Large deviation principle for certain spatially lifted Gaussian rough path

TL;DR

This paper establishes a large deviation principle for the spatial lift of Gaussian processes solving stochastic heat equations, advancing the understanding of rough path analysis in stochastic PDEs.

Contribution

It proves a Schilder type large deviation principle for the spatial lift of Gaussian processes in the context of rough stochastic PDEs, specifically for solutions of the stochastic heat equation.

Findings

Law of the spatial lift satisfies a large deviation principle

Provides a rigorous mathematical framework for rough path analysis in stochastic PDEs

Enhances understanding of probabilistic behavior of Gaussian rough paths

Abstract

In rough stochastic PDE theory of Hairer type, rough path lifts with respect to the space variable of two-parameter continuous Gaussian processes play a main role. A prominent example of such processes is the solution of the stochastic heat equation under the periodic condition. The main objective of this paper is to show that the law of the spatial lift of this process satisfies a Schilder type large deviation principle on the continuous path space over a geometric rough path space.