Solution theory to semilinear stochastic equations of Schr\"odinger type on curved spaces I -- Operators with uniformly bounded coefficients

TL;DR

This paper establishes existence and uniqueness of mild solutions for semilinear Schrödinger-type stochastic PDEs on curved spaces with bounded coefficients, expanding the theoretical understanding of such equations.

Contribution

It provides new conditions ensuring well-posedness of Schrödinger stochastic PDEs on curved spaces with bounded coefficients, a novel extension in the field.

Findings

Unique mild solutions exist under specified conditions.

Conditions on coefficients and spectral measure ensure well-posedness.

The framework applies to curved spaces with bounded coefficients.

Abstract

We study the Cauchy problem for Schr\"odinger type stochastic partial differential equations with uniformly bounded coefficients on a curved space. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy data, and on the spectral measure associated with the noise, such that the Cauchy problem admits a unique function-valued mild solution in the sense of Da Prato and Zabczyc.