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The stochastic Klausmeier system and a stochastic Schauder-Tychonoff type theorem

Erika Hausenblas, Jonas M. Tölle

Abstract

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.

The stochastic Klausmeier system and a stochastic Schauder-Tychonoff type theorem

Abstract

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.

Paper Structure

This paper contains 18 sections, 23 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. The stochastic Schauder-Tychonoff type theorem
  3. Existence of a solution to the stochastic Klausmeier system
  4. Proof of the stochastic Schauder-Tychonoff theorem
  5. Results on regularity and technical propositions
  6. Assumptions on the noise and consequences
  7. Properties of equation \ref{['eq:cutoffuu']}
  8. Properties of equation \ref{['eq:cutoffvv']}
  9. The system \ref{['eq:cutoffu']}--\ref{['eq:cutoffv']} and uniform bounds on the stopping time
  10. Pathwise uniqueness of the solution
  11. Step I.
  12. Step II.
  13. Step III.
  14. Step IV.
  15. Step V.
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $H$ be a Hilbert space, $Q:H\to H$ such that $Q$ is linear, symmetric, nonnegative definite and of trace class, let $U$ be a Banach space, and let us assume that we have a compact and dense embedding $\mathbb{X}^{\prime}\hookrightarrow\mathbb{X}$ as above. Let $m>1$. Suppose that for any filtere Let us assume that the operator ${\mathcal{V}}_{{\mathfrak A},W}$, defined by spdes, restricted to

Theorems & Definitions (57)

  • Theorem 2.1
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Theorem 3.6
  • Definition 3.7
  • Remark 3.9
  • Theorem 3.10
  • Remark 3.11
  • proof
  • ...and 47 more