Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients
Stefan Tappe
TL;DR
The paper addresses existence and uniqueness of mild solutions for semilinear SPDEs with locally monotone and coercive coefficients driven by a cylindrical Wiener process, allowing space-, time-, and randomness-dependent coefficients. It introduces the moving frame method and Nagy dilation to convert the SPDE into an infinite-dimensional SDE on an enlarged Hilbert space, enabling a variational-analytic treatment and derivation of quantitative solution bounds. A key result is a comprehensive existence-uniqueness theorem with moment estimates and, in the deterministic-coefficient case, Lipschitz continuity of the solution map. Additionally, the authors establish the Markov property for solutions with nonrandom coefficients by transferring the Markov structure from the associated SDE in the enlarged space back to the original SPDE, ensuring a coherent semigroup framework. The work broadens SPDE theory beyond global Lipschitz conditions and provides a robust toolkit for analyzing stochastic dynamics with locally monotone nonlinearities.
Abstract
We provide an existence and uniqueness result for mild solutions to semilinear stochastic partial differential equations in the framework of the semigroup approach with locally monotone coefficients. An important component of the proof is an application of the dilation theorem of Nagy, which allows us to reduce the problem to infinite dimensional stochastic differential equations on a larger Hilbert space. Properties of the solutions like the Markov property are discussed as well.