Stability of solutions of semilinear evolution equations with integro-differential operators
Andrzej Rozkosz, Leszek Słomiński
TL;DR
The work develops a general stability framework for semilinear evolution equations with Lévy-type and Lévy operators by leveraging probabilistic representations via BSDEs. By proving stability of BSDEs under weak filtration convergence and forward-process convergence, the authors show that solutions u_n corresponding to L^n converge to a limit u when the symbols converge. They further relate probabilistic solutions to weak and renormalized PDE notions under Hartman–Wintner density conditions, establishing convergence in pointwise, L^1, and capacity-based senses for a broad class of data and operator families. The results are illustrated through concrete examples including fractional and relativistic Lévy operators and state-dependent Lévy-type operators, highlighting practical convergence guarantees for semilinear PDEs with nonlocal terms.
Abstract
We consider solutions of the Cauchy problem for semilinear equations with (possibly) different Lévy operators. We provide various results on their convergence under the assumption that symbols of the involved operators converge to the symbol of some Lévy operator. Some results are proved for a more general class of pseudodifferential operators.