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Local moduli of continuity for permanental processes that are zero at zero

Michael B. Marcus, Jay Rosen

Abstract

Let $u(s,t)$ be a continuous potential density of a symmetric Lévy process or diffusion with state space $T$ killed at $T_{0}$, the first hitting time of $0$, or at $λ\wedge T_{0}$, where $λ$ is an independent exponential time. Let \[ f(t)=\int_{T} u(t,v)\,dμ(v), \] where $μ$ is a finite positive measure on $T$. Let $X_α=\{X_α(t),t\in T \}$ be an $α-$permanental process with kernel \[ v(s,t)=u(s,t)+f(t). \] Then when $\lim_{t\to 0}u(t,t)=0$, \[ \limsup_{t\downarrow 0}\frac{X_α(t )}{u(t,t)\log \log 1/t }\ge 1 ,\qquad \text{a.s.} \] and \[ \limsup_{t\downarrow 0}\frac{X_α(t )}{u(t,t)\log \log 1/t }\le 1+C_{u,h} ,\qquad \text{a.s.} \] where $C_{u,μ}\le |μ|$ is a constant that depends on both $u$ and $μ$, which is given explicitly, and is different in the different examples.

Local moduli of continuity for permanental processes that are zero at zero

Abstract

Let be a continuous potential density of a symmetric Lévy process or diffusion with state space killed at , the first hitting time of , or at , where is an independent exponential time. Let where is a finite positive measure on . Let be an permanental process with kernel Then when , and where is a constant that depends on both and , which is given explicitly, and is different in the different examples.
Paper Structure (7 sections, 16 theorems, 209 equations)

This paper contains 7 sections, 16 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['theo-7.2nn']}
  3. Proof of Theorem \ref{['theo-1.3q']}
  4. Proof of Theorem \ref{['theo-1.4']}
  5. Proof of Theorem \ref{['theo-1.5']}
  6. Proof of Theorem \ref{['theo-1.6']}
  7. Additional observations

Key Result

Theorem 1.1

Let $Y$ be a transient Borel right process with state space $T$, as described above. Then for any left potential $f$ for $Y$, there exists a transient Borel right process $\widetilde{Y} \!=\! ( \Omega, {\cal F}_{t}, \widetilde{Y}_t, \theta_{t}, P^x)$ with state space $S=T\cup \{\ast\}$, wher with respect to the measure $\widetilde{m}$ on $S$ which is equal to $m$ on $T$ and assigns a unit

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Example 1.1
  • Theorem 1.4
  • Remark 1.2
  • Example 1.2
  • Theorem 1.5
  • Example 1.3
  • ...and 20 more