets of $\Bbb{R}^d$. I also know the action of $\mu$ on $A\in Cly$ is given by: $$\mu[A] =\prod^n_{i=1} (2\pi(t_i-t_{i-1}))^{-d/2} \int_{A_1}...\int_{A_n} \exp\left( -\frac{1}{2} \sum^n_{i=1} \frac{|x_i-x_{i-1}|^2}{t_i-t_{i-1}}\right) dx_n...dx_1$$ My question is how do we go from this to the integral of a function $f(\omega)$ to (as indicated in (1)): $$\lim_{n\rightarrow \infty}\left(\frac{n}{4D\pi t}\right)^{nd/2}\int_{\Bbb{R}^d}...\int_{\Bbb{R}^d}\hat f(x_1,...,x_n)\exp\left( - \sum^n_{i=1} \frac{|x_i-x_{i-1}|^2}{4Dt/n}\right) dx_n...dx_1$$ (1) also explains that the existence of the measure is to do with the Riesz Representation Theorem - how is this the case?

Sources

(1) Nathanson, E.S., 2014. Path integration with non-positive distributions and applications to the Schrödinger equation. The University of Iowa. pdf pg$\sim$27 actual pg$\sim$17

$\endgroup$
4
$\begingroup$

The trick is to regard the Wiener measure as a random sample function $f(x,t)$ where $x\in (\Omega, \mathscr{F},P)$ and $t\in \mathscr{T}$ is the time index set. Then the whole stochastic process can be regarded as a curve/function when $x$ is fixed. Along with probability measure $P$ this consists of a random function. When you look at the curve of path generated by a Wiener measure, it is actually a realization of a function when the time index $t$ varies, see [1].

The reason why we need to invoke Riesz representation theorem is that when the path is $L^2$ integrable(and for a Wiener measure this is the case because its path is smooth with probability $P$ 1) we have a nice duality between function space and the time index $\mathscr{T}$,which is neatly explained in [2]. When the sample path $g_x(t):=f(x,t)$ is not $L^2$ integrable this perspective will not hold, that is why we need some conditions on covariance function of $\mu$ to make the sample path smooth with probability $P$ 1.

Reference

[1]Cramér, Harald. "Stochastic processes as curves in Hilbert space." Theory of Probability & Its Applications 9.2 (1964): 169-179.

[2]Enchev, Ognian, and Daniel W. Stroock. "Towards a Riemannian geometry on the path space over a Riemannian manifold." Journal of Functional Analysis 134.2 (1995): 392-416.

$\endgroup$
2
$\begingroup$

It seems that what you are looking for is the notion of "abstract Wiener space", which provides a rigorous setting for putting a white noise on various spaces of functions. Long story short, you define the white noise as a random distribution via a larger Banach space of functions; this distribution makes sense once taken against elements of the dual of this Banach space (hence the use of the Riesz representation theorem for seeing this dual space as a subspace of your initial Hilbert space of functions). A nice introduction can be found in Stroock's "Abstract Wiener space, revisited".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.