Study on the Well-Posedness of Solutions to Stochastic Evolution Equations Based on Stochastic Convolution

Abstract

This paper conducts an in-depth study on the well-posedness of solutions to stochastic evolution equations based on stochastic convolutions. By constructing appropriate operator semigroups and integrating Bochner-Itintegral theory, a systematic analysis of the properties
of stochastic convolutions is performed. Using the Burkholder-Davis-Gundy inequality, the contraction properties of semigroups, and Sobolev
space theory, the existence and uniqueness of mild solutions to stochastic evolution equations are rigorously proven in spaces, providing a
solid mathematical foundation for theoretical research and practical applications in related fields.