Stochastic processes and boundary value problems

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In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns out that for a large class of semi-elliptic second-order partial differential equations the associated Dirichlet boundary value problem can be solved using an Itō process that solves an associated stochastic differential equation.

Introduction: Kakutani's solution to the classical Dirichlet problem

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Let   be a domain (an open and connected set) in  . Let   be the Laplace operator, let   be a bounded function on the boundary  , and consider the problem:

 

It can be shown that if a solution   exists, then   is the expected value of   at the (random) first exit point from   for a canonical Brownian motion starting at  . See theorem 3 in Kakutani 1944, p. 710.

The Dirichlet–Poisson problem

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Let   be a domain in   and let   be a semi-elliptic differential operator on   of the form:

 

where the coefficients   and   are continuous functions and all the eigenvalues of the matrix   are non-negative. Let   and  . Consider the Poisson problem:

 

The idea of the stochastic method for solving this problem is as follows. First, one finds an Itō diffusion   whose infinitesimal generator   coincides with   on compactly-supported   functions  . For example,   can be taken to be the solution to the stochastic differential equation:

 

where   is n-dimensional Brownian motion,   has components   as above, and the matrix field   is chosen so that:

 

For a point  , let   denote the law of   given initial datum  , and let  denote expectation with respect to  . Let   denote the first exit time of   from  .

In this notation, the candidate solution for (P1) is:

 

provided that   is a bounded function and that:

 

It turns out that one further condition is required:

 

For all  , the process   starting at   almost surely leaves   in finite time. Under this assumption, the candidate solution above reduces to:

 

and solves (P1) in the sense that if   denotes the characteristic operator for   (which agrees with   on   functions), then:

 

Moreover, if   satisfies (P2) and there exists a constant   such that, for all  :

 

then  .

References

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  • Kakutani, Shizuo (1944). "Two-dimensional Brownian motion and harmonic functions". Proc. Imp. Acad. Tokyo. 20 (10): 706–714. doi:10.3792/pia/1195572706.
  • Kakutani, Shizuo (1944). "On Brownian motions in n-space". Proc. Imp. Acad. Tokyo. 20 (9): 648–652. doi:10.3792/pia/1195572742.
  • Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Section 9)