Differentiability of the transition semigroup of the stochastic Burgers-Huxley equation and application to optimal control

In this present paper we consider the transition semigroup {Pt}t & GE;0 related to the stochastic Burgers-Huxley equation which describes a prototype model for describing the interaction between reaction mechanisms, convection effects and diffusion transports. We are proving that it has a regularizing effect in the Banach space of continuous functions C(0, 1), that is, the transition semigroup {Pt}t & GE;0 possesses strong Feller property. Some estimates on derivative of the transition semigroup related to stochastic Burgers-Huxley equation are established based on the exponential moment estimates of the solution of stochastic Burgers-Huxley equation and its first variation equation. Finally, certain application in Hamilton-Jacobi-Bellman equation arising from stochastic optimal control problem is provided to illustrate our results.(c) 2022 Published by Elsevier Inc.

Automated summary

This paper considers the transition semigroup related to the stochastic Burgers-Huxley equation, which describes the interaction between reaction mechanisms, convection effects, and diffusion transports. It is proved that the transition semigroup possesses a regularizing effect in the Banach space of continuous functions and some estimates on the derivative of the transition semigroup are established. An application in the Hamilton-Jacobi-Bellman equation arising from a stochastic optimal control problem is provided.