# Harnack and Shift Harnack Inequalities for Degenerate (Functional) SPDEs with Singular Drifts

## TL;DR

This paper establishes Harnack and shift Harnack inequalities for degenerate functional SPDEs with singular, Hölder-Dini continuous drifts, proving existence, uniqueness, and non-explosion of solutions, and deriving new inequalities even in non-degenerate cases.

## Contribution

It introduces novel Harnack and shift Harnack inequalities for a class of degenerate SPDEs with singular drifts, extending existing results to more general cases.

## Key findings

- Proved existence and uniqueness of solutions.
- Established non-explosion under certain conditions.
- Derived new shift Harnack inequality for non-delay equations.

## Abstract

The existence and uniqueness of the mild solutions for a class of degenerate functional SPDEs are obtained, where the drift is assumed to be H\"{o}lder-Dini continuous. Moreover, the non-explosion of the solution is proved under some reasonable conditions. In addition, the Harnack is derived by the 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.