- Stochastic processes and boundary value problems
In
mathematics , someboundary value problem s can be solved using the methods of stochastic analysis. Perhaps the most celebrated example isShizuo Kakutani 's 1944 solution of theDirichlet problem for theLaplace operator usingBrownian motion . However, it turns out that for a large class of semi-elliptic second-orderpartial differential equations the associated Dirichlet boundary value problem can be solved using anItō process that solves an associatedstochastic differential equation .Introduction: Kakutani's solution to the classical Dirichlet problem
Let "D" be a domain (an open and connected set) in R"n". Let Δ be the
Laplace operator , let "g" be abounded function on the boundary ∂"D", and consider the problem:
It can be shown that if a solution "u" exists, then "u"("x") is the
expected value of the (random) first exit time from "D" for a canonicalBrownian motion starting at "x".The Dirichlet-Poisson problem
Let "D" be a domain in R"n" and let "L" be a semi-elliptic differential operator on "C"2(R"n"; R) of the form
:
where the coefficients "b""i" and "a""ij" are
continuous function s and all theeigenvalue s of the matrix "a"("x") = ("a""ij"("x")) are non-negative. Let "f" ∈ "C"("D"; R) and "g" ∈ "C"(∂"D"; R). Consider the Poisson problem:
The idea of the stochastic method for solving this problem is as follows. First, one finds an
Itō diffusion "X" whose infinitesimal generator "A" coincides with "L" on compactly-supported "C"2 functions "f" : R"n" → R. For example, "X" can be taken to be the solution to the stochastic differential equation:
where "B" is "n"-dimensional Brownian motion, "b" has components "b""i" as above, and the matrix field "σ" is chosen so that
:
For a point "x" ∈ R"n", let P"x" denote the law of "X" given initial datum "X"0 = "x", and let E"x" denote expectation with respect to P"x". Let "τ""D" denote the first exit time of "X" from "D".
In this notation, the candidate solution for (P1) is
:
provided that "g" is a
bounded function and that:
It turns out that one further condition is required:
:
i.e., for all "x", the process "X" starting at "x"
almost surely leaves "D" 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 "X" (which agrees with "A" on "C"2 functions), then
:
Moreover, if "v" ∈ "C"2("D"; R) satisfies (P2) and there exists a constant "C" such that, for all "x" ∈ "D",
:
then "v" = "u".
References
* cite journal
last = Kakutani
first = Shizuo
authorlink= Shizuo Kakutani
title = Two-dimensional Brownian motion and harmonic functions
journal = Proc. Imp. Acad. Tokyo
volume = 20
year = 1944
pages = 706–714
* cite journal
last = Kakutani
first = Shizuo
authorlink= Shizuo Kakutani
title = On Brownian motions in "n"-space
journal = Proc. Imp. Acad. Tokyo
volume = 20
year = 1944
pages = 648–652
* cite book
last = Øksendal
first = Bernt K.
authorlink = Bernt Øksendal
title = Stochastic Differential Equations: An Introduction with Applications
edition = Sixth edition
publisher=Springer
location = Berlin
year = 2003
isbn = 3-540-04758-1 (See Section 9)
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