- Stochastic processes and boundary value problems
In

mathematics , some. Perhaps the most celebrated example isboundary value problem s can be solved using the methods of stochastic analysisShizuo 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 theLaplace operator , let "g" be abounded function on the boundary ∂"D", and consider the problem:$egin\{cases\}\; -\; Delta\; u(x)\; =\; 0,\; x\; in\; D;\; \backslash \; displaystyle\{lim\_\{y\; o\; x\}\; u(y)\}\; =\; g(x),\; x\; in\; partial\; D.\; end\{cases\}$

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 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:$L\; =\; sum\_\{i\; =\; 1\}^\{n\}\; b\_\{i\}\; (x)\; frac\{partial\}\{partial\; x\_\{i\; +\; sum\_\{i,\; j\; =\; 1\}^\{n\}\; a\_\{ij\}\; (x)\; frac\{partial^\{2\{partial\; x\_\{i\}\; ,\; partial\; x\_\{j,$

where the coefficients "b"

_{"i"}and "a"_{"ij"}arecontinuous 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:$egin\{cases\}\; -\; L\; u(x)\; =\; f(x),\; x\; in\; D;\; \backslash \; displaystyle\{lim\_\{y\; o\; x\}\; u(y)\}\; =\; g(x),\; x\; in\; partial\; D.\; end\{cases\}\; quad\; mbox\{(P1)\}$

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:$egin\{cases\}\; -\; mathcal\{A\}\; u(x)\; =\; f(x),\; x\; in\; D;\; \backslash \; displaystyle\{lim\_\{t\; uparrow\; au\_\{D\; u(X\_\{t\})\}\; =\; g\; ig(\; X\_\{\; au\_\{D\; ig),\; mathbf\{P\}^\{x\}\; mbox\{-a.s.,\; for\; all\; \}\; x\; in\; D.\; end\{cases\}\; quad\; mbox\{(P2)\}$

Moreover, if "v" ∈ "C"

^{2}("D";**R**) satisfies (P2) and there exists a constant "C" such that, for all "x" ∈ "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".

**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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