Local Operators, Regular Sets, and Evolution Equations of Diffusion Type

Abstract We consider operator semigroups generated by Feller processes killed upon leaving a given domain. These semigroups correspond to Cauchy–Dirichlet type initial-exterior value problems in this domain for a class of evolution equations with (possibly non-local) operators. The considered semigroups are approximated by means of the Chernoff theorem. For a class of killed Feller processes, the constructed Chernoff approximation leads to a representation of the solution of the corresponding Cauchy–Dirichlet type problem by a Feynman formula, i.e. by a limit of n-fold iterated integrals of certain functions as n → ∞. Feynman formulae can be used for direct calculations, modelling of underlying dynamics, simulation of underlying stochastic processes. Further, a method to approximate solutions of time-fractional evolution equations is suggested. The method is based on connections between time-fractional and time-non-fractional evolution equations as well as on Chernoff approximations for the latter ones. This method leads to Feynman formulae for solutions of time-fractional evolution equations. A class of distributed order time-fractional equations is considered; Feynman formulae for solutions of the corresponding Cauchy and Cauchy–Dirichlet type problems are obtained.

Download Full-text

Random Evolution Equations: Well-Posedness, Asymptotics, and Applications to Graphs

Applied Mathematics & Optimization ◽

10.1007/s00245-020-09732-w ◽

2021 ◽

Author(s):

Stefano Bonaccorsi ◽

Francesca Cottini ◽

Delio Mugnolo

Keyword(s):

Evolution Equations ◽

Differential Operators ◽

Sufficient Conditions ◽

Asymptotic Behavior Of Solutions ◽

Difference Operators ◽

Well Posedness ◽

Metric Graphs ◽

Diffusion Type ◽

Convergence In Norm ◽

Combinatorial Graphs

AbstractWe study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for pathwise convergence in norm of the (random) propagator towards a (deterministic) steady state. We apply our findings in two environments with randomly evolving features: ensembles of difference operators on combinatorial graphs, or else of differential operators on metric graphs.

Download Full-text