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).

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    $\begingroup$ Your requirement (2) seems wrong; you are essentially conditioning on your process being inside $U$ at time $t$, but it seems what you want is to condition on it staying inside $U$ all the way up to time 1. I have to say that unfortunately, I think answering this question properly requires rewriting a book on Brownian motion. $\endgroup$ – Nate Eldredge Jan 21 '15 at 18:59
  • $\begingroup$ @Nate: Suppose $n=1$, $U:= (0,L)$ and $\overline{U}:= [0,L]$. Suppose now I consider a random walk starting from some point $x \in (0,L)$ and I only look at paths that stay inside $(0,L)$ for all time. Assuming the weak limit of these measures exist, what should be the distribution of the evaluation map wrt to this measure? Suppose $A$ is some subset inside $(0,L)$. What ought to be the right condition? What I thought is that the equation I have written in 2) should hold, where $G(x,y,t)$ is the heat kernel for a finite rod and the integration is from $0$ to $L$ in the denominator. $\endgroup$ – Ritwik Jan 21 '15 at 19:21

As Nate already pointed out, the Brownian motion conditioned to stay inside $D$ up to time $1$ (which is what the limit in "Added later" gives you) will not satisfy (2). I suspect (and this what there's a nice theory for) that you really want to consider Brownian motion conditioned to stay inside $U$ forever. (That's of course a measure $0$ event, but most reasonable ways you can invent to approximate this will give the same weak limit.) This will be a Markov process with transition probabilities given by $$ P_t(x,y) = e^{-\lambda t}{h(y) \over h(x)}G_t(x,y)\;, $$ where $\lambda$ is the smallest eigenvalue of the Dirichlet Laplacian on $D$, $h$ is the corresponding (necessarily positive) eigenfunction, and $G$ is the Dirichlet heat kernel. If you prefer SDEs, this is the solution to $$ dx = {1\over 2} \nabla \log h(x)\,dt + dB(t)\;, $$ which has invariant measure $h$. Look for "Doob's $h$-transform" in the literature for more details. There are similar formulae for the Brownian motion conditioned to stay inside $D$ up to times $1$, but it's a bit less clean.


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