Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parabolic Anderson model in bounded domains of recurrent metric measure spaces

Fabrice Baudoin, Li Chen, Che-Hung Huang, Cheng Ouyang, Samy Tindel, Jing Wang

Abstract

A metric measure space equipped with a Dirichlet form is called recurrent if its Hausdorff dimension is less than its walk dimension. In bounded domains of such spaces we study the parabolic Anderson models \[ \partial_{t} u(t,x) = Δu(t,x) + βu(t,x) \, \dot{W}_α(t,x) \] where the noise $W_α$ is white in time and colored in space when $α>0$ while for $α=0$ it is also white in space. Both Dirichlet and Neumann boundary conditions are considered. Besides proving existence and uniqueness in the Itô sense we also get precise $L^p$ estimates for the moments and intermittency properties of the solution as a consequence. Our study reveals new exponents which are intrinsically associated to the geometry of the underlying space and the results for instance apply in metric graphs or fractals like the Sierpiński gasket for which we prove scaling invariance properties of the models.

Parabolic Anderson model in bounded domains of recurrent metric measure spaces

Abstract

A metric measure space equipped with a Dirichlet form is called recurrent if its Hausdorff dimension is less than its walk dimension. In bounded domains of such spaces we study the parabolic Anderson models where the noise is white in time and colored in space when while for it is also white in space. Both Dirichlet and Neumann boundary conditions are considered. Besides proving existence and uniqueness in the Itô sense we also get precise estimates for the moments and intermittency properties of the solution as a consequence. Our study reveals new exponents which are intrinsically associated to the geometry of the underlying space and the results for instance apply in metric graphs or fractals like the Sierpiński gasket for which we prove scaling invariance properties of the models.
Paper Structure (29 sections, 37 theorems, 258 equations, 3 figures)

This paper contains 29 sections, 37 theorems, 258 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Recurrent metric measure spaces with sub-Gaussian heat kernel estimates
  4. Dirichlet boundary conditions
  5. Heat semigroup with Dirichlet boundary conditions
  6. Fractional Laplacian
  7. Neumann boundary conditions
  8. Heat semigroup with Neumann boundary conditions
  9. Fractional Laplacian
  10. Examples of application
  11. Example 1: Compact intervals
  12. Example 2: Metric graphs
  13. Example 3: Fractals, Sierpiński gasket
  14. Dirichlet parabolic Anderson model in inner uniform domains
  15. Fractional noise
  16. ...and 14 more sections

Key Result

Theorem 1.1

On any bounded open domain $U$ of $(X,d,\mu)$, with either Neumann or Dirichlet boundary conditions, equation PAM intro admits a unique solution in the Itô sense for any value $\alpha\in[0,\infty)$. In addition, for $\alpha>d_h/2d_w$, the solution admits a Feynman-Kac representation.

Figures (3)

  • Figure 1: Examples of metric graphs (picture from Wikipedia)
  • Figure 2: Sierpiński gasket.
  • Figure 3: Sierpiński gasket graphs $G_0$, $G_1$, $G_2$ and $G_3$

Theorems & Definitions (93)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Corollary 2.9
  • ...and 83 more