Solving stochastic equations with unbounded nonlinear perturbations

TL;DR

This paper develops a framework for solving semilinear stochastic equations with unbounded nonlinear perturbations, establishing well-posedness, regularity, and the Feller property, with applications to heat, Schrödinger, and neutral stochastic equations.

Contribution

It introduces a novel approach using Yosida extensions to define mild solutions for equations with unbounded perturbations and non-analytic semigroups, expanding the class of solvable stochastic equations.

Findings

Proved regularity of stochastic convolution with unbounded operators.

Established well-posedness and Feller property for the equations.

Provided examples including heat and Schrödinger equations with nonlocal perturbations.

Abstract

This paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the chosen type of unbounded perturbations. To overcome this problem, we first proved a regularity property of the stochastic convolution with respect to the domain of "admissible" unbounded linear operators (not necessarily closed or closable). This is done using Yosida extensions of such unbounded linear operators. After proving the well-posedness of these equations, we also establish the Feller property for the corresponding transition semigroups. Several examples like heat equations and schr\"odinger equations with nonlocal perturbations terms are given. Finally, we give an application to a general class of semilinear neutral stochastic equations.