Papiers de recherche
Juillet 2022
Scaling limit of a kinetic inhomogeneous stochastic system in the quadratic potential, avec Thomas Cavallazzi, soumis. .
Disponible sur HAL.
We consider a particle evolving in the quadratic potential and subject to a time-inhomogeneous frictional force and to a random force. The couple of its velocity and position is solution to a stochastic differential equation driven by an 𝛼-stable Lévy process with \(\alpha \in (1,2]\) and the frictional force is of the form \(t^{-\beta}sgn(v)|v|^{\gamma}\). We identify three regimes for the behavior in long-time of the couple velocity-position with a suitable rescaling, depending on the balance between the frictional force and the index of stability \(\alpha\) of the noise.
Décembre 2021
Kinetic time-inhomogeneous Lévy-driven model, avec Mihai Gradinaru, soumis.
Disponible sur HAL.
We study a one-dimensional kinetic stochastic model driven by a Lévy process, with a non-linear time-inhomogeneous drift. More precisely, the process \((V,X)\) is considered, where \(X\) is the position of the particle and its velocity \(V\) is the solution of a stochastic differential equation with a drift of the form \(t^{-\beta}F(v)\). The driving noise can be a stable Lévy process of index \(\alpha\) or a general Lévy process under appropriate assumptions. The function \(F\) satisfies a homogeneity condition and \(\beta\) is non-negative. The behavior in large time of the process \((V,X)\) is proved and the precise rate of convergence is pointed out by using stochastic analysis tools. To this end, we compute the moment estimates of the velocity process.
Mars 2021
Asymptotic behavior for a time-inhomogeneous Kolmogorov type diffusion, avec Mihai Gradinaru, soumis.
Disponible sur HAL.
We study a kinetic stochastic model with a non-linear time-inhomogeneous drag force and a Brownian-type random force. More precisely, the Kolmogorov type diffusion \( (V,X) \) is considered. Here \(X\) is the position of the particle and \(V\) is its velocity and is solution of a stochastic differential equation driven by a one-dimensional Brownian motion, with the drift of the form \(t^{-\beta}F(v)\). The function \(F\) satisfies some homogeneity condition and \(\beta\) is positive. The behaviour of the process \((V,X)\) in large time is proved and the precise rate of convergence is pointed out by using stochastic analysis tools.