arXiv subjects

Jonas Arista

Publications and source records attributed to Jonas Arista.

6 recordsLinked to original sources

arXiv2025

Myopic non-intersection in a periodic potential

We introduce a class of Markov processes conditioned to avoid intersection over a moving time window of length T>0, a setting we refer to as myopic non-intersection. In particular, we study a system of myopic non-intersecting Brownian motions subject to a periodic potential. Our focus lies in understanding the interplay between the confining effect of the potential and the repulsion induced by the non-intersection constraint. We show that, in the long time limit, and as both T and the strength of the potential become large, the model converges to a system of myopic non-intersecting random walks, which transitions between standard non-intersection dynamics and exclusion behavior. The main technical contribution of the paper is the introduction of an algorithm, based on a modification of the acceptance-rejection sampling scheme, that provides an explicit construction of myopically constrained systems.

math.PR↗

arXiv2023

Implicit renewal theory for exponential functionals of Lévy processes

We establish a new integral equation for the probability density of the exponential functional of a Lévy process and provide a three-term (Wiener-Hopf type) factorisation of its law. We explain how these results complement the techniques used in the study of exponential functionals and, in some cases, provide quick proofs of known results and derive new ones. We explain how the factors appearing in the three-term factorisation determine the local and asymptotic behaviour of the law of the exponential functional. We describe the behaviour of the tail distribution at infinity and of the distribution at zero under some mild assumptions.

math.PR↗

arXiv2022

Matrix Whittaker processes

We study a discrete-time Markov process on triangular arrays of matrices of size $d\geq 1$, driven by inverse Wishart random matrices. The components of the right edge evolve as multiplicative random walks on positive definite matrices with one-sided interactions and can be viewed as a $d$-dimensional generalisation of log-gamma polymer partition functions. We establish intertwining relations to prove that, for suitable initial configurations of the triangular process, the bottom edge has an autonomous Markovian evolution with an explicit transition kernel. We then show that, for a special singular initial configuration, the fixed-time law of the bottom edge is a matrix Whittaker measure, which we define. To achieve this, we perform a Laplace approximation that requires solving a constrained minimisation problem for certain energy functions of matrix arguments on directed graphs.

math.PR↗

arXiv2021

Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices

We establish analogues of the geometric Pitman $2M-X$ theorem of Matsumoto and Yor and of the classical Dufresne identity, for a multiplicative random walk on positive definite matrices with Beta type II distributed increments. The Dufresne type identity provides another example of a stochastic matrix recursion, as considered by Chamayou and Letac (J. Theoret. Probab. 12, 1999), that admits an explicit solution.

math.PR↗

arXiv2020

Explicit expressions of the Hua-Pickrell semi-group

In this paper, we study the one-dimensional Hua-Pickrell diffusion. We start by revisiting the stationary case considered by E. Wong for which we supply omitted details and write down a unified expression of its semi-group density through the associated Legendre function in the cut. Next, we focus on the general (not necessarily stationary) case for which we prove an intertwining relation between Hua-Pickrell diffusions corresponding to different sets of parameters. Using Cauchy Beta integral on the one hand and Girsanov's Theorem on the other hand, we discuss the connection between the stationary and general cases. Afterwards, we prove our main result providing novel integral representations of the Hua-Pickrell semi-group density, answering a question raised by Alili, Matsumoto and Shiraishi (Séminaire de Probabilités, 35, 2001). To this end, we appeal to the semi-group density of the Maass Laplacian and extend it to purely-imaginary values of the magnetic field. In the last section, we use the Karlin-McGregor formula to derive an expression of the semi-group density of the multi-dimensional Hua-Pickrell particle system introduced by T. Assiotis.

math.PR↗

arXiv2019

Loop-erased walks and random matrices

It is well known that there are close connections between non-intersecting processes in one dimension and random matrices, based on the reflection principle. There is a generalisation of the reflection principle for more general (e.g. planar) processes, due to S. Fomin, in which the non-intersection condition is replaced by a condition involving loop-erased paths. In the context of independent Brownian motions in suitable planar domains, this also has close connections to random matrices. An example of this was first observed by Sato and Katori (Phys. Rev. E, 83, 2011). We present further examples which give rise to various Cauchy-type ensembles. We also extend Fomin's identity to the affine setting and show that in this case, by considering independent Brownian motions in an annulus, one obtains a novel interpretation of the circular orthogonal ensemble.

math.PR↗