# Schauder-type estimates for higher-order parabolic SPDEs

## TL;DR

This paper establishes Schauder-type estimates for higher-order parabolic SPDEs, revealing that solution regularity depends on a novel coercivity condition that varies with the order's parity, and proves existence and uniqueness in stochastic Hölder spaces.

## Contribution

It introduces a new coercivity condition for higher-order parabolic SPDEs, showing how solution regularity depends on parity and integrability, with proofs of existence and uniqueness.

## Key findings

- Regularity of solutions depends on a parity-based coercivity condition.
- Established Schauder estimates for solutions and derivatives up to order 2m.
- Demonstrated the sharpness of the coercivity condition with an example.

## Abstract

In this paper we consider the Cauchy problem for $2m$-order stochastic partial differential equations of parabolic type in a class of stochastic Hoelder spaces. The Hoelder estimates of solutions and their spatial derivatives up to order $2m$ are obtained, based on which the existence and uniqueness of solution is proved. An interesting finding of this paper is that the regularity of solutions relies on a coercivity condition that differs when $m$ is odd or even: the condition for odd $m$ coincides with the standard parabolicity condition in the literature for higher-order stochastic partial differential equations, while for even $m$ it depends on the integrability index $p$. The sharpness of the new-found coercivity condition is demonstrated by an example.