Samuel Herrmann

A singular large deviations phenomenon

Consider {Xte :t⩾0} (e>0), the solution starting from 0 of a stochastic differential equation, which is a small Brownian perturbation of the one-dimensional ordinary differential equation x′t=sgn(xt)|xt|γ (0<γ<1). Denote by pte(x) the density of Xte. We study the exponential decay of the density as e→0. We prove that, for the points (t,x) lying between the extremal solutions of the ordinary differential equation, the rate of the convergence is different from the rate of convergence in large deviations theory (although respected for the points (t,x) which does not lie between the extremals). Proofs are based on probabilistic (large deviations theory) and analytic (viscosity solutions for Hamilton–Jacobi equations) tools.

The exit problem for diffusions with time-periodic drift and stochastic resonance

Physical notions of stochastic resonance for potential diffusions in periodically changing double-well potentials such as the spectral power amplification have proved to be defective. They are not robust for the passage to their effective dynamics: continuous-time finite-state Markov chains describing the rough features of transitions between different domains of attraction of metastable points. In the framework of one-dimensional diffusions moving in periodically changing double-well potentials we design a new notion of stochastic resonance which refines Freidlins concept of quasi-periodic motion. It is based on exact exponential rates for the transition probabilities between the domains of attraction which are robust with respect to the reduced Markov chains. The quality of periodic tuning is measured by the probability for transition during fixed time windows depending on a time scale parameter. Maximizing it in this parameter produces the stochastic resonance points.

BARRIER CROSSINGS CHARACTERIZE STOCHASTIC RESONANCE

In a two-state Markov chain with time periodic dynamics, we study path properties such as the sojourn time in one state between two consecutive jumps or the distribution of the first jump. This is done in order to exhibit a resonance interval and an optimal tuning rate interpreting the phenomenon of stochastic resonance through quality notions related with interspike intervals. We consider two cases representing the reduced dynamics of particles diffusing in time periodic potentials: Markov chains with piecewise constant periodic infinitesimal generators and Markov chains with time-continuous periodic generators.

Hitting time for Bessel processes—walk on moving spheres algorithm (WoMS)

In this article we investigate the hitting time of some given boundaries for Bessel processes. The main motivation is coming from mathematical finance when dealing with volatility models but the results can also be used in optimal control problems. The aim is here to construct a new and efficient algorithm in order to approach this hitting time. As an application we will consider the hitting time of a given level for the Cox-Ingersoll-Ross process. The main tools we use are on one side an adaptation of the method of images to this particular situation and on the other side the connexion existing between Cox-Ingersoll-Ross processes and Bessel processes.

From persistent random walks to the telegraph noise

We study a family of memory-based persistent random walks and we prove weak convergences after space-time rescaling. The limit processes are not only Brownian motions with drift. We have obtained a continuous but non-Markov process

Stochastic resonance : a mathematical approach in the small noise limit

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Two Mathematical Approaches to Stochastic Resonance

which can be easely expressed in terms of a counting process

The walk on moving spheres: A new tool for simulating Brownian motion’s exit time from a domain

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The first-passage time of the Brownian motion to a curved boundary: an algorithmic approach

. In a particular case the counting process is a Poisson process, and

Simulation of hitting times for Bessel processes with non-integer dimension

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