Abstract : This minisymposium aims to bring together young researchers in PDEs, probability and applied mathematics to share recent progress and complementary perspectives in the growing field of interacting particle systems. Such systems are not only important models in physics, biology, and many applied sciences, but also rich in mathematical structure.
Our topics include mean-field limits and propagation of chaos for large systems with singular interactions, limit theorems and large deviations for interacting particle systems on random graphs, particle and numerical methods of McKean-Vlasov PDEs, convergence theorems of SPDEs, and quasi-stationary behavior of SPDEs and their dual particle systems.
[05450] Ergodic properties of rank-based diffusions
Format : Online Talk on Zoom
Author(s) :
Sayan Banerjee (University of North Carolina, Chapel Hill)
Amarjit Budhiraja (University of North Carolina, Chapel Hill)
Abstract : We investigate the long-time behavior of rank-based diffusions with infinitely many particles where the drift and diffusivity of each particle depends on its relative rank in the system. Unlike their finite dimensional analogues, such systems have infinitely many stationary measures and domains of attraction and extremality properties of such measures have been long-standing open questions. In this talk, we will explore some of these questions and provide answers to them in certain cases.
Based on joint works with Amarjit Budhiraja.
[04342] Systems with Riesz Interactions in the Mean-Field Regime
Format : Online Talk on Zoom
Author(s) :
Matthew Rosenzweig (MIT)
Sylvia Serfaty (Courant Institute, NYU)
Antonin Chodron de Courcel (Ecole Polytechnique)
Abstract : We present recent results on the large particle number and large time effective behavior of conservative or gradient dynamics for particle systems with mean-field interactions governed by a Coulomb or more general Riesz potential and subject to possible noise modeling thermal fluctuations. The talk will discuss modulated energy/free energy techniques for studying the rate of mean-field convergence, how the rate deteriorates with time, and how fluctuations around the mean-field limit behave.
[03026] Large Deviations for Multiscale Weakly Interacting Diffusions
Format : Online Talk on Zoom
Author(s) :
Zachary Bezemek (Boston University)
Konstantinos Spiliopoulos (Boston University)
Abstract : In this talk, we consider a collection of weakly interacting diffusion processes moving in a two-scale locally periodic environment. We study the large deviations principle of the empirical distribution of the particles' positions in the combined limit as the number of particles grow to infinity and the time-scale separation parameter goes to zero simultaneously. We derive several equivalent formulations of the rate function, making connections between a mean-field control formulation and the formulation of Dawson-Gärtner.
[05239] Hydrodynamic Limits of non-Markovian Interacting Particle Systems on Sparse Graphs
Format : Online Talk on Zoom
Author(s) :
Ankan Ganguly (University of Pennsylvania)
Kavita Ramanan (Brown University)
Abstract : We consider hydrodynamic limits of non-Markovian interacting particle systems on large sparse graphs. Under mild conditions on the jump intensities and underlying graphs, it is shown that if the sequence of interaction graphs $G_n$ converges locally in probability to a limit graph $G$, then the corresponding sequence of empirical measures of the particle trajectories converges weakly to the law of the marginal dynamics at the root vertex of $G$.
[04551] Wave propagation for reaction-diffusion equations on infinite trees
Format : Online Talk on Zoom
Author(s) :
Grigory Terlov (UNC Chapel Hill )
Wai-Tong (Lous) Fan (Indiana University)
Wenqing Hu (Missouri University of S&T)
Abstract : The asymptotic speed of the wavefront of the solution to FKPP equation on $\mathbb{R}$ is well understood. I will present a probabilistic approach to the same problem on infinite metric trees. When the reaction rate is large enough we show that a travelling wavefront emerges. Its speed is slower than that of the same equation on the real line, and we can estimate this slow-down in terms of the structure of the tree.
[03975] From the KPZ equation to the directed landscape
Format : Online Talk on Zoom
Author(s) :
XUAN WU (University of Illinois Urbana-Champaign)
Abstract : This talk presents the convergence of the KPZ equation to the directed landscape, which is the central object in the KPZ universality class. This convergence result is the first to the directed landscape among the positive temperature models.
[05115] Longtime behaviour of the stochastic FKPP equation conditioned on non-fixation
Format : Online Talk on Zoom
Author(s) :
Oliver Kelsey Tough (University of Bath)
Wai-Tong (Louis) Fan (Indiana University)
Abstract : In population genetics, fixation is the phenomenon in which all members of a population have the same copy of a given gene. Whilst most genes are fixed (most genes are shared by all individuals), not all genes are fixed (individuals aren't identical). The prototypical continuum model for the spread of a genetic type in a spatially distributed population under the effects of genetic drift, selection and migration is the stochastic Fisher-Kolmogorov-Petrovsky-Piscunov (FKPP) equation. We consider the stochastic FKPP equation on the circle. We establish existence and uniqueness of the quasi-stationary distribution (QSD) for solutions of the stochastic FKPP, considered to be absorbed upon fixation. We show that the distribution of the solution conditioned on non-fixation converges to this unique QSD as time $t\rightarrow \infty$, for any initial distribution. Moreover we characterise the leading-order asymptotics for the tail distribution of the fixation time. This is based on joint work with Wai-Tong Fan.
[03336] The interacting multiplicative coalescent and Levy-like random fields
Format : Online Talk on Zoom
Author(s) :
David J Clancy, Jr. (University of Wiscons)
Abstract : The multiplicative coalescent describes the evolution of blocks where blocks of masses $x$ and $y$ form a single block of mass $x+y$ at rate $xy$. This process naturally appears when studying many random graph models at criticality. Their marginal laws are described using (mixtures of) Levy-type processes. Using stochastic blockmodels, we show that one can describe the marginal law of two interacting multiplicative coalescences using a Levy-type random field. Based on work with V. Konarovskyi and V. Limic.
Abstract : We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with (random) weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Well-posedness of the graphon particle system is established. A law of large numbers result is proved as the system size increases and the underlying graphons converge.
[03358] Strong convergence of propagation of chaos for McKean-Vlasov SDEs with singular interactions
Format : Online Talk on Zoom
Author(s) :
Zimo Hao (Bielefeld University)
Abstract : In this work we show the strong convergence of propagation of chaos for the particle approximation of McKean-Vlasov SDEs with singular $L^p$-interactions as well as for the moderate interaction particle systems on the level of particle trajectories. One of the main obstacles is to establish the strong well-posedness of the SDEs for particle systems with singular interaction. To this end, we extend the results on strong well-posedness of Krylov and R\"ockner to the case of mixed $L^{\boldsymbol{p}}$-drifts, where the heat kernel estimates play a crucial role. Moreover, when the interaction kernel is bounded measurable, we also obtain the optimal rate of strong convergence, which is partially based on Jabin and Wang's entropy method and Zvonkin's transformation.
[03721] Nonlocal approximation of nonlinear diffusion equations
Format : Talk at Waseda University
Author(s) :
José Antonio Carrillo (University of Oxford)
Antonio Esposito (University of Oxford)
Jeremy S.-H. Wu (UCLA)
Abstract : Nonlinear diffusion equations are ubiquitous in several real world applications. They were introduced to analyse gas expansion in a porous medium, groundwater infiltration, and heat conduction in plasmas, to name a few applications in physics. In this talk, I will present recent joint work with José A. Carrillo and Antonio Esposito concerning a nonlocal approximation inspired by the theory of gradient flows for a general family of equations closely related to the porous medium equation with m>1. Our approximation is inspired by recent ideas to use (nonlocal) interaction equations to approximate (local) diffusion equations. We prove under very general assumptions that weak solutions to our nonlocal approximation converge to weak solutions of the original local equation. One byproduct of our analysis is the development of a deterministic particle method for numerically approximating solutions to nonlinear diffusion equations.
[04156] Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEs
Format : Online Talk on Zoom
Author(s) :
Zebang Shen (ETH Zürich)
Zhenfu Wang (Peking University)
Abstract : We extend the concept of self-consistency for the Fokker-Planck equation (FPE)(Shen et al., 2022) to the more general McKean-Vlasov equation (MVE). While FPE describes the macroscopic behavior of particles under drift and diffusion, MVE accounts for the additional inter-particle interactions, which are often highly singular in physical systems. Two important examples considered in this paper are the MVE with Coulomb interactions and the vorticity formulation of the 2D Navier-Stokes equation. We show that a generalized self-consistency potential controls the KL-divergence between a hypothesis solution to the ground truth, through entropy dissipation. Built on this result, we propose to solve the MVEs by minimizing this potential function, while utilizing the neural networks for function approximation. We validate the empirical performance of our approach by comparing with state-of-the-art NN-based PDE solvers on several example problems.
