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<h2 id="introduction-kakutanis-solution-to-the-classical-dirichlet-problem">Introduction: Kakutani's solution to the classical Dirichlet problem</h2> <p>Let D {\displaystyle D} be a domain (an <a href="/facts/Open_set/QfTCIqMu">open</a> and <a href="/facts/Connected_space/5CUTxfoA">connected set</a>) in R n {\textstyle \mathbb {R} ^{n}} . Let Δ {\displaystyle \Delta } be the <a href="/facts/Laplace_operator/kov9u3PT">Laplace operator</a>, let g {\displaystyle g} be a <a href="/facts/Bounded_function/whVjbtuk">bounded function</a> on the <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a> ∂ D {\displaystyle \partial D} , and consider the problem: </p> { − Δ u ( x ) = 0 , x ∈ D lim y → x u ( y ) = g ( x ) , x ∈ ∂ D {\displaystyle {\begin{cases}-\Delta u(x)=0,&x\in D\\\displaystyle {\lim _{y\to x}u(y)}=g(x),&x\in \partial D\end{cases}}} <p>It can be shown that if a solution u {\displaystyle u} exists, then u ( x ) {\displaystyle u(x)} is the <a href="/facts/Expected_value/1XV0JKL8">expected value</a> of g ( x ) {\displaystyle g(x)} at the (random) first exit point from D {\displaystyle D} for a canonical <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a> starting at x {\displaystyle x} . See theorem 3 in Kakutani 1944, p. 710. </p> <h2 id="the-dirichletpoisson-problem">The Dirichlet–Poisson problem</h2> <p>Let D {\displaystyle D} be a domain in R n {\textstyle \mathbb {R} ^{n}} and let L {\displaystyle L} be a semi-elliptic differential operator on C 2 ( R n ; R ) {\textstyle C^{2}(\mathbb {R} ^{n};\mathbb {R} )} of the form: </p> L = ∑ i = 1 n b i ( x ) ∂ ∂ x i + ∑ i , j = 1 n a i j ( x ) ∂ 2 ∂ x i ∂ x j {\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}}}} <p>where the coefficients <i> b i {\displaystyle b_{i}} </i> and <i> a i j {\displaystyle a_{ij}} </i> are <a href="/facts/Continuous_function/sKbl02pB">continuous functions</a> and all the <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> of the <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> <i> α ( x ) = a i j ( x ) {\displaystyle \alpha (x)=a_{ij}(x)} </i> are non-negative. Let <i> f ∈ C ( D ; R ) {\textstyle f\in C(D;\mathbb {R} )} </i> and <i> g ∈ C ( ∂ D ; R ) {\textstyle g\in C(\partial D;\mathbb {R} )} </i>. Consider the <a href="/facts/Poisson_equation/Pr8zkCXB">Poisson problem</a>: </p> { − L u ( x ) = f ( x ) , x ∈ D lim y → x u ( y ) = g ( x ) , x ∈ ∂ D (P1) {\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)}}} <p>The idea of the stochastic method for solving this problem is as follows. First, one finds an <a href="/facts/It%C5%8D_diffusion/vlwAqWSU">Itō diffusion</a> X {\displaystyle X} whose <a href="/facts/Infinitesimal_generator_(stochastic_processes)/EUHczpU5">infinitesimal generator</a> A {\displaystyle A} coincides with L {\displaystyle L} on <a href="/facts/Compact_support/BTuMGbsb">compactly-supported</a> C 2 {\displaystyle C^{2}} functions f : R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } . For example, X {\displaystyle X} can be taken to be the solution to the stochastic differential equation: </p> d X t = b ( X t ) d t + σ ( X t ) d B t {\displaystyle \mathrm {d} X_{t}=b(X_{t})\,\mathrm {d} t+\sigma (X_{t})\,\mathrm {d} B_{t}} <p>where B {\displaystyle B} is <i>n</i>-dimensional Brownian motion, <i> b {\displaystyle b} </i> has components <i> b i {\displaystyle b_{i}} </i> as above, and the <a href="/facts/Tensor_field/cAC6F6rF">matrix field</a> <i> σ {\displaystyle \sigma } </i> is chosen so that: </p> 1 2 σ ( x ) σ ( x ) ⊤ = a ( x ) , ∀ x ∈ R n {\displaystyle {\frac {1}{2}}\sigma (x)\sigma (x)^{\top }=a(x),\quad \forall x\in \mathbb {R} ^{n}} <p>For a point x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} , let P x {\displaystyle \mathbb {P} ^{x}} denote the law of X {\displaystyle X} given initial datum X 0 = x {\displaystyle X_{0}=x} , and let E x {\displaystyle \mathbb {E} ^{x}} denote expectation with respect to P x {\displaystyle \mathbb {P} ^{x}} . Let <i> τ D {\displaystyle \tau _{D}} </i> denote the first exit time of X {\displaystyle X} from D {\displaystyle D} . </p><p>In this notation, the candidate solution for (P1) is: </p> u ( x ) = E x [ g ( X τ D ) ⋅ χ { τ D < + ∞ } ] + E x [ ∫ 0 τ D f ( X t ) d t ] {\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]} <p>provided that g {\displaystyle g} is a <a href="/facts/Bounded_function/whVjbtuk">bounded function</a> and that: </p> E x [ ∫ 0 τ D | f ( X t ) | d t ] < + ∞ {\displaystyle \mathbb {E} ^{x}\left[\int _{0}^{\tau _{D}}{\big |}f(X_{t}){\big |}\,\mathrm {d} t\right]<+\infty } <p>It turns out that one further condition is required: </p> P x ( τ D < ∞ ) = 1 , ∀ x ∈ D {\displaystyle \mathbb {P} ^{x}{\big (}\tau _{D}<\infty {\big )}=1,\quad \forall x\in D} <p>For all x {\displaystyle x} , the process X {\displaystyle X} starting at x {\displaystyle x} <a href="/facts/Almost_surely/fWaxzpBA">almost surely</a> leaves D {\displaystyle D} in finite time. Under this assumption, the candidate solution above reduces to: </p> u ( x ) = E x [ g ( X τ D ) ] + E x [ ∫ 0 τ D f ( X t ) d t ] {\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]} <p>and solves (P1) in the sense that if A {\displaystyle {\mathcal {A}}} denotes the characteristic operator for X {\displaystyle X} (which agrees with A {\displaystyle A} on C 2 {\displaystyle C^{2}} functions), then: </p> { − A u ( x ) = f ( x ) , x ∈ D lim t ↑ τ D u ( X t ) = g ( X τ D ) , P x -a.s., ∀ x ∈ D (P2) {\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)}}} <p>Moreover, if v ∈ C 2 ( D ; R ) {\textstyle v\in C^{2}(D;\mathbb {R} )} satisfies (P2) and there exists a constant C {\displaystyle C} such that, for all x ∈ D {\displaystyle x\in D} : </p> | v ( x ) | ≤ C ( 1 + E x [ ∫ 0 τ D | g ( X s ) | d s ] ) {\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)} <p>then v = u {\displaystyle v=u} . </p> <ul><li><a href="/facts/Shizuo_Kakutani/WGLekK88">Kakutani, Shizuo</a> (1944). <a href="https://doi.org/10.3792%2Fpia%2F1195572706">"Two-dimensional Brownian motion and harmonic functions"</a>. <i>Proc. Imp. Acad. Tokyo</i>. 20 (10): 706–714. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.3792%2Fpia%2F1195572706">10.3792/pia/1195572706</a>.</li> <li><a href="/facts/Shizuo_Kakutani/WGLekK88">Kakutani, Shizuo</a> (1944). <a href="https://doi.org/10.3792%2Fpia%2F1195572742">"On Brownian motions in <i>n</i>-space"</a>. <i>Proc. Imp. Acad. Tokyo</i>. 20 (9): 648–652. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.3792%2Fpia%2F1195572742">10.3792/pia/1195572742</a>.</li> <li><a href="/facts/Bernt_%C3%98ksendal/Au4oSbnG">Øksendal, Bernt K.</a> (2003). <i>Stochastic Differential Equations: An Introduction with Applications</i> (Sixth ed.). Berlin: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-04758-1. (See Section 9)</li></ul>