Stability of Solutions to Semilinear Exterior Dirichlet Problems under Convergence of Lévy Symbols
Abstract
We study stability of semilinear exterior Dirichlet problems driven by possibly nonsymmetric Levy operators. Let Ln and L have Lévy-Khintchine symbols ψn and ψ, respectively. Our basic assumption on the operators is that ψn(ξ)→ψ(ξ) for every ξ∈ℝd. Under mild regularity assumptions on the domain and an absolute-continuity condition for the resolvent of the limiting process, we prove pointwise and Lp-stability of solutions. No symmetry, uniform ellipticity, comparability of Lévy 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 Lévy 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 C0 domains and give a counterexample showing that whole-space Mosco convergence need not pass to part forms on arbitrary open sets.
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BibTeX
@misc{klimsiak2026stability,
title = {{Stability of Solutions to Semilinear Exterior Dirichlet Problems under Convergence of Lévy Symbols}},
author = {Tomasz Klimsiak and Andrzej Rozkosz and Leszek Słomiński},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07963},
eprint = {2610.07963},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07963},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stability of Solutions to Semilinear Exterior Dirichlet Problems under Convergence of Lévy Symbols
AU - Tomasz Klimsiak
AU - Andrzej Rozkosz
AU - Leszek Słomiński
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07963
UR - https://arxiv.org/abs/2610.07963
AB - We study stability of semilinear exterior Dirichlet problems driven by possibly nonsymmetric Levy operators. Let L^n and L have Lévy-Khintchine symbols ψ_n and ψ, respectively. Our basic assumption on the operators is that ψ_n(ξ)→ψ(ξ) for every ξ∈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 Lévy 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 Lévy 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.
ER -