Institute of Applied Analysis
University Ulm
Helmholtzstraße 18
89081 Ulm
Room E.10
Tel.: +49 731 50-23560
andre.schlichting@uni-ulm.de
News and Updates
- Extended updated preprint: Variational convergence for an irreversible exchange-driven stochastic particle system
Together with Jasper Hoeksema and Chin Yin Lam, we considerably extended and updated the previous version of the arXiv preprint 2401.06696.
We show the variational convergence of an irreversible Markov jump process describing a finite stochastic particle system to the solution of a countable infinite system of deterministic time-inhomogeneous quadratic differential equations known as the exchange-driven growth model, which has two conserved quantities. As a bounded perturbation of the reversible kernel, the variational formulation is a generalization of the gradient flow formulation of the reversible process and can be interpreted as the large deviation functional of the Markov jump process. As a consequence of the variational convergence result, we show the propagation of chaos of the Markov processes to the limiting equation and the -convergence of the energy functional. The latter convergence is consistent with related results for reversible coagulation-fragmentation equations and reveals the connection of stochastic processes to the long-time condensation phenomena in the limit equation. - 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. This can be viewed as both a continuous generalization of the exchange-driven growth model and a coagulation-fragmentation equation that generalizes the continuous Smoluchowski equation.
Starting from a Markov jump process that describes a finite stochastic interacting particle system with exchange dynamics, we prove the weak law of large numbers for this process for sublinearly growing kernels in the mean-field limit. We establish the tightness of the stochastic process on a measure-valued Skorokhod space induced by the 1-Wasserstein metric, from which we deduce the existence of solutions to the (CGEDG) system. The solution is shown to have a Lebesgue density under suitable assumptions on the initial data. Moreover, within the class of solutions with density, we establish the uniqueness under slightly more restrictive conditions on the kernel.
arXiv:2503.21572 - 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. This observation is made rigorous for a suitable spatial discretization of the considered PDEs: these are semi-contractive gradient flows with respect to an adapted diffusive transport distance for measures on the point lattice. The main result is that the modulus of convexity is uniform with respect to the lattice spacing. Particularly for the quadratic porous medium equation, this is in contrast to what has been observed for discretizations of the Wasserstein gradient flow structure.
arXiv:2501.14527 - Preprint: Gradient flows on metric graphs with reservoirs: Microscopic derivation and multiscale limits
Together with Georg Heinze and Jan-Frederik Pietschmann, we study evolution equations on metric graphs with reservoirs, that is graphs where a one-dimensional interval is associated to each edge and, in addition, the vertices are able to store and exchange mass with these intervals. Focusing on the case where the dynamics are driven by an entropy functional defined both on the metric edges and vertices, we provide a rigorous understanding of such systems of coupled ordinary and partial differential equations as (generalized) gradient flows in continuity equation format. Approximating the edges by a sequence of vertices, which yields a fully discrete system, we are able to establish existence of solutions in this formalism. Furthermore, we study several scaling limits using the recently developed framework of EDP convergence with embeddings to rigorously show convergence to gradient flows on reduced metric and combinatorial graphs. Finally, numerical studies confirm our theoretical findings and provide additional insights into the dynamics under rescaling.
arXiv:2412.16775 - 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. On a sphere, we employ the properties of spherical convolution to study the bifurcation branches around the uniform state. We also give a sufficient condition for an existence of a discontinuous transition point in terms of the interaction kernel and compare it to the Euclidean setting. We illustrate our results on a range of system, including the particle system arising from the transformer models and the Onsager model of liquid crystals.
arXiv:2412.14813