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P. J. Fitzsimmons

Publications and source records attributed to P. J. Fitzsimmons.

13 recordsLinked to original sources
arXiv2026

Exact moduli of continuity for the local times of Feller Brownian motions

We examine the modulus of continuity, in the spatial variable, of the local time process of Feller Brownian motion (FBM) on the half-line $[0,\infty)$. Briefly, a FBM is a strong Markov process on $[0,\infty)$ that moves like standard Brownian motion on $[0,\infty)$ up until it first encounters the state $0$. The process returns to $(0,\infty)$, either continuously (like reflecting Brownian motion) or by jumping to a (random) positive state chosen according to a specified measure. The present work is a continuation and application of our earlier work with Michael Marcus on the moduli of continuity for the local times of a Markov process built by piecing together (``rebirthing") the paths of another Markov process with finite lifetime. We first establish a general result on the resolvent and local times for a rebirthed process with a special holding state (the state $0$ for FBM). We show how our earlier approach using the Eisenbaum Isomorphism Theorem on an assemblage of excursions works out in this context. This knowledge is then used as an approximation device to obtain our main result on the exact uniform moduli of continuity for the local time of FBM on a spatial interval of the form $(0,1]$. Extensions are made, under certain conditions, to the more delicate situation of the spatial interval $[0,1]$. We also consider briefly the case of more general diffusions on $[0,\infty)$.

math.PR↗
arXiv2026

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

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.

math.PR↗
arXiv2026

Kemeny's Constant for Markov Processes

The mean time taken by an irreducible Markov chain on a finite state space to hit a target chosen at random according to the stationary distribution does not depend on the initial state of the chain. This mean time is known as Kemeny's constant. I present a new approach, based on time reversal and a mean occupation time formula. The method is used to prove an analogous result for continuous-time Markov processes. We also present a second approach, based on work of N.~Eisenbaum and H.~Kaspi, when all states are regular. Examples are provided.

math.PR↗
arXiv2024

Moduli of continuity for the local times of rebirthed Markov processes

Let $S$ a be locally compact space with a countable base. Let $\cal Y$ be a transient symmetric Borel right process with state space $S$ and continuous strictly positive $p$--potential densities $u^p(x,y)$. Local and uniform moduli of continuity are obtained for the local times of both fully and partially rebirthed versions of $\cal Y$. A fully rebirthed version of $\cal Y$ is an extension of $\cal Y$ so that instead of terminating at the end of its lifetime it is immediately ``reborn'' with a probability measure $μ$, on $S$. I.e., the process goes to the set $B\subset S$ with probability $μ(B) $, after which it continues to evolve the way $\YY$ did, being reborn with probability $μ$ each time it dies. This rebirthed version of $\cal Y$ is a recurrent Borel right process with state space $S$ and $p$-potential densities of form, \[ u^p(x,y)+h(x,y),\qquad x,y\in S,\,\, p>0, \] where $h(x,y)$ is not symmetric. The local times of the rebirthed process are given in terms of the local times of $\cal Y$ and isomorphism theorems in the spirit of Dynkin, Eisenbam and Kaspi are obtained that relate these local times to generalized chi--square processes formed by Gaussian processes with covariances $u^{q}(x,y)$ for different values of $q$. These isomorphisms allow one to obtain exact local and uniform moduli of continuity for the local times of the rebirthed process. Several explicit examples are given in which $\cal Y$ is either a modified Lévy process or a diffusion. Analogous results are obtained for partially rebirthed versions of $\cal Y$. This is obtained by starting $\cal Y$ in $S$ and when it dies returning it to $S$ with a sub-probability measure $Ξ$. (With probability $1-|Ξ|$ it is sent to a disjoint state space $S'$, where it remains.)

math.PR↗
arXiv2022

A Proof of Lehoczky's Theorem on Drawdowns

We give a proof of Lehoczky's drawdown formula for one-dimensional diffusion processes, using the Poisson structure of the excursions of the diffusion below its running maximum.

math.PR↗
arXiv2014

Markovian loop soups: permanental processes and isomorphism theorems

We construct loop soups for general Markov processes without transition densities and show that the associated permanental process is equal in distribution to the loop soup local time. This is used to establish isomorphism theorems connecting the local time of the original process with the associated permanental process. Further properties of the loop measure are studied.

math.PR↗
arXiv2013

Excursions Above the Minimum for Diffusions

We demonstrate the existence of a "Lévy system" for the excursions of a one-dimensional diffusion process above its past-minimum process. As applications we provide a direct proof of D. Williams' decomposition (in both a global and a local form) and of a theorem of W. Vervaat.

math.PR↗
arXiv2013

On a result of D.W. Stroock

We show how to modify an argument of D. W. Stroock to show that an additive map from one separable Banach space to another, that is "universally Gaussian measurable", must be continuous (hence linear).

math.PR↗
arXiv2008

Stochastic calculus for symmetric Markov processes

Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an Itô formula for Dirichlet processes is obtained.

math.PR↗
arXiv2006

Time change approach to generalized excursion measures, and its application to limit theorems

It is proved that generalized excursion measures can be constructed via time change of Ito's Brownian excursion measure. A tightness-like condition on strings is introduced to prove a convergence theorem of generalized excursion measures. The convergence theorem is applied to obtain a conditional limit theorem, a kind of invariance principle where the limit is the Bessel meander.

math.PR↗
arXiv2004

Absolute continuity of symmetric Markov processes

We study Girsanov's theorem in the context of symmetric Markov processes, extending earlier work of Fukushima-Takeda and Fitzsimmons on Girsanov transformations of ``gradient type.'' We investigate the most general Girsanov transformation leading to another symmetric Markov process. This investigation requires an extension of the forward-backward martingale method of Lyons-Zheng, to cover the case of processes with jumps.

math.PR↗
arXiv1998

Markov Processes with Identical Bridges

Let X and Y be time-homogeneous Markov processes with common state space E, and assume that the transition kernels of X and Y admit densities with respect to suitable reference measures. We show that if there is a time t>0 such that, for each x\in E, the conditional distribution of (X_s)_{0 < s < t}, given X_0 = x = X_t, coincides with the conditional distribution of (Y_s)_{0 < s < t}, given Y_0 = x = Y_t, then the infinitesimal generators of X and Y are related by [L^Y]f = ψ^{-1}[L^X](ψf)-λf, where ψis an eigenfunction of L^X with eigenvalue λ. Under an additional continuity hypothesis, the same conclusion obtains assuming merely that X and Y share a ``bridge'' law for one triple (x,t,y). Our work entends and clarifies a recent result of I. Benjamini and S. Lee.

math.PR↗