Author Archives: André

Preprint: Convergence of a Stochastic Particle System to the Continuous Generalized Exchange-Driven Growth Model

Together with Chun Yin Lam, we study the continuous generalized exchange-driven growth model (CGEDG) is a system of integro-differential equations describing the evolution of cluster mass under mass exchange. The rate of exchange depends on the masses of the clusters involved and the mass being exchanged.… Read the rest

Preprint: Diffusive transport on the real line: semi-contractive gradient flows and their discretization

Together with Daniel Matthes and Eva-Maria Rott, we introduce the diffusive transport distance, a novel pseudo-metric between probability measures on the real line. It generalizes Martingale optimal transport, and forms a hierarchy with the Hellinger and the Wasserstein metrics. We observe that certain classes of parabolic PDEs, among them the porous medium equation of exponent two, are formally semi-contractive metric gradient flows in the new distance.… Read the rest

Preprint: Solutions of stationary McKean-Vlasov equation on a high-dimensional sphere and other Riemannian manifolds

Together with Anna Shalova, we study stationary solutions of McKean-Vlasov equation on a high-dimensional sphere and other compact Riemannian manifolds. We extend the equivalence of the energetic problem formulation to the manifold setting and characterize critical points of the corresponding free energy functional.… Read the rest

Preprint: Singular-limit analysis of gradient descent with noise injection

Together with Anna Shalova and Mark Peletier, we study the limiting dynamics of a large class of noisy gradient descent systems in the overparameterized regime. In this regime the set of global minimizers of the loss is large, and when initialized in a neighbourhood of this zero-loss set a noisy gradient descent algorithm slowly evolves along this set.… Read the rest

Preprint: Variational convergence of exchange-driven stochastic particle systems in the thermodynamic limit

Together with Chun Yin Lam we consider the thermodynamic limit of mean-field stochastic particle systems on a complete graph. The evolution of occupation number at each vertex is driven by particle exchange with its rate depending on the population of the starting vertex and the destination vertex, including zero-range and misanthrope process.… Read the rest

Preprint: Modelling and numerical analysis for a durotaxis model in “Piezo1-induced durotaxis of pancreatic stellate cells depends on TRPC1 and TRPV4 channels”

In this interdisciplinary project with a group from the Institute of Physiology II in Münster and the Department of Medicine at Duke, together with Angela Stevens we contributed a mathematical model and numerical analysis for a phenomenological model of durotaxis with mechanosensitive ion channels.… Read the rest

Preprint: A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport:

Together with Daniel Matthes, Eva-Maria Rott and Giuseppe Savaré we propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle. Our choice of discretization is motivated by a novel gradient flow formulation with respect to a metric that generalizes martingale transport.… Read the rest

Conference announcement: PDE & Probability in interaction: functional inequalities, optimal transport and particle systems

Together with Pierre Monmarché (Sorbonne Université), Julien Reygner (École des Ponts ParisTech), and Marielle Simon (Université de Lille), we are delighted to announce the upcoming workshop “PDE & Probability in interaction: functional inequalities, optimal transport and particle systems”.

The event will be held from January 22 to 26, 2024, at CIRM in Marseille.… Read the rest