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Camille Tardif

Publications and source records attributed to Camille Tardif.

12 recordsLinked to original sources
arXiv2024

On joint returns to zero of Bessel processes

In this article, we consider joint returns to zero of $n$ Bessel processes ($n\geq 2$): our main goal is to estimate the probability that they avoid having joint returns to zero for a long time. More precisely, considering $n$ independent Bessel processes $(X_t^{(i)})_{1\leq i \leq n}$ of dimension $δ\in (0,1)$, we are interested in the first joint return to zero of any two of them: \[ H_n := \inf\big\{ t>0, \exists 1\leq i t) = t^{-θ_n+o(1)}$ as $t\to\infty$, and we provide some non-trivial bounds on $θ_n$. In particular, when $n=3$, we show that $2(1-δ)\leq θ_3 \leq 2 (1-δ) + f(δ)$ for some (explicit) function $f(δ)$ with $\sup_{[0,1]} f(δ) \approx 0.079$.

math.PR↗
arXiv2023

Persistence problems for additive functionals of one-dimensional Markov processes

In this article, we consider additive functionals $ζ_t = \int_0^t f(X_s)\mathrm{d} s$ of a càdlàg Markov process $(X_t)_{t\geq 0}$ on $\mathbb{R}$. Under some general conditions on the process $(X_t)_{t\geq 0}$ and on the function $f$, we show that the persistence probabilities verify $\mathbb{P}(ζ_s < z \text{ for all } s\leq t ) \sim \mathcal{V}(z) ς(t) t^{-θ}$ as $t\to\infty$, for some (explicit) $\mathcal{V}(\cdot)$, some slowly varying function $ς(\cdot)$ and some $θ\in (0,1)$. This extends results in the literature, which mostly focused on the case of a self-similar process $(X_t)_{t\geq 0}$ (such as Brownian motion or skew-Bessel process) with a homogeneous functional $f$ (namely a pure power, possibly asymmetric). In a nutshell, we are able to deal with processes which are only asymptotically self-similar and functionals which are only asymptotically homogeneous. Our results rely on an excursion decomposition of $(X_t)_{t\geq 0}$, together with a Wiener--Hopf decomposition of an auxiliary (bivariate) Lévy process, with a probabilistic point of view. This provides an interpretation for the asymptotic behavior of the persistence probabilities, and in particular for the exponent $θ$, which we write as $θ= ρβ$, with $β$ the scaling exponent of the local time of $(X_{t})_{t\geq 0}$ at level $0$ and $ρ$ the (asymptotic) positivity parameter of the auxiliary Lévy process.

math.PR↗
arXiv2020

Simulated Annealing In $\mathbf{R}^d$ With Slowly Growing Potentials

We use a localization procedure to weaken the growth assumptions of Royer [8], Miclo [4] and Zitt [9] concerning the continuous-time simulated annealing in $\mathbf{R}^d$. We show that a transition occurs for potentials growing like $a \log \log |x|$ at infinity. We also study a class of potentials with possibly unbounded sets of local minima.

math.PR↗
arXiv2020

On The Simulated Annealing In $\mathbf{R}^d$

Using a localization procedure and the result of Holley-Kusuoka-Stroock [7] in the torus, we widely weaken the usual growth assumptions concerning the success of the continuous-time simulated annealing in $\mathbf{R}^d$. Our only assumption is the existence of an invariant probability measure for a sufficiently low temperature. We also prove, in an appendix, a non-explosion criterion for a class of time-inhomogeneous diffusions.

math.PR↗
arXiv2018

On the Poisson boundary of the relativistic Brownian motion

In this paper, we determine the Poisson boundary of the relativistic Brownian motion in two classes of Lorentzian manifolds, namely model manifolds of constant scalar curvature and Robertson--Walker space-times, the latter constituting a large family of curved manifolds. Our objective is two fold: on the one hand, to understand the interplay between the geometry at infinity of these manifolds and the asymptotics of random sample paths, in particular to compare the stochastic compactification given by the Poisson boundary to classical purely geometric compactifications such as the conformal or causal boundaries. On the other hand, we want to illustrate the power of the dévissage method introduced by the authors in [AT16], method which we show to be particularly well suited in the geometric contexts under consideration here.

math.PR↗
arXiv2018

Anomalous diffusion for multi-dimensional critical Kinetic Fokker-Planck equations

We consider a particle moving in $d\geq 2$ dimensions, its velocity being a reversible diffusion process, with identity diffusion coefficient, of which the invariant measure behaves, roughly, like $(1+|v|)^{-β}$ as $|v|\to \infty$, for some constant $β>0$. We prove that for large times, after a suitable rescaling, the position process resembles a Brownian motion if $β\geq 4+d$, a stable process if $β\in [d,4+d)$ and an integrated multi-dimensional generalization of a Bessel process if $β\in (d-2,d)$. The critical cases $β=d$, $β=1+d$ and $β=4+d$ require special rescalings.

math.PR↗
arXiv2018

One dimensional critical Kinetic Fokker-Planck equations, Bessel and stable processes

We consider a particle moving in one dimension, its velocity being a reversible diffusion process, with constant diffusion coefficient, of which the invariant measure behaves like $(1+|v|)^{-β}$ for some $β>0$. We prove that, under a suitable rescaling, the position process resembles a Brownian motion if $β\geq 5$, a stable process if $β\in [1,5)$ and an integrated symmetric Bessel process if $β\in (0,1)$. The critical cases $β=1$ and $β=5$ require special rescaling. We recover some results of G.Lebeau and M.Puel [LP17], P.Cattiaux, E.Nasreddine and M. Puel [CNP16], and E.Barkai, E.Aghion and D. A. Kesslerwith [BAK14] with an alternative approach.

math.PR↗
arXiv2016

Hypocoercive estimates on foliations and velocity spherical Brownian motion

By further developing the generalized $Γ$-calculus for hypoelliptic operators, we prove hypocoercive estimates for a large class of Kolmogorov type operators which are defined on non necessarily totally geodesic Riemannian foliations. We study then in detail the example of the velocity spherical Brownian motion, whose generator is a step-3 generating hypoelliptic Hörmander's type operator. To prove hypocoercivity in that case, the key point is to show the existence of a convenient Riemannian foliation associated to the diffusion. We will then deduce, under suitable geometric conditions, the convergence to equilibrium of the diffusion in $H^1$ and in $L^2$.

math.AP↗
arXiv2015

Kinetic Brownian motion on Riemannian manifolds

We consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle $T^1 \mathcal M$ of a Riemannian manifold $(\mathcal M,g)$, collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter $σ$ quantifying the size of the noise. Projection on $\mathcal M$ of these processes provides random $C^1$ paths in $\mathcal M$. We show, both qualitively and quantitatively, that the laws of these $\mathcal M$-valued paths provide an interpolation between geodesic and Brownian motions. This qualitative description of kinetic Brownian motion as the parameter $σ$ varies is complemented by a thourough study of its long time asymptotic behaviour on rotationally invariant manifolds, when $σ$ is fixed, as we are able to give a complete description of its Poisson boundary in geometric terms.

math.PR↗
arXiv2013

Dévissage of a Poisson boundary under equivariance and regularity conditions

We present a method that allows, under suitable equivariance and regularity conditions, to determine the Poisson boundary of a diffusion starting from the Poisson boundary of a sub-diffusion of the original one. We then give two examples of application of this dévissage method. Namely, we first recover the classical result that the Poisson boundary of Brownian motion on a rotationally symmetric manifolds is generated by its escape angle, and we then give an "elementary" probabilistic proof of the delicate result of [Bai08], i.e. the determination of the Poisson boundary of the relativistic Brownian motion in Minkowski space-time.

math.PR↗
arXiv2013

A Poincaré cone condition in the Poincaré group

Gradinaru and Ben Arous described the behavior of the Green function of a sub-elliptic diffusion. We adapt their proof in the general setting of an hypoelliptic diffusion and deduce a Wiener criterion and a Poincaré cone condition of thinness for a relativistic diffusion in the Poincaré group introduced by Dudley.

math.PR↗