An integrable bound for semilinear rough partial differential equations with unbounded diffusion coefficients

TL;DR

This work advances the theory of semilinear rough PDEs driven by low-regularity rough paths with unbounded diffusion. By introducing a novel control-function based framework on a monotone Banach scale, the authors derive integrable moment bounds for the mild solution's controlled rough-path norm even when the diffusion is unbounded and time-regularity is as low as $\gamma\in(1/4,1/2)$. The analysis combines a parabolic evolution family with a customized sewing lemma to define rough integrals in infinite dimensions, and it leverages greedy-point techniques alongside Borell’s inequality to obtain Gaussian-tail bounds and, under further assumptions, uniform $L^q$ bounds. These results extend prior bounded-diffusion theories to unbounded linear diffusion and lay groundwork for long-time/dynamical-systems analyses in rough stochastic PDEs. They also provide a probabilistic-dine framework via translations in the Cameron–Martin space, enabling sharp integrable bounds for random input paths such as fractional Brownian motion.

Abstract

This work develops moment bounds for the controlled rough path norm of the solution of semilinear rough partial differential equations.~The novel aspects are two-fold: first we consider rough paths of low time regularity $γ\in(1/4,1/2)$ and second treat unbounded diffusion coefficients. To this aim we introduce a suitable notion of a controlled rough path according to a monotone scale of Banach spaces and innovative control functions.

Theorems & Definitions (60)

• Theorem 3.1
• Lemma 3.2
• Definition 3.3
• Definition 3.4
• Lemma 3.5
• proof
• Definition 3.6
• Remark 3.7
• Definition 3.8
• Lemma 3.9