Dirichlet problem for diffusions with jumps
Zhen-Qing Chen, Jun Peng
TL;DR
Addresses the Dirichlet problem for a nonlocal operator $L = L^0 + \int (u(y)-u(x)) J(x,dy)$ with $L^0 = \frac{1}{2} \nabla \cdot (A(x) \nabla) + b \cdot \nabla$, under Kato class conditions on $b$ and the jump kernel $J$. The authors construct a unique Feller process $X$ with strong Feller by a piecing-together construction from the diffusion $X^0$ killed at rate $\kappa$, and prove a resolvent relation $(\alpha - L) G_\alpha f = f$. For bounded regular domains $D$, they show that $u(x) = \mathbb E_x[ \varphi(X_{\tau_D}) ]$ gives the unique bounded continuous weak solution to $L u=0$ in $D$ with exterior data $u=\varphi$ on $D^c$, providing a probabilistic representation. The approach avoids Dirichlet form methods due to the lack of sector condition and possible atoms in $J$, and extends to Schrödinger-type perturbations $q$ via similar resolvent techniques.
Abstract
In this paper, we study Dirichlet problem for non-local operator on bounded domains in ${\mathbb R}^d$ $$ {\cal L}u = {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x) + \int_{{\mathbb R}^d} (u(y)-u(x) ) J(x, dy) , $$ where $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on ${\mathbb R}^d$ that is uniformly elliptic and bounded, $b$ is an ${\mathbb R}^d$-valued function so that $|b|^2$ is in some Kato class ${\mathbb K}_d$, for each $x\in {\mathbb R}^d$, $J(x, dy)$ is a finite measure on ${\mathbb R}^d$ so that $x\mapsto J(x, {\mathbb R}^d)$ is in the Kato class ${\mathbb K}_d$. We show there is a unique Feller process $X$ having strong Feller property associated with ${\cal L}$, which can be obtained from the diffusion process having generator $ {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x) $ through redistribution. We further show that for any bounded connected open subset $D\subset{\mathbb R}^d$ that is regular with respect to the Laplace operator $Δ$ and for any bounded continuous function $\varphi $ on $D^c$, the Dirichlet problem ${\cal L} u=0$ in $D$ with $u=\varphi$ on $D^c$ has a unique bounded continuous weak solution on ${\mathbb R}^d$. This unique weak solution can be represented in terms of the Feller process associated with ${\cal L}$.