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On the density of the supremum of nonlinear SPDEs

Georgia Karali, Alexandra Stavrianidi, Konstantinos Tzirakis, Pavlos Zoubouloglou

Abstract

We study the one-dimensional stochastic partial differential equation \begin{equation*} \frac{\partial u}{\partial t}(t,x) = -κ\frac{\partial^4 u}{\partial x^4}(t,x) + ρ\frac{\partial^2 u}{\partial x^2}(t,x) + b(u(t,x)) + σ(u(t,x))\, \dot W(t,x), \end{equation*} posed on a bounded spatial domain, where $u$ is understood in the random field sense. Depending on the value of $κ$, this equation includes the nonlinear stochastic heat equation with Dirichlet or Neumann boundary conditions, as well as the linearized stochastic Cahn-Hilliard equation with Neumann boundary conditions. We prove that the supremum of the solution admits a density with respect to Lebesgue measure. Our approach is based on Malliavin calculus, and in particular on the version of the Bouleau-Hirsch criterion for suprema developed by Nualart and Vives. One of the main difficulties lies in the analysis of the argmax set of the solution and in showing that the Malliavin derivative is almost surely nondegenerate on this set. As a byproduct of our arguments, we also establish Hölder continuity properties for the Malliavin derivative of the solution as an $L^2-$valued process in the regimes considered in this work.

On the density of the supremum of nonlinear SPDEs

Abstract

We study the one-dimensional stochastic partial differential equation \begin{equation*} \frac{\partial u}{\partial t}(t,x) = -κ\frac{\partial^4 u}{\partial x^4}(t,x) + ρ\frac{\partial^2 u}{\partial x^2}(t,x) + b(u(t,x)) + σ(u(t,x))\, \dot W(t,x), \end{equation*} posed on a bounded spatial domain, where is understood in the random field sense. Depending on the value of , this equation includes the nonlinear stochastic heat equation with Dirichlet or Neumann boundary conditions, as well as the linearized stochastic Cahn-Hilliard equation with Neumann boundary conditions. We prove that the supremum of the solution admits a density with respect to Lebesgue measure. Our approach is based on Malliavin calculus, and in particular on the version of the Bouleau-Hirsch criterion for suprema developed by Nualart and Vives. One of the main difficulties lies in the analysis of the argmax set of the solution and in showing that the Malliavin derivative is almost surely nondegenerate on this set. As a byproduct of our arguments, we also establish Hölder continuity properties for the Malliavin derivative of the solution as an valued process in the regimes considered in this work.
Paper Structure (29 sections, 21 theorems, 304 equations)

This paper contains 29 sections, 21 theorems, 304 equations.

Table of Contents

  1. Introduction
  2. Main results and assumptions
  3. Literature review and proof technique
  4. Notation
  5. Organization of the paper
  6. Preliminaries
  7. Elements of Malliavin calculus
  8. Anisotropic Kolmogorov criterion and a Bouleau-Hirsch type criterion
  9. Green's function estimates
  10. Auxiliary Lemmas
  11. For the case $\kappa = 0$
  12. Estimates for the kernels
  13. Estimates for the Malliavin derivative
  14. For the case $\kappa > 0$
  15. Estimates for the kernel
  16. ...and 14 more sections

Key Result

Theorem 1.3

Let $u$ be the solution to sCH with $\kappa=0$ and $\rho=1$, subject to the boundary conditions of either Regime :dirichlet or Regime case:neumann. Suppose that Assumption assum:b-s holds. Then and the law of $\sup_{(t,x)\in K}u(t,x)$ is absolutely continuous with respect to $\lambda$ for every nonempty compact set $K\subset(0,T]\times(0,1)$ in Regime :dirichlet or $K\subset(0,T]\times [0,1]$ in

Theorems & Definitions (39)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: Anisotropic Kolmogorov criterion
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof : Proof of item (iii)
  • Lemma 3.1
  • ...and 29 more