Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ray-Knight theorems for the local times of rebirthed Markov processes

P. J. Fitzsimmons, Jay Rosen

Abstract

We prove generalizations of the first and second Ray-Knight theorems, for a large class of non-symmetric strong Markov processes. These results link the local times of the Markov process with the squares of associated Gaussian processes. This connection allows us to establish results about the exact modulus of continuity (in the spatial variable) of the local times. Our approach is different from earlier treatments which were based on associated permanental processes rather than Gaussian processes. The type of process with which we work can be described as follows. Start with a symmetric Markov process with finite lifetime; upon its death resurrect it at a place in the state space chosen at random, independent of the past. Continue in this way, resurrecting at each death, to obtain a recurrent process. The rebirthing procedure destroys the symmetry of the original process, leading to a large class of non-symmetric processes. The main results are illustrated by many examples.

Ray-Knight theorems for the local times of rebirthed Markov processes

Abstract

We prove generalizations of the first and second Ray-Knight theorems, for a large class of non-symmetric strong Markov processes. These results link the local times of the Markov process with the squares of associated Gaussian processes. This connection allows us to establish results about the exact modulus of continuity (in the spatial variable) of the local times. Our approach is different from earlier treatments which were based on associated permanental processes rather than Gaussian processes. The type of process with which we work can be described as follows. Start with a symmetric Markov process with finite lifetime; upon its death resurrect it at a place in the state space chosen at random, independent of the past. Continue in this way, resurrecting at each death, to obtain a recurrent process. The rebirthing procedure destroys the symmetry of the original process, leading to a large class of non-symmetric processes. The main results are illustrated by many examples.
Paper Structure (6 sections, 16 theorems, 130 equations)

This paper contains 6 sections, 16 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. A Generalized First Ray-Knight Theorem for rebirthed Markov Processes
  3. Case 2: A Generalized First Ray-Knight Theorem for rebirthed Markov Processes which do not hit $0$
  4. Behavior of $\widetilde{L}^{x}_{T_{0}}$ at $x$ near $0$ for rebirthed diffusions
  5. A Generalized Second Ray-Knight Theorem for rebirthed Markov Processes
  6. Application: Exact moduli of continuity at $\widetilde{\tau} (t)$ for the local times of rebirthed Markov processes

Key Result

Lemma 2.1

For each $r\geq 2$, and all Borel sets $B_1, B_2,\ldots, B_r$ in $C(S, R^1)$,

Theorems & Definitions (16)

  • Lemma 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.2
  • Theorem 2.5
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 6 more