Stochastic Partial Differential Equations Associated with Pseudo-Differential Operators and Hilbert Space-Valued Gaussian Processes

TL;DR

The paper develops a comprehensive $L^{p}$-regularity theory for stochastic PDEs in Hilbert spaces driven by Hilbert-space-valued Gaussian processes, using pseudo-differential operators as the spatial generator. It combines Malliavin calculus for $Q$-Gaussian noises with generalized Littlewood–Paley and Fourier multiplier techniques to obtain existence, uniqueness, and sharp $L^{p}$-regularity results for SPDE solutions in Banach-space-valued function spaces. The main contributions include a $p$-th moment maximal inequality for Skorohod integrals with respect to $Q$-Gaussian processes, a vector-valued Littlewood–Paley inequality, and a robust SPDE well-posedness theory under precise covariance kernel conditions (R1)–(R2). The framework is validated through concrete covariance kernel examples, such as $Q$-Wiener and $Q$-fBm, demonstrating practical applicability to a broad class of infinite-dimensional Gaussian noises. These results extend the SPDE theory to settings with general pseudo-differential operators and Hilbert-space-valued noises, with implications for quantitative regularity and stability analyses in applied contexts.

Abstract

In this paper, we prove the unique existence and investigate the $L^{p}$-regularity of solutions to stochastic partial differential equations in Hilbert spaces associated with pseudo-differential operators, driven by Hilbert space-valued Gaussian processes that satisfy certain regularity conditions for the covariance kernels of the Gaussian processes. For our purposes, we develop an $L^{p}$-regularity framework for the solutions to the stochastic partial differential equations associated with pseudo-differential operators. As the main tools, we establish the $p$-th moment maximal inequality for stochastic integrals with respect to a Hilbert space-valued Gaussian process and a Littlewood-Paley type inequality for Banach space-valued functions. Additionally, during our study, we improved the sufficient conditions for Fourier multipliers and examined the covariance kernels for Gaussian processes.

Theorems & Definitions (54)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Lemma 2.5
• Proposition 2.6
• Proposition 2.7
• Proposition 2.8
• Remark 2.9
• Theorem 2.10