128.15.195.87: /* Formal description */

2013-08-01T19:58:02Z

Formal description

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In [[mathematics]], some '''[[boundary value problem]]s can be solved using the methods of [[stochastic processes|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 operator|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 set|open]] and [[connected space|connected set]]) in '''R'''''n''. Let Δ be the [[Laplace operator]], let ''g'' be a [[bounded function]] on the [[boundary (topology)|boundary]] ∂''D'', and consider the problem

:<math>\begin{cases} - \Delta u(x) = 0, & x \in D; \\ \displaystyle{\lim_{y \to x} u(y)} = g(x), & x \in \partial D. \end{cases}</math>

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

==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

:<math>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}},</math>

where the coefficients ''b''''i'' and ''a''''ij'' are [[continuous function]]s and all the [[eigenvalue]]s of the [[matrix (mathematics)|matrix]] ''a''(''x'') = (''a''''ij''(''x'')) are non-negative. Let ''f'' ∈ ''C''(''D''; '''R''') and ''g'' ∈ ''C''(∂''D''; '''R'''). Consider the [[Poisson equation|Poisson problem]]

:<math>\begin{cases} - L u(x) = f(x), & x \in D; \\ \displaystyle{\lim_{y \to x} u(y)} = g(x), & x \in \partial D. \end{cases} \quad \mbox{(P1)}</math>

The idea of the stochastic method for solving this problem is as follows. First, one finds an [[Itō diffusion]] ''X'' whose [[infinitesimal generator (stochastic processes)|infinitesimal generator]] ''A'' coincides with ''L'' on [[compact support|compactly-supported]] ''C''2 functions ''f'' : '''R'''''n'' → '''R'''. For example, ''X'' can be taken to be the solution to the stochastic differential equation

:<math>\mathrm{d} X_{t} = b(X_{t}) \, \mathrm{d} t + \sigma (X_{t}) \, \mathrm{d} B_{t},</math>

where ''B'' is ''n''-dimensional Brownian motion, ''b'' has components ''b''''i'' as above, and the [[tensor field|matrix field]] ''σ'' is chosen so that

:<math>\frac1{2} \sigma (x) \sigma(x)^{\top} = a(x) \mbox{ for all } x \in \mathbf{R}^{n}.</math>

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

:<math>u(x) = \mathbf{E}^{x} \left[ g \big( X_{\tau_{D}} \big) \cdot \chi_{\{ \tau_{D} < + \infty \}} \right] + \mathbf{E}^{X} \left[ \int_{0}^{\tau_{D}} f(X_{t}) \, \mathrm{d} t \right]</math>

provided that ''g'' is a [[bounded function]] and that

:<math>\mathbf{E}^{x} \left[ \int_{0}^{\tau_{D}} \big| f(X_{t}) \big| \, \mathrm{d} t \right] < + \infty.</math>

It turns out that one further condition is required:

:<math>\mathbf{P}^{x} \big[ \tau_{D} < + \infty \big] = 1 \mbox{ for all } x \in D,</math>

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

:<math>u(x) = \mathbf{E}^{x} \left[ g \big( X_{\tau_{D}} \big) \right] + \mathbf{E}^{x} \left[ \int_{0}^{\tau_{D}} f(X_{t}) \, \mathrm{d} t \right]</math>

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

:<math>\begin{cases} - \mathcal{A} u(x) = f(x), & x \in D; \\ \displaystyle{\lim_{t \uparrow \tau_{D}} u(X_{t})} = g \big( X_{\tau_{D}} \big), & \mathbf{P}^{x} \mbox{-a.s., for all } x \in D. \end{cases} \quad \mbox{(P2)}</math>

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

:<math>| v(x) | \leq C \left( 1 + \mathbf{E}^{x} \left[ \int_{0}^{\tau_{D}} \big| g(X_{s}) \big| \, \mathrm{d} s \right] \right),</math>

then ''v'' = ''u''.

==References==

* {{cite journal
| doi = 10.3792/pia/1195572706
| last = Kakutani
| first = Shizuo
|authorlink= Shizuo Kakutani
| title = Two-dimensional Brownian motion and harmonic functions
| journal = Proc. Imp. Acad. Tokyo
| volume = 20
| issue = 10
| year = 1944
| pages = 706–714
}}
* {{cite journal
| doi = 10.3792/pia/1195572742
| last = Kakutani
| first = Shizuo
|authorlink= Shizuo Kakutani
| title = On Brownian motions in ''n''-space
| journal = Proc. Imp. Acad. Tokyo
| volume = 20
| issue = 9
| 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
| publisher=Springer
| location = Berlin
| year = 2003
| isbn = 3-540-04758-1
}} (See Section 9)

[[Category:Boundary conditions]]
[[Category:Partial differential equations]]
[[Category:Stochastic differential equations]]

Eikonal approximation - Revision history

128.15.195.87: /* Formal description */