Let $U$ be an open subset of $\mathbb{R}^n$ such that $\partial \overline{U}$, the boundary of $\overline{U}$ is ''nice'' (for simplicity you can assume piecewise smooth). I also want to allow the possibility of $U$ being the whole of $\mathbb{R}^n$, which means there is no boundary. Let $G(x,y,t)$ be the fundamental solution to the Heat equation with Dirichlet boundary condition (for example, if $U$ was the whole of $\mathbb{R}^n$, then $G(x,y,t)$ would be the Gaussian).

Given an $x \in U$, let me define the following spaces $$ \Omega_x := \{ \gamma:[0,1] \rightarrow \mathbb{R}^n: \gamma(0) =x, ~~\gamma ~~ \textsf{is continuous} \}, $$ $$ \Omega_x (\overline{U}) := \{ \gamma \in \Omega_x, ~~\gamma(t) \in \overline{U} ~~~\forall t \}.$$ To say it in words, $\Omega_x$ is the space of all continuous paths starting at $x$ and $\Omega_x (\overline{U})$ is the space of all such paths that never go outside $\overline{U}$. These are both metric spaces with respect to the supremum norm. Now given any $t \in [0,1]$, there is an evaluation map $$ ev_t: \Omega_x \rightarrow \mathbb{R}^n \qquad ev_t(\gamma):= \gamma(t).$$

Now my question is the following: is there a probability measure $\mu_W $ on $\Omega_x$ with the following property:

1) $\mu_W$ is zero outside $\Omega_x(\overline{U})$,

2) For every $t \in [0,1]$ and every set $~A \in \mathbb{R}^n$ that is a product of intervals (i.e. $A$ is of the form $[a_1,b_1] \times \ldots \times [a_n, b_n]$), we have $$ \mu_W(ev_t^{-1}(A)) = \frac{\int_{A\cap U} G(x,y,t) dy}{\int_{U} G(x,y,t) dy}.$$

$\textbf{Added Later:}$ I realized later that I have to be more specific. I do not simply want such a measure to exist; I want that measure to be a weak limit of the following sequence of measures: Divide $[0,1]$ into $k$ intervals of equal length and consider the following finite set of $(2n)^k$ distinct continuous paths; at time $t=0, ~1/k, ~2/k, \ldots$ one can move in any of the $2n$ directions by $\frac{1}{\sqrt{k}}$ units. Join the time in between by a straight line. Choose any of the paths that are inside $\overline{U}$ with equal probability and the rest with zero probability. Call this measure $\mu_k$. I want this final $\mu_W$ to be a weak limit of these $\mu_k$.

Note that I am asking for two things; first that the discrete measure I defined should have a weak limit (existence) and secondly that limit should be something specific I want.

I believe this fact is true if $U$ is the whole of $\mathbb{R}^n$ (i.e. no boundary). I also believe this fact is true with $\overline{U} := [0,L]$ inside $\mathbb{R}$. I want to know if there is any general result for domains with boundary.

$\textbf{Note:}$ When I say "measure on $\Omega_x$" I mean a measure on the Borel sigma algebra of open sets (Borel sigma algebra is the smallest sigma algebra that contains the open sets and all such sets have to measurable; sigma algebra means the collection has to be closed under taking complements, intersections and countable unions).