Stochastic processes and boundary value problems

Stochastic processes and boundary value problems

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

Let "D" be a domain (an open and connected set) in R"n". Let Δ be the Laplace operator, let "g" be a bounded function on the boundary ∂"D", and consider the problem

:egin{cases} - Delta u(x) = 0, & x in D; \ displaystyle{lim_{y o x} u(y)} = g(x), & x in partial D. end{cases}

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 canonical Brownian 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

:L = sum_{i = 1}^{n} b_{i} (x) frac{partial}{partial x_{i + sum_{i, j = 1}^{n} a_{ij} (x) frac{partial^{2{partial x_{i} , partial x_{j,

where the coefficients "b""i" and "a""ij" are continuous functions and all the eigenvalues 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

:egin{cases} - L u(x) = f(x), & x in D; \ displaystyle{lim_{y o x} u(y)} = g(x), & x in partial D. end{cases} quad mbox{(P1)}

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

:mathrm{d} X_{t} = b(X_{t}) , mathrm{d} t + sigma (X_{t}) , mathrm{d} B_{t},

where "B" is "n"-dimensional Brownian motion, "b" has components "b""i" as above, and the matrix field "σ" is chosen so that

:frac1{2} sigma (x) sigma(x)^{ op} = a(x) mbox{ for all } x in mathbf{R}^{n}.

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

:u(x) = mathbf{E}^{x} left [ g ig( X_{ au_{D ig) cdot chi_{{ au_{D} < + infty ight] + mathbf{E}^{X} left [ int_{0}^{ au_{D f(X_{t}) , mathrm{d} t ight]

provided that "g" is a bounded function and that

:mathbf{E}^{x} left [ int_{0}^{ au_{D ig| f(X_{t}) ig| , mathrm{d} t ight] < + infty.

It turns out that one further condition is required:

:mathbf{P}^{x} ig [ au_{D} < + infty ig] = 1 mbox{ for all } x in D,

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

:u(x) = mathbf{E}^{x} left [ g ig( X_{ au_{D ig) ight] + mathbf{E}^{x} left [ int_{0}^{ au_{D f(X_{t}) , mathrm{d} t ight]

and solves (P1) in the sense that if mathcal{A} denotes the characteristic operator for "X" (which agrees with "A" on "C"2 functions), then

:egin{cases} - mathcal{A} u(x) = f(x), & x in D; \ displaystyle{lim_{t uparrow au_{D u(X_{t})} = g ig( X_{ au_{D ig), & mathbf{P}^{x} mbox{-a.s., for all } x in D. end{cases} quad mbox{(P2)}

Moreover, if "v" &isin; "C"2("D"; R) satisfies (P2) and there exists a constant "C" such that, for all "x" &isin; "D",

:| v(x) | leq C left( 1 + mathbf{E}^{x} left [ int_{0}^{ au_{D ig| g(X_{s}) ig| , mathrm{d} s ight] ight),

then "v" = "u".


* 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&ndash;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&ndash;652

* cite book
last = Øksendal
first = Bernt K.
authorlink = Bernt Øksendal
title = Stochastic Differential Equations: An Introduction with Applications
edition = Sixth edition
location = Berlin
year = 2003
isbn = 3-540-04758-1
(See Section 9)

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Boundary value problem — In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the… …   Wikipedia

  • Comparison of analog and digital recording — This article compares the two ways in which sound is recorded and stored. Actual sound waves consist of continuous variations in air pressure. Representations of these signals can be recorded using either digital or analog techniques. An analog… …   Wikipedia

  • Partial differential equation — A visualisation of a solution to the heat equation on a two dimensional plane In mathematics, partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Differential equation — Not to be confused with Difference equation. Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state… …   Wikipedia

  • Cuckoo search — (CS) is an optimization algorithm developed by Xin she Yang and Suash Deb in 2009.[1][2] It was inspired by the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds (of other species). Some host… …   Wikipedia

  • probability theory — Math., Statistics. the theory of analyzing and making statements concerning the probability of the occurrence of uncertain events. Cf. probability (def. 4). [1830 40] * * * Branch of mathematics that deals with analysis of random events.… …   Universalium

  • analysis — /euh nal euh sis/, n., pl. analyses / seez /. 1. the separating of any material or abstract entity into its constituent elements (opposed to synthesis). 2. this process as a method of studying the nature of something or of determining its… …   Universalium

  • Mathematical optimization — For other uses, see Optimization (disambiguation). The maximum of a paraboloid (red dot) In mathematics, computational science, or management science, mathematical optimization (alternatively, optimization or mathematical programming) refers to… …   Wikipedia

  • Optimal control — theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union[1] and Richard Bellman in… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”