Regularity of solutions of abstract linear evolution equations

TL;DR

This paper investigates the regularity properties of solutions to linear evolution equations in Banach and Hilbert spaces, including cases with additive noise, and applies findings to a neurophysiological model.

Contribution

It provides new regularity results for linear evolution equations with sectorial operators, including stochastic cases, and applies these to a neurophysiology model.

Findings

Regularity results for deterministic evolution equations with sectorial operators.

Extension of regularity results to stochastic evolution equations with additive noise.

Application of theoretical results to a neurophysiological model.

Abstract

In this paper, we study regularity of solutions to linear evolution equations of the form $dX+AXdt=F(t)dt$ in a Banach space $H$, where $A$ is a sectorial operator in $H$ and $A^{-\alpha} F \, (\alpha>0)$ belongs to a weighted H\"{o}lder continuous function space. Similar results are obtained for linear evolution equations with additive noise of the form $dX+AXdt=F(t)dt+G(t)dW(t)$ in a separable Hilbert space $H$, where $W(t)$ is a cylindrical Wiener process. Our results are applied to a model arising in neurophysiology, which has been proposed by Walsh \cite{Walsh}.