Harnack and Shift Harnack Inequalities for Degenerate (Functional) Stochastic Partial Differential Equations with Singular Drifts

Abstract

The existence and uniqueness of the mild solutions for a class of degenerate functional stochastic partial differential equations (SPDEs) are obtained, where the drift is assumed to be Holder–Dini continuous. Moreover, the non-explosion of the solution is proved under some reasonable conditions. In addition, the Harnack inequality is derived by the method of coupling by change of measure. Finally, the shift Harnack inequality is obtained for the equations without delay, which is new even in the non-degenerate case. An example is presented in the final part of the paper.