Dirichlet problem for supercritical nonlocal operators

Abstract

Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(\alpha)}_\kappa u+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \ u|_{\mathbb R_+\times D^c}=0,\ u(0,\cdot)|_{D}=\varphi, $$ where $\alpha\in(0,2)$ and $\mathscr L^{(\alpha)}_\kappa$ is an $\alpha$-stable-like nonlocal operator with kernel function $\kappa(x,z)$ bounded from above and below by positive constants, and $b:\mathbb R^d\to\mathbb R^d$ is a bounded $C^\beta$-function with $\alpha+\beta>1$, $f: \mathbb R_+\times D\to\mathbb R$ is a $C^\gamma$-function in $D$ uniformly in $t$ with $\gamma\in((1-\alpha)\vee 0,\beta]$, $\varphi\in C^{\alpha+\gamma}(D)$. Under some H\"older assumptions on $\kappa$, we show the existence of a unique classical solution $u\in L^\infty_{loc}(\mathbb R_+; C^{\alpha+\gamma}_{loc}(D))\times C(\mathbb R_+; C_b(D))$ to the above problem. Moreover, we establish the following probabilistic representation for $u$ $$ u(t,x)=\mathbb E_x \Big(\varphi(X_{t}){\bf 1}_{\tau_{D}>t}\Big)+\mathbb E_x\left(\int^{t\wedge\tau_{D}}_0f(t-s,X_s){\rm d} s\right),\ t\geq 0,\ x\in D, $$ where $((X_t)_{t\geq 0},\mathbb P_x; x\in\mathbb R^d)$ is the Markov process associated with the operator $\mathscr L^{(\alpha)}_\kappa+b\cdot \nabla$, and $\tau_D$ is the first exit time of $X$ from $D$. In the sub and critical case $\alpha\in[1,2)$, the kernel function $\kappa$ can be rough in $z$. In the supercritical case $\alpha\in(0,1)$, we classify the boundary points according to the sign of $b(z)\cdot\vec{n}(z)$, where $z\in\partial D$ and $\vec{n}(z)$ is the unit outward normal vector. Finally, we provide an example and simulate it by Monte-Carlo method to show our results.