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SPDEs driven by standard symmetric $α$-stable cylindrical Lévy processes: existence, Lyapunov functionals and Itô formula

Gergely Bodó, Ondřej Týbl, Markus Riedle

Abstract

We investigate several aspects of solutions to stochastic evolution equations in Hilbert spaces driven by a standard symmetric $α$-stable cylindrical noise. Similarly to cylindrical Brownian motion or Gaussian white noise, standard symmetric $α$-stable noise exists only in a generalised sense in Hilbert spaces. The main results of this work are the existence of a mild solution, long-term regularity of the solutions via Lyapunov functional approach, and an Itô formula for mild solutions to evolution equations under consideration. The main tools for establishing these results are Yosida approximations and an Itô formula for Hilbert space-valued semi-martingales where the martingale part is represented as an integral driven by cylindrical $α$-stable noise. While these tools are standard in stochastic analysis, due to the cylindrical nature of our noise, their application requires completely novel arguments and techniques.

SPDEs driven by standard symmetric $α$-stable cylindrical Lévy processes: existence, Lyapunov functionals and Itô formula

Abstract

We investigate several aspects of solutions to stochastic evolution equations in Hilbert spaces driven by a standard symmetric -stable cylindrical noise. Similarly to cylindrical Brownian motion or Gaussian white noise, standard symmetric -stable noise exists only in a generalised sense in Hilbert spaces. The main results of this work are the existence of a mild solution, long-term regularity of the solutions via Lyapunov functional approach, and an Itô formula for mild solutions to evolution equations under consideration. The main tools for establishing these results are Yosida approximations and an Itô formula for Hilbert space-valued semi-martingales where the martingale part is represented as an integral driven by cylindrical -stable noise. While these tools are standard in stochastic analysis, due to the cylindrical nature of our noise, their application requires completely novel arguments and techniques.
Paper Structure (12 sections, 26 theorems, 197 equations)

This paper contains 12 sections, 26 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standard symmetric $\alpha$-stable cylindrical Lévy processes
  4. Stochastic integration
  5. Random measures and compensators
  6. Predictable compensator
  7. Quadratic variation of the integral process
  8. Strong Itô formula
  9. Mild solutions for stochastic evolution equations
  10. Moment boundedness for evolution equations
  11. Mild Itô formula
  12. Appendix

Key Result

Lemma 2.1

Let $\lambda$ be the cylindrical Lévy measure of an $\alpha$-stable cylindrical Lévy process for $\alpha\in (1,2)$. For every $m\in\mathbb{N}$ there exists $d_\alpha^m<\infty$, depending only on $\alpha$ and $m$, such that for all $\Phi\in\mathcal{L}_2(U, H)$. Moreover, we have $\lim_{m \rightarrow \infty}d_\alpha^m=0$.

Theorems & Definitions (61)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4: Stochastic Fubini Theorem
  • proof
  • Definition 2.5: Random measure
  • Example 2.6
  • ...and 51 more