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Strong Feller property, irreducibility, and uniqueness of the invariant measure for stochastic PDEs with degenerate multiplicative noise

Luca Scarpa, Margherita Zanella

Abstract

We establish strong Feller property and irreducibility for the transition semigroup associated to a class of nonlinear stochastic partial differential equations with multiplicative degenerate noise. As a by-product, we prove uniqueness of the invariant measure under no strong-dissipativity assumptions. The drift of the equation diverges exactly where the noise coefficient vanishes, resulting in a competition between the dissipative effects and the degeneracy of the noise. We propose a method to measure the accumulation of the solution towards the potential barriers, allowing to give rigorous meaning to the inverse of the degenerate noise coefficient. From the mathematical perspective, this is one of the first contributions in the literature establishing strong Feller properties and irreducibility in the multiplicative degenerate case, and opens up novel investigation paths in the direction of regularisation effects and ergodicity in the degenerate-noise framework. From the application perspective, the models cover interesting scenarios in physics, in the context of evolution of relative concentrations of mixtures, under the influence of thermodynamically-relevant potentials of Flory-Huggins type.

Strong Feller property, irreducibility, and uniqueness of the invariant measure for stochastic PDEs with degenerate multiplicative noise

Abstract

We establish strong Feller property and irreducibility for the transition semigroup associated to a class of nonlinear stochastic partial differential equations with multiplicative degenerate noise. As a by-product, we prove uniqueness of the invariant measure under no strong-dissipativity assumptions. The drift of the equation diverges exactly where the noise coefficient vanishes, resulting in a competition between the dissipative effects and the degeneracy of the noise. We propose a method to measure the accumulation of the solution towards the potential barriers, allowing to give rigorous meaning to the inverse of the degenerate noise coefficient. From the mathematical perspective, this is one of the first contributions in the literature establishing strong Feller properties and irreducibility in the multiplicative degenerate case, and opens up novel investigation paths in the direction of regularisation effects and ergodicity in the degenerate-noise framework. From the application perspective, the models cover interesting scenarios in physics, in the context of evolution of relative concentrations of mixtures, under the influence of thermodynamically-relevant potentials of Flory-Huggins type.

Paper Structure

This paper contains 30 sections, 28 theorems, 320 equations.

Table of Contents

  1. Introduction
  2. State of the art and open problems
  3. Presentation of the model
  4. Heuristic idea
  5. Main results and technical strategy
  6. Structure of the paper
  7. Setting and main results
  8. Notation
  9. Functional and probabilistic setting
  10. Operators
  11. The linear operator and its fractional powers
  12. The potential and mobility operators
  13. The multiplicative noise operator
  14. Main results
  15. Consequences on the random separation property
  16. ...and 15 more sections

Key Result

Lemma 2.1

Let $s\ge 0$, $s_1\geq s$, and $s_2\ge s$ be real numbers satisfying Then, the pointwise multiplication of functions extends uniquely to a continuous bilinear map namely there exists a constant $C$, depending on $s, s_1, s_2, d$ such that, for every $f_1\in H^{s_1}(\mathcal{O})$ and $f_2\in H^{s_2}(\mathcal{O})$ it holds that $f_1f_2\in H^{s}(\mathcal{O})$ and

Theorems & Definitions (63)

  • Lemma 2.1: Multiplication of Sobolev functions
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7: Well-posedness
  • ...and 53 more