Wiener Measure measure on functions?

Question

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

Answer 1

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.

Answer 2

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".