Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Occupation times for superprocesses in random environments

Ziling Cheng, Jieliang Hong, Dan Yao

Abstract

Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment governed by a Gaussian noise $W=\{W(t, x),t\geq 0,x\in\mathbb{R}^d\}$ white in time and colored in space with correlation kernel $g$. We consider the occupation time process of the model starting from a finite measure. It is shown that the occupation time process of $X$ is absolutely continuous with respect to Lebesgue measure in $d\leq 3$, whereas it is singular with respect to Lebesgue measure in $d\geq 4$. Regarding the absolutely continuous case in $d\leq 3$, we further prove that the associated density function is jointly Hölder continuous based on the Tanaka formula and moment formulas, and derive the Hölder exponents with respect to the spatial variable $x$ and the time variable $t$.

Occupation times for superprocesses in random environments

Abstract

Let be a superprocess in a random environment governed by a Gaussian noise white in time and colored in space with correlation kernel . We consider the occupation time process of the model starting from a finite measure. It is shown that the occupation time process of is absolutely continuous with respect to Lebesgue measure in , whereas it is singular with respect to Lebesgue measure in . Regarding the absolutely continuous case in , we further prove that the associated density function is jointly Hölder continuous based on the Tanaka formula and moment formulas, and derive the Hölder exponents with respect to the spatial variable and the time variable .

Paper Structure

This paper contains 20 sections, 48 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Moment formulas and bounds
  3. Moment formulas of $Y_t$
  4. Moment bounds of $Y_t$
  5. Moment formulas of $X_t$
  6. Moment bounds of $X_t$
  7. Existence of the density
  8. Tanaka formula
  9. Local joint Hölder continuity
  10. Estimates on the spatial difference
  11. Estimates on the time difference
  12. The singularities in high dimensions
  13. Moment bounds
  14. Bounds on $V^{\phi}_n(t,x)$
  15. Bounds on $Y_t$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\mu\in M_F(\mathbb{R}^d)$ with $d\geq 2$. With $\mathbb{P}_{\mu}$-probability one, $X_t$ is singular with respect to Lebesgue measure for almost every $t>0$.

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Lemma 2.4
  • ...and 38 more