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Malliavin Calculus for the stochastic Cahn-Hilliard equation driven by fractional noise

Dimitrios Dimitriou, Dimitris Farazakis, Georgia Karali

TL;DR

This work analyzes the stochastic Cahn-Hilliard equation in one spatial dimension driven by additive fractional white noise $\dot{W}_{H}$. It combines Malliavin calculus with a careful localization strategy to prove that the mild solution $u(x,t)$ has a Malliavin derivative locally and that its law has a density with respect to Lebesgue measure. Central contributions include constructing a localizing sequence $\{(\Omega_n,u_n)\}$, deriving sharp $L^{\infty}$-norm moment estimates for the stochastic integral and the associated kernel integral, and proving the absolute continuity of the law of $u_n$ and, via localization, of $u$ itself. The results extend density existence for SPDEs driven by fractional noise and provide concrete analytical tools for studying probabilistic properties of the stochastic CH dynamics.

Abstract

The stochastic partial differential equation analyzed in this work is the Cahn-Hilliard equation perturbed by an additive fractional white noise (fractional in time and white in space). We work in the case of one spatial dimension and apply Malliavin calculus to investigate the existence of a density for the stochastic solution $u$. In particular, we show that $u$ admits continuous paths almost surely and construct a localizing sequence through which we prove that its Malliavin derivative exists locally, and that its law is absolutely continuous with respect to the Lebesgue measure on $\bf R$, establishing thus that a density exists. A key contribution of this work is the analysis of the stochastic integral appearing in the mild formulation: we derive sharp estimates for the expectation of the $p$-th power ($p \geq 2$) of the $L^{\infty}(D)$-norm of this stochastic integral as well as for the integral involving the $L^{\infty}(D)$-norm of the operator associated with the kernel appearing in the integral representation of the fractional noise, all of which are essential for this study.

Malliavin Calculus for the stochastic Cahn-Hilliard equation driven by fractional noise

TL;DR

This work analyzes the stochastic Cahn-Hilliard equation in one spatial dimension driven by additive fractional white noise . It combines Malliavin calculus with a careful localization strategy to prove that the mild solution has a Malliavin derivative locally and that its law has a density with respect to Lebesgue measure. Central contributions include constructing a localizing sequence , deriving sharp -norm moment estimates for the stochastic integral and the associated kernel integral, and proving the absolute continuity of the law of and, via localization, of itself. The results extend density existence for SPDEs driven by fractional noise and provide concrete analytical tools for studying probabilistic properties of the stochastic CH dynamics.

Abstract

The stochastic partial differential equation analyzed in this work is the Cahn-Hilliard equation perturbed by an additive fractional white noise (fractional in time and white in space). We work in the case of one spatial dimension and apply Malliavin calculus to investigate the existence of a density for the stochastic solution . In particular, we show that admits continuous paths almost surely and construct a localizing sequence through which we prove that its Malliavin derivative exists locally, and that its law is absolutely continuous with respect to the Lebesgue measure on , establishing thus that a density exists. A key contribution of this work is the analysis of the stochastic integral appearing in the mild formulation: we derive sharp estimates for the expectation of the -th power () of the -norm of this stochastic integral as well as for the integral involving the -norm of the operator associated with the kernel appearing in the integral representation of the fractional noise, all of which are essential for this study.
Paper Structure (15 sections, 8 theorems, 182 equations)

This paper contains 15 sections, 8 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. The stochastic model
  3. Main results
  4. Preliminaries
  5. Basic concepts from Malliavin calculus
  6. Fractional white noise and related properties
  7. Weak and mild solutions
  8. Localization of $u(x,t)$ in $\mathbb{D}^{1,2}$
  9. Estimates for the stochastic integrals
  10. Regularity of the solution $u$
  11. Existence of a localizing sequence $\{(\Omega_{n},u_{n}(x,t))\}_{n \in \mathbb{N}_{*}}$ (Part 1)
  12. Existence of a localizing sequence $\{(\Omega_{n},u_{n}(x,t))\}_{n \in \mathbb{N}_{*}}$ (Part 2)
  13. Existence of a density for $u(x,t)$
  14. Absolute continuity of the law of $u_{n}(x,t)$ with respect to the Lebesgue measure on $\mathbb{R}$
  15. Absolute continuity of the law of $u(x,t)$ with respect to the Lebesgue measure on $\mathbb{R}$

Key Result

Theorem 1.1

Let $u$ be the unique solution of $(stoch. model)$ subjected to the homogeneous Neumann boundary conditions $(Neumann b.c.)$ and with a deterministic initial condition $u_{0} \in C(D)$. Then, the Malliavin derivative of $u(x,t)$ exists locally. Furthermore, for any $(x,t) \in D \times (0,T]$, the la

Theorems & Definitions (26)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 16 more