Grégoire Véchambre

Bio: Grégoire Véchambre is an academic researcher from Chinese Academy of Sciences. The author has contributed to research in topics: Lévy process & Smoothness (probability theory). The author has an hindex of 3, co-authored 10 publications receiving 17 citations.

Renewal structure and local time for diffusions in random environment

Abstract: We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W_\kappa$, with $ 0 0$. In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite sites. These limits can be written explicitly from a two dimensional stable Levy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of $X$.

Exponential functionals of spectrally one-sided lévy processes conditioned to stay positive

Abstract: We study the properties of the exponential functional $\int_0^{+ \infty} e^{- X^{\uparrow} (t)}dt$ where $X^{\uparrow}$ is a spectrally one-sided Levy process conditioned to stay positive. In particular, we study finiteness, self-decomposability, existence of finite exponential moments, asymptotic tail at $0$ and smoothness of the density.

Moran model and Wright-Fisher diffusion with selection and mutation in a one-sided random environment

Abstract: Consider a two-type Moran population of size $N$ subject to selection and mutation, which is immersed in a varying environment. The population is susceptible to exceptional changes in the environment, which accentuate the selective advantage of the fit individuals. In this setting, we show that the type-composition in the population is continuous with respect to the environment. This allows us to replace the deterministic environment by a random one, which is driven by a subordinator. Assuming that selection, mutation and the environment are weak in relation to $N$, we show that the type-frequency process, with time speed up by $N$, converges as $N\to\infty$ to a Wright--Fisher-type SDE with a jump term modeling the effect of the environment. Next, we study the asymptotic behavior of the limiting model in the far future and in the distant past, both in the annealed and in the quenched setting. Our approach builds on the genealogical picture behind the model. The latter is described by means of an extension of the ancestral selection graph (ASG). The formal relation between forward and backward objects is given in the form of a moment duality between the type-frequency process and the line-counting process of a pruned version of the ASG. This relation yields characterizations of the annealed and the quenched moments of the asymptotic type distribution. A more involved pruning of the ASG allows us to obtain annealed and quenched results for the ancestral type distribution. In the absence of mutations, one of the types fixates and our results yield expressions for the fixation probabilities.

Path decomposition of a spectrally negative Lévy process, and local time of a diffusion in this environment

Abstract: We study the convergence in distribution of the supremum of the local time and of the favorite site for a transient diffusion in a spectrally negative Levy potential. To do so, we study the h-valleys of a spectrally negative Levy process, and we prove in partiular that the renormalized sequence of the h-minima converges to the jumping times sequence of a standard Poisson process.

Almost sure behavior for the local time of a diffusion in a spectrally negative Lévy environment

Abstract: We study the almost sure asymptotic behavior of the supremum of the local time for a transient diffusion in a spectrally negative Levy environment. In particular, we link this behavior with the left tail of an exponential functional of the environment conditioned to stay positive.

Lévy processes and infinitely divisible distributions

Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Localization and number of visited valleys for a transient diffusion in random environment

Abstract: We consider a transient diffusion in a $(-\kappa/2)$-drifted Brownian potential $W_{\kappa}$ with $0<\kappa<1$. We prove its localization at time $t$ in the neighborhood of some random points depending only on the environment, which are the positive $h_t$-minima of the environment, for $h_t$ a bit smaller than $\log t$. We also prove an Aging phenomenon for the diffusion, a renewal theorem for the hitting time of the farthest visited valley, and provide a central limit theorem for the number of valleys visited up to time $t$. The proof relies on a decomposition of the trajectory of $W_{\kappa}$ in the neighborhood of $h_t$-minima, with the help of results of A. Faggionato, and on a precise analysis of exponential functionals of $W_{\kappa}$ and of $W_{\kappa}$ Doob-conditioned to stay positive.

The Maximum of the Local Time of a Diffusion Process in a Drifted Brownian Potential

Abstract: We consider a one-dimensional diffusion process X in a (−κ∕2)-drifted Brownian potential for κ ≠ 0. We are interested in the maximum of its local time, and study its almost sure asymptotic behaviour, which is proved to be different from the behaviour of the maximum local time of the transient random walk in random environment. We also obtain the convergence in law of the maximum local time of X under the annealed law after suitable renormalization when κ ≥ 1. Moreover, we characterize all the upper and lower classes for the hitting times of X, in the sense of Paul Levy, and provide laws of the iterated logarithm for the diffusion X itself. To this aim, we use annealed technics.

Renewal structure and local time for diffusions in random environment

Abstract: We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W_\kappa$, with $ 0 0$. In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite sites. These limits can be written explicitly from a two dimensional stable Levy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of $X$.

A slow transient diffusion in a drifted stable potential

Abstract: We consider a diffusion process $X$ in a random potential $\V$ of the form $\V_x = §_x -\delta x$ where $\delta$ is a positive drift and $§$ is a strictly stable process of index $\alpha\in (1,2)$ with positive jumps. Then the diffusion is transient and $X_t / \log^\alpha t$ converges in law towards an exponential distribution. This behaviour contrasts with the case where $\V$ is a drifted Brownian motion and provides an example of a transient diffusion in a random potential which is as "slow" as in the recurrent setting.