arXiv subjects

Thomas Cavallazzi

Publications and source records attributed to Thomas Cavallazzi.

6 recordsLinked to original sources

arXiv2025

Quantitative approximation of a Keller--Segel PDE by a branching moderately interacting particle system and suppression of blow-up

The Keller--Segel PDE is a model for chemotaxis known to exhibit possible finite-time blow-up. Following a seminal work by Tello and Winkler, a logistic damping term is added in this PDE and local well-posedness of mild solutions is proven. When the space dimension is $2$ or when the damping is strong enough, the solution is global in time. In the second part of this work, a microscopic description of this model is introduced in terms of a system of stochastic moderately interacting particles. This system features two main characteristics: the interaction between particles happens through a singular (Coulomb-type) kernel which is attractive; and the particles are subject to demographic events, birth and death due to local competition with other particles. The latter induces a branching structure of the particle system. Then the main result of this work is the convergence of the empirical measure of the particle system towards the Keller--Segel PDE with logistic damping, with a rate of order $N^{-\frac{1}{2(d+1)}}$.

math.PR↗

arXiv2025

Well-posedness and propagation of chaos for L{é}vy-driven McKean-Vlasov SDEs under Lipschitz assumptions

The first goal of this note is to prove the strong well-posedness of McKean-Vlasov SDEs driven by L{é}vy processes on $\mathbb{R}^d$ having a finite moment of order $β\in [1,2]$ and under standard Lipschitz assumptions on the coefficients. Then, we prove a quantitative propagation of chaos result at the level of paths for the associated interacting particle system, with constant diffusion coefficient. Finally, we improve the rates of convergence obtained for linear interactions with respect to the measure and when the noise is a $α$-stable process with $α\in (1,2)$, for which we have $β< α$.

math.PR↗

arXiv2024

Quantitative weak propagation of chaos for stable-driven McKean-Vlasov SDEs

We consider a general McKean-Vlasov stochastic differential equation driven by a rotationally invariant $α$-stable process on $\mathbb{R}^d$ with $α\in (1,2)$. We assume that the diffusion coefficient is the identity matrix and that the drift is bounded and H{ö}lder continuous in some precise sense with respect to both space and measure variables. The main goal of this work is to prove new propagation of chaos estimates, at the level of semigroup, for the associated mean-field interacting particle system. Our study relies on the regularizing properties and the dynamics of the semigroup associated with the McKean-Vlasov stochastic differential equation, which acts on functions defined on $\mathcal{P}_β(\mathbb{R}^d)$, the space of probability measures on $\mathbb{R}^d$ having a finite moment of order $β\in (1,α)$. More precisely, the dynamics of the semigroup is described by a backward Kolmogorov partial differential equation defined on the strip $[0,T] \times \mathcal{P}_β(\mathbb{R}^d)$.

math.AP↗

arXiv2023

Scaling limit of a kinetic inhomogeneous stochastic system in the quadratic potential

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 L{é}vy process with $α\in (1,2]$ and the frictional force is of the form $t^{-β}\text{sgn}(v)|v|^γ$. 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 $α$ of the noise.

math.PR↗

arXiv2022

It{ô}'s formula for the flow of measures of Poisson stochastic integrals and applications

We prove It{ô}'s formula for the flow of measures associated with a jump process defined by a drift, an integral with respect to a Poisson random measure and with respect to the associated compensated Poisson random measure. We work in $\mathcal{P}_β(\mathbb{R}^d)$, the space of probability measures on $\mathbb{R}^d$ having a finite moment of order $β\in (0, 2]$. As an application, we exhibit the backward Kolmogorov partial differential equation stated on $[0,T] \times \mathcal{P}_β(\mathbb{R}^d)$ associated with a McKean-Vlasov stochastic differential equation driven by a Poisson random measure. It describes the dynamics of the semigroup associated with the McKean-Vlasov stochastic differential equation, under regularity assumptions on it. Finally, we use the semigroup and the backward Kolmogorov equation to prove new quantitative weak propagation of chaos results for a mean-field system of interacting Ornstein-Uhlenbeck processes driven by i.i.d. $α$-stable processes with $α\in (1,2)$.

math.PR↗

arXiv2022

It{ô}-Krylov's formula for a flow of measures

We prove It{ô}'s formula for the flow of measures associated with an It{ô} process having a bounded drift and a uniformly elliptic and bounded diffusion matrix, and for functions in an appropriate Sobolev-type space. This formula is the almost analogue, in the measure-dependent case, of the It{ô}-Krylov formula for functions in a Sobolev space on $\mathbf{R}^+ \times \mathbf{R}^d $.

math.PR↗