A Cox Process Involved in the Bose–Einstein Condensation

The point process corresponding to the configurations of bosons in standard conditions is a Cox process driven by the square norm of a centered Gaussian process. This point process is infinitely divisible. We point out the fact that this property is preserved by the Bose–Einstein condensation phenomenon and show that the obtained point process after such a condensation occured, is still a Cox process but driven by the square norm of a shifted Gaussian process, the shift depending on the density of the particles. This law provides an illustration of a “super”- Isomorphism Theorem existing above the usual Isomorphism Theorem of Dynkin available for Gaussian processes. Submitted: February 8, 2008. Accepted: March 5, 2008.     

Dynkin's isomorphism theorem and the Ray-Knight theorems

Summary In the case of diffusions, we show that the isomorphism theorem of Dynkin and the Ray-Knight theorems can be derived from each other. Our proof uses additivity properties of squared Bessel processes and an absolute continuity relation between squared Bessel processes of dimensions one and three.Research supported in part by Air Force Office of Scientific Research (USAFOSR 89-0261)This work was carried out while visiting the Technion     

Measuring the Rarely Visited Sites of Brownian Motion

We study the almost sure asymptotic behaviors of the Lebesgue measure of the points which are hardly visited, in the sense of Földes and Révész,⁽⁷⁾ by a linear Wiener process.     

Isomorphism theorems for Markov chains

The isomorphism theorem of Dynkin is definitely an important tool to investigate the problems raised in terms of local times of Markov processes. This theorem concerns continuous time Markov processes. We give here an equivalent version for Markov chains.     

Local time–space stochastic calculus for Lévy processes

We develop a stochastic calculus on the plane with respect to the local times of a large class of Lévy processes. We can then extend to these Lévy processes an Itô formula that was established previously for Brownian motion. Our method provides also a multidimensional version of the formula. We show that this formula generates many “Itô formulas” that fit various problems. In the special case of a linear Brownian motion, we recover a recently established Itô formula that involves local times on curves. This formula is already used in financial mathematics.     

Dynkin’s Isomorphism Theorem and the Stochastic Heat Equation

Consider the stochastic heat equation \(\partial_t u = \mathcal{L} u + \dot{W}\), where \(\mathcal{L}\) is the generator of a [Borel right] Markov process in duality. We show that the solution is locally mutually absolutely continuous with respect to a smooth perturbation of the Gaussian process that is associated, via Dynkin’s isomorphism theorem, to the local times of the replica-symmetric process that corresponds to \(\mathcal{L}\). In the case that \(\mathcal{L}\) is the generator of a Lévy process on R ^d , our result gives a probabilistic explanation of the recent findings of Foondun et al. (Trans Am Math Soc, 2007).     

Stochastic order for alpha-permanental point processes

We establish relations of stochastic comparison among point processes elements of the set of alpha-permanental point processes. This set contains in particular, the determinantal point processes, the Poisson point processes and the permanental point processes. We show that these three classes of point processes can be ordered according to the increasing stochastic order. Elementary particles provide illustrations of some of the obtained relations of stochastic comparison.     

On permanental processes

Permanental processes can be viewed as a generalization of squared centered Gaussian processes. We analyze the connections of these processes with the local time process of general Markov processes. The obtained results are related to the notion of infinite divisibility.     

Exponential Burkholder Davis Gundy Inequalities

Klass has established decoupling inequalities for discrete timeprocesses with independent increments. We extend his resultto continuous time processes with independent increments. Thisextension enables us to obtain BDG inequalities for exponentialfunctions instead of moderate functions.