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Inmathematics, someboundary value problems can be solved using the methods ofstochastic analysis. Perhaps the most celebrated example isShizuo Kakutani's 1944 solution of theDirichlet problem for theLaplace operator usingBrownian motion.^[1] However, it turns out that for a large class ofsemi-elliptic second-orderpartial differential equations the associated Dirichlet boundary value problem can be solved using anItō process that solves an associatedstochastic differential equation.

The link between semi-elliptic operators and stochastic processes, followed by their use to solve boundary value problems, is repeatedly and independently rediscovered in the early-mid-20th century.

The connection that Kakutani makes between stochastic differential equations and the Itō process is effectively the same asKolmogorov's forward equation, made in 1931, which is only later recognized as theFokker–Planck equation, first presented in 1914-1917. The solution of a boundary value problem by means of expectation values over stochastic processes is now more commonly known not under Kakutani's name, but as theFeynman–Kac formula, developed in 1947.

These results are founded on the use of theItō integral, required to integrate a stochastic process. But this is also independently rediscovered as theStratonovich integral; the two forms can be translated into one-another by an offset.

Let${\displaystyle D}$ be a domain (anopen andconnected set) in${\mathbb{R}}^n$. Let${\displaystyle \mathrm{\Delta }}$ be theLaplace operator, let${\displaystyle g}$ be abounded function on theboundary${\displaystyle \mathrm{\partial }D}$, and consider the problem:

It can be shown that if a solution${\displaystyle u}$ exists, then${\displaystyle u(x)}$ is theexpected value of${\displaystyle g(x)}$ at the (random) first exit point from${\displaystyle D}$ for a canonicalBrownian motion starting at${\displaystyle x}$. See theorem 3 in Kakutani 1944, p. 710.

Let${\displaystyle D}$ be a domain in${\mathbb{R}}^n$ and let${\displaystyle L}$ be a semi-ellipticdifferential operator on$C^2({\mathbb{R}}^n;\mathbb{R})$ of the form:

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

where the coefficients${\displaystyle b_i}$ and${\displaystyle a_{ij}}$ arecontinuous functions and all theeigenvalues of thematrix${\displaystyle \alpha (x)=a_{ij}(x)}$ are non-negative. Let$f\in C(D;\mathbb{R})$ and$g\in C(\mathrm{\partial }D;\mathbb{R})$. Consider thePoisson problem:

The idea of the stochastic method for solving this problem is as follows. First, one finds anItō diffusion${\displaystyle X}$ whoseinfinitesimal generator${\displaystyle A}$ coincides with${\displaystyle L}$ oncompactly-supported${\displaystyle C^2}$ functions${\displaystyle f:{\mathbb{R}}^n\to \mathbb{R}}$. For example,${\displaystyle X}$ can be taken to be the solution to the stochastic differential equation:

${\displaystyle \mathrm{d}{X}_t=b(X_t)\mathrm{d}t+\sigma (X_t)\mathrm{d}{B}_t}$

where${\displaystyle B}$ isn-dimensional Brownian motion,${\displaystyle b}$ has components${\displaystyle b_i}$ as above, and thematrix field${\displaystyle \sigma }$ is chosen so that:

${\displaystyle \frac{1}{2}\sigma (x)\sigma (x{)}^{\mathrm{\top }}=a(x),\mathrm{\forall }x\in {\mathbb{R}}^n}$

For a point${\displaystyle x\in {\mathbb{R}}^n}$, let${\displaystyle {\mathbb{P}}^x}$ denote the law of${\displaystyle X}$ given initial datum${\displaystyle X_0=x}$, and let${\displaystyle {\mathbb{E}}^x}$denote expectation with respect to${\displaystyle {\mathbb{P}}^x}$. Let${\displaystyle {\tau }_D}$ denote thefirst exit time of${\displaystyle X}$ from${\displaystyle D}$.

In this notation, thecandidate solution for (P1) is:

provided that${\displaystyle g}$ is abounded function and that:

It turns out that one further condition is required:

For all${\displaystyle x}$, the process${\displaystyle X}$ starting at${\displaystyle x}$almost surely leaves${\displaystyle D}$ in finite time. Under this assumption, the candidate solution above reduces to:

${\displaystyle u(x)={\mathbb{E}}^x\left[g(X_{{\tau }_D})\right]+{\mathbb{E}}^x\left[{\int }_0^{{\tau }_D}f(X_t)\mathrm{d}t\right]}$

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

Moreover, if$v\in C^2(D;\mathbb{R})$ satisfies (P2) and there exists a constant${\displaystyle C}$ such that, for all${\displaystyle x\in D}$:

${\displaystyle |v(x)|\le C\left(1+{\mathbb{E}}^x\left[{\int }_0^{{\tau }_D}|g(X_s)|\mathrm{d}s\right]\right)}$

then${\displaystyle v=u}$.

• 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 inn-space".Proc. Imp. Acad. Tokyo.20 (9):648–652.doi:10.3792/pia/1195572742.

• Boundary value problems
• Partial differential equations
• Stochastic differential equations

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