Schauder estimates for stationary and evolution equations associated to stochastic reaction-diffusion equations driven by colored noise

Abstract

We consider stochastic reaction-diffusion equations with colored noise on the space of real-valued and continuous functions on a compact subset of ℝ^d for $d=1,2,3$. We prove Schauder-type estimates, which will depend on the color of the noise, for the stationary and evolution problems associated with the corresponding transition semigroup.

Keywords:

• Colored noise
• evolution equations
• infinite dimensional analysis
• Schauder estimates
• stationary equations
• stochastic reaction-diffusion equations

2020 Mathematics Subject Classification:

• 35R15
• 47D06
• 47D07
• 60H30

Acknowledgments

The authors would like to express their gratitude to the anonymous referees for the meticulous review of the manuscript and for providing numerous valuable comments and suggestions that have significantly improved the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Fundings

The authors are members of GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of the Italian Istituto Nazionale di Alta Matematica (INdAM).