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Stochastic Kimura Equations

Roland Riachi, Linan Chen

Abstract

In this work we study the one-dimensional stochastic Kimura equation $\partial_{t}u\left(z,t\right)=z\partial_{z}^{2}u\left(z,t\right)+u\left(z,t\right)\dot{W}\left(z,t\right)$ for $z,t>0$ equipped with a Dirichlet boundary condition at $0$, with $\dot{W}$ being a Gaussian space-time noise. This equation can be seen as a degenerate analog of the parabolic Anderson model. We combine the Wiener chaos theory from Malliavin calculus, the Duhamel perturbation technique from PDEs, and the kernel analysis of (deterministic) degenerate diffusion equations to develop a solution theory for the stochastic Kimura equation. We establish results on existence, uniqueness, moments, and continuity for the solution $u\left(z,t\right)$. In particular, we investigate how the stochastic potential and the degeneracy in the diffusion operator jointly affect the properties of $u\left(z,t\right)$ near the boundary. We also derive explicit estimates on the comparison under the $L^{2}-$ norm between $u\left(z,t\right)$ and its deterministic counterpart for $\left(z,t\right)$ within a proper range.

Stochastic Kimura Equations

Abstract

In this work we study the one-dimensional stochastic Kimura equation for equipped with a Dirichlet boundary condition at , with being a Gaussian space-time noise. This equation can be seen as a degenerate analog of the parabolic Anderson model. We combine the Wiener chaos theory from Malliavin calculus, the Duhamel perturbation technique from PDEs, and the kernel analysis of (deterministic) degenerate diffusion equations to develop a solution theory for the stochastic Kimura equation. We establish results on existence, uniqueness, moments, and continuity for the solution . In particular, we investigate how the stochastic potential and the degeneracy in the diffusion operator jointly affect the properties of near the boundary. We also derive explicit estimates on the comparison under the norm between and its deterministic counterpart for within a proper range.
Paper Structure (23 sections, 27 theorems, 238 equations, 1 figure)

This paper contains 23 sections, 27 theorems, 238 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Kimura Diffusion and Parabolic Anderson Model
  3. Stochastic Kimura Equation and Main Results
  4. Contents and Notations
  5. Background
  6. The Malliavin Calculus
  7. The Kimura Equation
  8. Stochastic Kimura Equation
  9. Existence and Uniqueness of a Solution
  10. Space-Time White Noise
  11. Colored Noise
  12. Ratio Comparison in $L^2$
  13. Space-Time White Noise
  14. Colored Noise
  15. $L^p-$ Bounds
  16. ...and 8 more sections

Key Result

Proposition 2.1

Let $u \in L^2(\Omega; H)$ with the expansion (eqn:wienerChaosDecomposition3). As a function $\mathbb{R}_+^{2(n+1)} \to \mathbb{R}$, $f_n \in H^{\otimes (n+1)}$. Then $u \in \text{Dom}(\delta)$ if and only if the series converges in $L^2(\Omega)$.

Figures (1)

  • Figure 3.1: The left child of each vertex corresponds to the bound obtained by handling the integral over $A_\varepsilon$ while the right child corresponds to the integral over $A_\varepsilon^c$. Each $1 \leq m \leq n$ corresponds to the $m$th subtree from the left whose respective roots are the right children of the leftmost vertex at each depth. Moreover, for any given path in a subtree, each $0 \leq k \leq m - 1$ corresponds to the number of rights in the path.

Theorems & Definitions (49)

  • Proposition 2.1: Proposition 1.3.7 of nualartBook
  • Proposition 2.2
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.3
  • ...and 39 more