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Stochastic processes and boundary value problems - Wikiwand

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

{\displaystyle 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 {\displaystyle b_{i}} and {\displaystyle a_{ij}} are continuous functions and all the eigenvalues of the matrix {\displaystyle \alpha (x)=a_{ij}(x)} are non-negative. Let {\textstyle f\in C(D;\mathbb {R} )} and {\textstyle g\in C(\partial D;\mathbb {R} )}. Consider the Poisson problem:

{\displaystyle {\begin{cases}-Lu(x)=f(x),&x\in D\\\displaystyle {\lim _{y\to 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 {\displaystyle X} whose infinitesimal generator {\displaystyle A} coincides with {\displaystyle L} on compactly-supported {\displaystyle C^{2}} functions {\displaystyle f:\mathbb {R} ^{n}\rightarrow \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} is n-dimensional Brownian motion, {\displaystyle b} has components {\displaystyle b_{i}} as above, and the matrix field {\displaystyle \sigma } is chosen so that:

{\displaystyle {\frac {1}{2}}\sigma (x)\sigma (x)^{\top }=a(x),\quad \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 the first exit time of {\displaystyle X} from {\displaystyle D}.

In this notation, the candidate solution for (P1) is:

{\displaystyle u(x)=\mathbb {E} ^{x}\left[g{\big (}X_{\tau _{D}}{\big )}\cdot \chi _{\{\tau _{D}<+\infty \}}\right]+\mathbb {E} ^{x}\left[\int _{0}^{\tau _{D}}f(X_{t})\,\mathrm {d} t\right]}

provided that {\displaystyle g} is a bounded function and that:

{\displaystyle \mathbb {E} ^{x}\left[\int _{0}^{\tau _{D}}{\big |}f(X_{t}){\big |}\,\mathrm {d} t\right]<+\infty }

It turns out that one further condition is required:

{\displaystyle \mathbb {P} ^{x}{\big (}\tau _{D}<\infty {\big )}=1,\quad \forall x\in D}

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{\big (}X_{\tau _{D}}{\big )}\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:

{\displaystyle {\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 )},&\mathbb {P} ^{x}{\mbox{-a.s.,}}\;\forall x\in D\end{cases}}\quad {\mbox{(P2)}}}

Moreover, if {\textstyle 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)|\leq C\left(1+\mathbb {E} ^{x}\left[\int _{0}^{\tau _{D}}{\big |}g(X_{s}){\big |}\,\mathrm {d} s\right]\right)}

then {\displaystyle v=u}.