Harnack and shift Harnack inequalities for degenerate (functional) stochastic partial differential equations with singular drifts. (English) Zbl 1483.60084
Summary: 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 Hölder-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.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60E15 | Inequalities; stochastic orderings |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 34K26 | Singular perturbations of functional-differential equations |
| 39B72 | Systems of functional equations and inequalities |
Keywords:
Hölder-Dini continuous; degenerate SPDEs; Zvonkin-type transform; functional SPDEs; Harnack inequalitiesReferences:
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