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In the case of $n=3$ (by the procedure outlined by fedja) the problem boils down to study two dimensional BM in the yellow domain. Let now $G$ be a group generated by the reflections in the blue lines, then the transition density of the BM killed on the hitting boundary, h_t(x,y), $h_t(x,y)$, is

$h_t(x,y) := \sum_{g\in G} (-1)^{r(g)}p_t(x,g(x)),$

where $p_t(x,y) = (2\pi t)^{-1} e^{-|x-y|^2/(2t)}$ is the transition density of the BM in $\mathbb{R}^2$ and $r$ is "the rank" of $g$ (to be explained in a moment).

I know almost nothing about the group theory but it seems to me that $G$ is what is called "a reflection group" or a special case of a Coxeter group. $r(g)$ as far as I understand is the length of the shortest way in which $g$ can be represented using the generator only.

So, the questions are:

1. Is there a way of presenting the above sum in by a closed formula.

2. What is asymptotic behaviour of $h_t(x,y)/\int h_t(x,y)$?

P.S. It is probably not very common to answer own questions but I think it is more then a comment and I do not know how to paste images into comments.

P.P.S. Following Omer's comment, it should be not hard to prove that

$l(y):=\lim_{t\rightarrow +\infty} h_t(0,y)/h_t(0,0)$

exists and $l$ is the leading eigenvalue of the Laplacian in the considered domain.

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In the case of $n=3$ (by the procedure outlined by fedja) the problem boils down to study two dimensional BM in the yellow domain. Let now $G$ be a group generated by the reflections in the blue lines, then the transition density of the BM killed on the hitting boundary, h_t(x,y), is

$h_t(x,y) := \sum_{g\in G} (-1)^{r(g)}p_t(x,g(x)),$

where $p_t(x,y) = (2\pi t)^{-1} e^{-|x-y|^2/(2t)}$ is the transition density of the BM in $\mathbb{R}^2$ and $r$ is "the rank" of $g$ (to be explained in a moment).

I know almost nothing about the group theory but it seems to me that $G$ is what is called "a reflection group" or a special case of a Coxeter group. $r(g)$ as far as I understand is the length of the shortest way in which $g$ can be represented using the generator only.

So, the questions are:

1. Is there a way of presenting the above sum in by a closed formula.

2. What is asymptotic behaviour of $h_t(x,y)/\int h_t(x,y)$?

P.S. It is probably not very common to answer own questions but I think it is more then a comment and I do not know how to paste images into comments.