Representation of solution for elliptic PDE using
stochastic process.

roposition

Let
be a bounded subset of
with a
-boundary
.
Let
and
,
.
The solution of the boundary
problem
is given
by
where
is standard Brownian motion in
and
is the first time when the process
exits
.

where
.
Thus
,
.
We calculate as in the section (
Backward
equation
) and under assumption that
is away from the boundary
:
It remains to note that
Indeed,
for any
.