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: $${\partial }_{t}u={\mathcal{L}}_{\kappa }^{(\alpha )}u+b\cdot \nabla u+f\mathrm{in}{\mathbb{R}}_{+}\times D,u{|}_{{\mathbb{R}}_{+}\times D^c}=0,u(0,\cdot ){|}_D=\phi ,$$ where $\alpha \in (0,2)$ and ${\mathcal{L}}_{\kappa }^{(\alpha )}$ 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 $\gamma \in ((1-\alpha )\vee 0,\beta ]$, $\phi \in C^{\alpha +\gamma }(D)$. Under some Hölder assumptions on $\kappa$, we show the existence of a unique classical solution $u\in L_{loc}^{\infty }({\mathbb{R}}_{+};C_{loc}^{\alpha +\gamma }(D))\times C({\mathbb{R}}_{+};C_b(D))$ to the above problem. Moreover, we establish the following probabilistic representation for $u$ $$u(t,x)={\mathbb{E}}_x(\phi (X_t)1_{{\tau }_D>t})+{\mathbb{E}}_x\left({\int }_0^{t\wedge {\tau }_D}f(t-s,X_s)\mathrm{d}s\right),t\ge 0,x\in D,$$ where $((X_t{)}_{t\ge 0},{\mathbb{P}}_x;x\in {\mathbb{R}}^d)$ is the Markov process associated with the operator ${\mathcal{L}}_{\kappa }^{(\alpha )}+b\cdot \nabla$, 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.

50 pages, 6 figures

References in corpus (2)

• A new Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation
• Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts

Cited by in corpus (6)

• Nonlocal elliptic equation in Hölder space and the martingale problem
• Dynamical behavior of a nonlocal Fokker-Planck equation for a stochastic system with tempered stable noise
• Schauder's estimate for nonlocal kinetic equations and its applications
• Schauder's estimates for nonlocal equations with singular Lévy measures
• Heat kernel of supercritical SDEs with unbounded drifts
• Regularity properties of jump diffusions with irregular coefficients