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SPDEs on narrow channels and graphs: convergence and large deviations in case of non smooth noise

Sandra Cerrai, Wen-Tai Hsu

Abstract

We investigate a class of stochastic partial differential equations of reaction-diffusion type defined on graphs, which can be derived as the limit of SPDEs on narrow planar channels. In the first part, we demonstrate that this limit can be achieved under less restrictive assumptions on the regularity of the noise, compared to [4]. In the second part, we establish the validity of a large deviation principle for the SPDEs on the narrow channels and on the graphs, as the width of the narrow channels and the intensity of the noise are jointly vanishing.

SPDEs on narrow channels and graphs: convergence and large deviations in case of non smooth noise

Abstract

We investigate a class of stochastic partial differential equations of reaction-diffusion type defined on graphs, which can be derived as the limit of SPDEs on narrow planar channels. In the first part, we demonstrate that this limit can be achieved under less restrictive assumptions on the regularity of the noise, compared to [4]. In the second part, we establish the validity of a large deviation principle for the SPDEs on the narrow channels and on the graphs, as the width of the narrow channels and the intensity of the noise are jointly vanishing.
Paper Structure (13 sections, 14 theorems, 167 equations, 1 figure)

This paper contains 13 sections, 14 theorems, 167 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. The domain $G$, the narrow channel $G_\epsilon$ and the graph $\Gamma$
  4. Limiting results for the transition semigroup
  5. Functions and operators on the graph $\Gamma$
  6. Limiting results for the SPDE on $G$
  7. Convergence of solutions in case of rough noise
  8. Proof of \ref{['star-2']}
  9. The large deviation principle
  10. From weak convergence to strong convergence
  11. Proof of Theorem \ref{['LDPT']}
  12. Proof of Claim 1.
  13. Proof of Claim 2.

Key Result

Theorem 2.1

fw12 For any bounded and continuous functional $F$ on $C([0,T];\Gamma)$ and $z \in G$ it holds

Figures (1)

  • Figure 1: The domain $G$ and the graph $\Gamma$

Theorems & Definitions (24)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • ...and 14 more