Approximate calculation of operator semigroups by perturbation of generators

TL;DR

This paper develops methods to approximate operator semigroups by perturbing their generators, deriving integral equations and proving continuous dependence, with applications to random walks on integers.

Contribution

It introduces a new approach to approximate semigroups via generator perturbations, including integral equations and a continuous dependence theorem.

Findings

Derived two integral equations for perturbed semigroups

Proved a theorem on continuous dependence of semigroups on generators

Applied results to random walk on integers

Abstract

Let $\Omega$ be an operator semigroup with generator $A$ in a sequentially complete locally convex topological vector space $E$. For a semigroup with generator $A+D$, where $D$ is a bounded linear operator on $E$, two integral equations are derived. A theorem on continuous dependence of a semigroup on its generator is proved. An application to random walk on $\mathbb{Z}$ is given.