Representation of solution for elliptic PDE using stochastic process.

Let $U$ be a bounded subset of $\QTR{cal}{R}^{n}$ with $C^{1}$ -boundary. Let MATH and $f|_{\partial U}=0$ , MATH . The solution of the boundary problem MATH is given by MATH where the $W_{t}$ is the standard Brownian motion and MATH is the first time when the process $x+W_{t}$ exits $U$ .

Proof

We verify for one dimension: MATH By the compatibility condition $f|_{\partial U}=0$ we have MATH . We continue MATH MATH Note that MATH hence MATH

Proof

We give another proof more inline with considerations of the section ( Backward_equation_section). Let MATH

where MATH . So that MATH . We calculate as in the section ( Backward_equation_section) and under assumption that $x$ is away from the boundary MATH : MATH MATH It remains to note that MATH Indeed, MATH for any $h$ .

Forum

Notation

Index

Contents