0
$\begingroup$

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).

$\endgroup$
  • 1
    $\begingroup$ 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
5
$\begingroup$

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.

$\endgroup$

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.