Well-posedness and Optimal Regularity of Stochastic Evolution Equations with Multiplicative Noises

TL;DR

This paper proves the well-posedness and optimal regularity of solutions to stochastic evolution equations with broad classes of multiplicative noises, including white and rough noises, under minimal assumptions on the linear operator.

Contribution

It establishes well-posedness and optimal regularity results for stochastic evolution equations driven by general multiplicative noises with minimal operator assumptions.

Findings

Solutions are well-posed under generalized Lipschitz conditions.

Optimal trajectory regularity is achieved for equations with analytic semigroup generators.

The results include cases with space-time white noise and rougher noises.

Abstract

In this paper, we establish the well-posedness and optimal trajectory regularity for the solution of stochastic evolution equations with generalized Lipschitz-type coefficients driven by general multiplicative noises. To ensure the well-posedness of the problem, the linear operator of the equations is only need to be a generator of a $\CC_0$-semigroup and the proposed noises are quite general, which include space-time white noise and rougher noises. When the linear operator generates an analytic $\CC_0$-semigroup, we derive the optimal trajectory regularity of the solution through a generalized criterion of factorization method.