arXiv subjects

Joaquin Fontbona

Publications and source records attributed to Joaquin Fontbona.

22 records · Page 2Linked to original sources

arXiv2012

Quantitative estimates for the long time behavior of an ergodic variant of the telegraph process

Motivated by stability questions on piecewise deterministic Markov models of bacterial chemotaxis, we study the long time behavior of a variant of the classic telegraph process having a non-constant jump rate that induces a drift towards the origin. We compute its invariant law and show exponential ergodicity, obtaining a quantitative control of the total variation distance to equilibrium at each instant of time. These results rely on an exact description of the excursions of the process away from the origin and on the explicit construction of an original coalescent coupling for both velocity and position. Sharpness of the obtained convergence rate is discussed.

math.PR↗

arXiv2010

Energy efficiency of consecutive fragmentation processes

We present a first study on the energy required to reduce a unit mass fragment by consecutively using several devices, as it happens in the mining industry. Two devices are considered, which we represent as different stochastic fragmentation processes. Following the self-similar energy model introduced by Bertoin and Martinez, we compute the average energy required to attain a size x with this two-device procedure. We then asymptotically compare, as x goes to 0 or 1, its energy requirement with that of individual fragmentation processes. In particular, we show that for certain range of parameters of the fragmentation processes and of their energy cost-functions, the consecutive use of two devices can be asymptotically more efficient than using each of them separately, or conversely.

math.PR↗

arXiv2008

Measurability of optimal transportation and strong coupling of martingale measures

We consider the optimal mass transportation problem in $\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure. A motivation is the construction of a strong coupling between orthogonal martingale measures. By this we mean that, given a martingale measure, we construct in the same probability space a second one with specified covariance measure. This is done by pushing forward one martingale measure through a predictable version of the optimal transport map between the covariance measures. This coupling allows us to obtain quantitative estimates in terms of the Wasserstein distance between those covariance measures.

math.PR↗

arXiv2007

Measurability of optimal transportation and convergence rate for Landau type interacting particle systems

In this paper, we consider nonlinear diffusion processes driven by space-time white noises, which have an interpretation in terms of partial differential equations. For a specific choice of coefficients, they correspond to the Landau equation arising in kinetic theory. A particular feature is that the diffusion matrix of this process is a linear function the law of the process, and not a quadratic one, as in the McKean-Vlasov model. The main goal of the paper is to construct an easily simulable diffusive interacting particle system, converging towards this nonlinear process and to obtain an explicit pathwise rate. This requires to find a significant coupling between finitely many Brownian motions and the infinite dimensional white noise process. The key idea will be to construct the right Brownian motions by pushing forward the white noise processes, through the Brenier map realizing the optimal transport between the law of the nonlinear process, and the empirical measure of independent copies of it. A striking problem then is to establish the joint measurability of this optimal transport map with respect to the space variable and the parameters (time and randomness) making the marginals vary. We shall prove a general measurability result for the mass transportation problem in terms of the support of the transfert plans, in the sense of set-valued mappings. This will allow us to construct the coupling and to obtain explicit convergence rates.

math.PR↗