New Approaches to the Monotonicity Inequality for Linear Stochastic PDEs

TL;DR

This work analyzes the deterministic Monotonicity inequality for a pair of adjoint operators $(L^*,A^*)$ acting on the Schwartz–temper distributions within the Hermite–Sobolev scale. It introduces and implements three explicit strategies—bounded-operator methods for affine multipliers, a Good terms framework with Finite Sums of Good Terms (FSGT) to handle smooth multipliers, and a Three Lines Lemma–based interpolation—to establish the inequality $2\langle\phi,L^*\phi\rangle_p+\|A^*\phi\|_p^2\le C\|\phi\|_p^2$ for $\phi$ in $\mathcal{S}(\mathbb{R})$ across the Hermite–Sobolev family $\{\mathcal{S}_p(\mathbb{R})\}_{p\in\mathbb{R}}$, including complex-valued spaces. The paper develops the complex extensions on $\mathcal{S}_p(\mathbb{R};\mathbb{C})$, establishes adjoint identities such as $\partial^*=-\partial+T$, and proves the Monotonicity inequality for affine $\sigma$ and $b$ for all $p$, with an interpolation-based extension to fractional $p$. Collectively, these results enrich deterministic monotonicity tools for linear SPDEs in dual countably Hilbertian nuclear spaces and provide explicit, verifiable inequalities that support existence and uniqueness analyses of strong solutions.

Abstract

The Monotonicity inequality is an important tool in the understanding of existence and uniqueness of strong solutions for Stochastic PDEs. In this article, we discuss three approaches to establish this deterministic inequality explicitly.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Lemma 1.4
• Lemma 1.5
• Theorem 1.6
• Theorem 1.7: Three Lines Lemma MR304972
• Theorem 1.8
• Proposition 2.1: Properties of the Hermite operator, MR1215939 and MR1999259
• Proposition 2.2: MR1999259