Stability of Solutions to Semilinear Exterior Dirichlet Problems under Convergence of L\'evy Symbols

Abstract

We study stability of semilinear exterior Dirichlet problems driven by possibly nonsymmetric Levy operators. Let $L^n$ and $L$ have L\'evy-Khintchine symbols $\psi_n$ and $\psi$, respectively. Our basic assumption on the operators is that $\psi_n(\xi)\to\psi(\xi)$ for every $\xi\in\mathbb R^d$. Under mild regularity assumptions on the domain and an absolute-continuity condition for the resolvent of the limiting process, we prove pointwise and $L^p$-stability of solutions. No symmetry, uniform ellipticity, comparability of L\'evy kernels, or common energy space is required; nonzero exterior data are allowed; and the monotone nonlinearity is subject to no growth restriction in the solution variable. The framework therefore covers local, nonlocal and mixed operators and, in particular, nonlocal-to-local limits, as well as fractional, relativistic, anisotropic and nonsymmetric stable-type operators. The proofs combine convergence of L\'evy processes with a detailed analysis of exit times and exit positions and a stability theorem for monotone backward stochastic differential equations with varying terminal times. In the symmetric case, we also obtain Mosco convergence of the corresponding part forms on $C^0$ domains and give a counterexample showing that whole-space Mosco convergence need not pass to part forms on arbitrary open sets.