André Schlichting
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View article: Derivation of the fourth-order DLSS equation with nonlinear mobility via chemical reactions
Derivation of the fourth-order DLSS equation with nonlinear mobility via chemical reactions Open
We provide a derivation of the fourth-order DLSS equation based on an interpretation as a chemical reaction network. We consider the rate equation on the discretized circle for a process in which pairs of particles occupying the same site …
View article: A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport
A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport Open
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. T…
View article: Convergence of a Stochastic Particle System to the Continuous Generalized Exchange-Driven Growth Model
Convergence of a Stochastic Particle System to the Continuous Generalized Exchange-Driven Growth Model Open
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…
View article: Piezo1-induced durotaxis of pancreatic stellate cells depends on TRPC1 and TRPV4 channels
Piezo1-induced durotaxis of pancreatic stellate cells depends on TRPC1 and TRPV4 channels Open
Pancreatic stellate cells (PSCs) are primarily responsible for producing the stiff tumor tissue in pancreatic ductal adenocarcinoma (PDAC). Thereby, PSCs generate a stiffness gradient between the healthy pancreas and the tumor. This gradie…
View article: Diffusive transport on the real line: semi-contractive gradient flows and their discretization
Diffusive transport on the real line: semi-contractive gradient flows and their discretization Open
The diffusive transport distance, a novel pseudo-metric between probability measures on the real line, is introduced. It generalizes Martingale optimal transport, and forms a hierarchy with the Hellinger and the Wasserstein metrics. We obs…
View article: Gradient flows on metric graphs with reservoirs: Microscopic derivation and multiscale limits
Gradient flows on metric graphs with reservoirs: Microscopic derivation and multiscale limits Open
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 …
View article: Solutions of stationary McKean-Vlasov equation on a high-dimensional sphere and other Riemannian manifolds
Solutions of stationary McKean-Vlasov equation on a high-dimensional sphere and other Riemannian manifolds Open
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 p…
View article: Covariance-Modulated Optimal Transport and Gradient Flows
Covariance-Modulated Optimal Transport and Gradient Flows Open
We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Ka…
View article: Variational convergence of the Scharfetter–Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit
Variational convergence of the Scharfetter–Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit Open
In this paper, we explore the convergence of the semi-discrete Scharfetter–Gummel scheme for the aggregation-diffusion equation using a variational approach. Our investigation involves obtaining a novel gradient structure for the finite vo…
View article: Singular-limit analysis of gradient descent with noise injection
Singular-limit analysis of gradient descent with noise injection Open
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 …
View article: Variational convergence for an irreversible exchange-driven stochastic particle system
Variational convergence for an irreversible exchange-driven stochastic particle system Open
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 k…
View article: Piezo1-induced durotaxis of pancreatic stellate cells depends on TRPC1 and TRPV4 channels
Piezo1-induced durotaxis of pancreatic stellate cells depends on TRPC1 and TRPV4 channels Open
Pancreatic stellate cells (PSCs) are primarily responsible for producing the stiff tumor tissue in pancreatic ductal adenocarcinoma (PDAC). Thereby, PSCs generate a stiffness gradient between the healthy pancreas and the tumor. This gradie…
View article: A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport
A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport Open
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. T…
View article: Graph‐to‐local limit for a multi‐species nonlocal cross‐interaction system
Graph‐to‐local limit for a multi‐species nonlocal cross‐interaction system Open
In this note we continue the study of nonlocal interaction dynamics on a sequence of infinite graphs, extending the results of Esposito, Heinze and Schlichting to an arbitrary number of species. Our analysis relies on the observation that …
View article: Graph-to-local limit for a multi-species nonlocal cross-interaction system
Graph-to-local limit for a multi-species nonlocal cross-interaction system Open
In this note we continue the study of nonlocal interaction dynamics on a sequence of infinite graphs, extending the results of [Esposito et. al 2023+] to an arbitrary number of species. Our analysis relies on the observation that the graph…
View article: Graph-to-local limit for the nonlocal interaction equation
Graph-to-local limit for the nonlocal interaction equation Open
We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equa…
View article: Variational convergence of the Scharfetter-Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit
Variational convergence of the Scharfetter-Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit Open
In this paper, we explore the convergence of the semi-discrete Scharfetter-Gummel scheme for the aggregation-diffusion equation using a variational approach. Our investigation involves obtaining a novel gradient structure for the finite vo…
View article: On a class of nonlocal continuity equations on graphs
On a class of nonlocal continuity equations on graphs Open
Motivated by applications in data science, we study partial differential equations on graphs. By a classical fixed-point argument, we show existence and uniqueness of solutions to a class of nonlocal continuity equations on graphs. We cons…
View article: Error estimates for a finite volume scheme for advection–diffusion equations with rough coefficients
Error estimates for a finite volume scheme for advection–diffusion equations with rough coefficients Open
We study the implicit upwind finite volume scheme for numerically approximating the advection–diffusion equation with a vector field in the low regularity DiPerna–Lions setting. That is, we are concerned with advecting velocity fields that…
View article: Covariance-modulated optimal transport and gradient flows
Covariance-modulated optimal transport and gradient flows Open
We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Ka…
View article: Cosh gradient systems and tilting
Cosh gradient systems and tilting Open
We review a class of gradient systems with dissipation potentials of hyperbolic-cosine type. We show how such dissipation potentials emerge in large deviations of jump processes, multi-scale limits of diffusion processes, and more. We show…
View article: Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients
Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients Open
We study the implicit upwind finite volume scheme for numerically approximating the advection-diffusion equation with a vector field in the low regularity DiPerna-Lions setting. That is, we are concerned with advecting velocity fields that…
View article: The Scharfetter–Gummel scheme for aggregation–diffusion equations
The Scharfetter–Gummel scheme for aggregation–diffusion equations Open
In this paper we propose a finite-volume scheme for aggregation–diffusion equations based on a Scharfetter–Gummel approximation of the quadratic, nonlocal flux term. This scheme is analyzed concerning well posedness and convergence towards…
View article: Optimal stability estimates and a new uniqueness result for advection-diffusion equations
Optimal stability estimates and a new uniqueness result for advection-diffusion equations Open
This paper contains two main contributions. First, it provides optimal stability estimates for advection-diffusion equations in a setting in which the velocity field is Sobolev regular in the spatial variable. This estimate is formulated w…
View article: Oscillations in a Becker-Döring model with injection and depletion
Oscillations in a Becker-Döring model with injection and depletion Open
We study the Becker-Döring bubblelator, a variant of the Becker-Döring coagulation-fragmentation system that models the growth of clusters by gain or loss of monomers. Motivated by models of gas evolution oscillators from physical chemistr…
View article: Long-Time Behaviour and Phase Transitions for the Mckean–Vlasov Equation on the Torus
Long-Time Behaviour and Phase Transitions for the Mckean–Vlasov Equation on the Torus Open
We study the McKean-Vlasov equation ∂t% = β −1∆% + κ ∇·(%∇(W ? %)) , with periodic boundary conditions on the torus. We first study the global asymptotic stability of the homogeneous steady state. We then focus our attention on the station…
View article: Poincaré and Log–Sobolev Inequalities for Mixtures
Poincaré and Log–Sobolev Inequalities for Mixtures Open
This work studies mixtures of probability measures on R n and gives bounds on the Poincaré and the log–Sobolev constants of two-component mixtures provided that each component satisfies the functional inequality, and both components are cl…
View article: A non-local problem for the Fokker-Planck equation related to the Becker-Döring model
A non-local problem for the Fokker-Planck equation related to the Becker-Döring model Open
This paper concerns a Fokker-Planck equation on the positive real line\nmodeling nucleation and growth of clusters. The main feature of the equation is\nthe dependence of the driving vector field and boundary condition on a\nnon-local orde…