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Yves Le Jan

Publications and source records attributed to Yves Le Jan.

At least 37 records · Page 2Linked to original sources
arXiv2015

Permanental fields, loop soups and continuous additive functionals

A permanental field, $ψ=\{ψ(ν),ν\in {\mathcal{V}}\}$, is a particular stochastic process indexed by a space of measures on a set $S$. It is determined by a kernel $u(x,y)$, $x,y\in S$, that need not be symmetric and is allowed to be infinite on the diagonal. We show that these fields exist when $u(x,y)$ is a potential density of a transient Markov process $X$ in $S$. A permanental field $ψ$ can be realized as the limit of a renormalized sum of continuous additive functionals determined by a loop soup of $X$, which we carefully construct. A Dynkin-type isomorphism theorem is obtained that relates $ψ$ to continuous additive functionals of $X$ (continuous in $t$), $L=\{L_t^ν,(ν,t)\in {\mathcal{V}}\times R_+\}$. Sufficient conditions are obtained for the continuity of $L$ on ${\mathcal{V}}\times R_+$. The metric on ${\mathcal{V}}$ is given by a proper norm.

math.PR↗
arXiv2014

Perturbation of the loop measure

The loop measure is associated with a Markov generator. We compute the variation of the loop measure induced by an in nitesimal variation of the generator a ecting the killing rates or the jumping rates.

math.PR↗
arXiv2014

Markov loops in discrete spaces

The main topic of these notes are Markov loops, studied in the context of continuous time Markov chains on discrete state spaces. We refer to [1] and [2] for the short "history" of the subject. In contrast with these references, symmetry is not assumed, and more attention is given to the infinite case. All results are presented in terms of the semigroup generator. In comparison with [1], some delicate proofs are given in more details or with a better method. We focus mostly on properties of the (multi)occupation field but also included some results about loop clusters (see [3] in the symmetric context) and spanning trees. [1] Markov paths, loops and fields, by Yves Le Jan, Lecture Notes in Mathematics, vol. 2026, Springer, Heidelberg, 2011 [2] Topics in occupation times and Gaussian free fields, by Alain-Sol Sznitman, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2012 [3] Markovian loop clusters on graphs, by Yves Le Jan and Sophie Lemaire, Preprint, arxiv:1211.0300 [math.PR], 2012

math.PR↗
arXiv2013

Intersection local times, loop soups and permanental Wick powers

Several stochastic processes related to transient Lévy processes with potential densities $u(x,y)=u(y-x)$, that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures $V$ endowed with a metric $d$. Sufficient conditions are obtained for the continuity of these processes on $(V,d)$. The processes include $n$-fold self-intersection local times of transient Lévy processes and permanental chaoses, which are `loop soup $n$-fold self-intersection local times' constructed from the loop soup of the Lévy process. Loop soups are also used to define permanental Wick powers, which generalizes standard Wick powers, a class of $n$-th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes. Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional properties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.

math.PR↗
arXiv2013

Markovian loop clusters on graphs

We study the loop clusters induced by Poissonian ensembles of Markov loops on a finite or countable graph (Markov loops can be viewed as excursions of Markov chains with a random starting point, up to re-rooting). Poissonian ensembles are seen as a Poisson point process of loops indexed by 'time'. The evolution in time of the loop clusters defines a coalescent process on the vertices of the graph. After a description of some general properties of the coalescent process, we address several aspects of the loop clusters defined by a simple random walk killed at a constant rate on three different graphs: the integer number line $\mathbb{Z}$, the integer lattice $\mathbb{Z}^d$ with $d\geq 2$ and the complete graph. These examples show the relations between Poissonian ensembles of Markov loops and other models: renewal process, percolation and random graphs.

math.PR↗
arXiv2011

Three examples of Brownian flows on $\RR$

We show that the only flow solving the stochastic differential equation (SDE) on $\RR$ $$dX_t = 1_{\{X_t>0\}}W_+(dt) + 1_{\{X_t<0\}}dW_-(dt),$$ where $W^+$ and $W^-$ are two independent white noises, is a coalescing flow we will denote $\p^{\pm}$. The flow $\p^\pm$ is a Wiener solution. Moreover, $K^+=\E[δ_{\p^\pm}|W_+]$ is the unique solution (it is also a Wiener solution) of the SDE $$K^+_{s,t}f(x)=f(x)+\int_s^t K_{s,u}(1_{\RR^+}f')(x)W_+(du)+(1/2) \int_s^t K_{s,u}f"(x) du$$ for $s

math.PR↗
arXiv2011

Curvature Diffusions in General Relativity

We define and study on Lorentz manifolds a family of covariant diffusions in which the quadratic variation is locally determined by the curvature. This allows the interpretation of the diffusion effect on a particle by its interaction with the ambient space-time. We will focus on the case of warped products, especially Robertson-Walker manifolds, and analyse their asymptotic behaviour in the case of Einstein-de Sitter-like manifolds.

math.PR↗
arXiv2010

Markov paths, loops and fields

This is an extended version of a series of lectures given in St Flour. It includes a discussion of relations between the occupation field of Markov loops with the corresponding free field.

math.PR↗
arXiv2007

Dynkin's isomorphism without symmetry

The purpose of this note is to extend Dynkin's isomorphim involving functionals of the occupation field of a symmetric Markov processes and of the associated Gaussian field to a suitable class of non symmetric Markov processes.

math.PR↗
arXiv2005

Relativistic Diffusions and Schwarzschild Geometry

The purpose of this article is to introduce and study a relativistic motion whose acceleration, in proper time, is given by a white noise. We deal with general relativity, and consider more closely the problem of the asymptotic behaviour of paths in the Schwarzschild geometry example.

math.PR↗
arXiv2004

Products of Beta matrices and sticky flows

A discrete model of Brownian sticky flows on the unit circle is described: it is constructed with products of Beta matrices on the discrete torus. Sticky flows are defined by their ``moments'' which are consistent systems of transition kernels on the unit circle. Similarly, the moments of the discrete model form a consistent system of transition matrices on the discrete torus. A convergence of Beta matrices to sticky kernels is shown at the level of the moments. As the generators of the n-point processes are defined in terms of Dirichlet forms, the proof is performed at the level of the Dirichlet forms. The evolution of a probability measure by the flow of Beta matrices is described by a measure-valued Markov process. A convergence result of its finite dimensional distributions is deduced.

math.PR↗
arXiv2004

Relativistic Diffusions

The purpose of this note is to introduce and study a relativistic motion whose acceleration, in proper time, is given by a white noise. We begin with the flat case of special relativity, continue with the case of general relativity, and finally consider more closely the example of the Schwarzschild space.

math.PR↗
arXiv2002

Sticky flows on the circle

The purpose of this note is to give an example of stochastic flows of kernels, which naturally interpolates between the Arratia coalescing flow associated with systems of coalescing independent Brownian particles on the circle and the deterministic diffusion flow (actually, the results are given in the slightly more general framework of symmetric Levy processes for which points are not polar). The construction is performed using Dirichlet form theory and the extension of De Finetti's theorem given in the previous work by the authors. The sticky flows of kernels are associated with systems of sticky independent Levy particles on the circle, for some fixed parameter of stickyness. Some elementary asymptotic properties of the flow are also given.

math.PR↗