This work focuses on the quantitative contraction rates for McKean-Vlasov stochastic differential equations (SDEs) with multiplicative noise. Under suitable conditions on the coefficients of the SDE, this paper derives explicit quantitative contraction rates for the convergence in Wasserstein distances of McKean-Vlasov SDEs using the coupling method. The contraction results are then used to prove a propagation of chaos uniformly in time, which provides quantitative bounds on convergence rate of interacting particle systems, and establishes exponential ergodicity for McKean-Vlasov SDEs.

Abstract In this work, we study the asymptotic behavior of mixture of experts (MoE) trained via gradient flow on supervised learning problems. Our main result establishes the propagation of chaos for a MoE as the number of experts diverges. We demonstrate that the corresponding empirical measure of their parameters is close to a probability measure that solves a nonlinear continuity equation, and we provide an explicit convergence rate that depends solely on the number of experts and on the dimensionality of the parameter space. We apply our results to a MoE generated by a quantum neural network.

Abstract We propose a novel approach to numerically approximate McKean–Vlasov stochastic differential equations (MV-SDEs) using stochastic gradient descent (SGD) while avoiding the use of interacting particle systems (IPSs) and the associated simulation costs required to achieve the ‘propagation of chaos’ limit. The SGD technique is deployed to solve a Euclidean minimization problem, obtained by first representing the MV-SDE as a minimization problem over the set of continuous functions of time, and then approximating the domain with a finite-dimensional sub-space. Convergence is established by proving certain intermediate stability and moment estimates of the relevant stochastic processes, including the tangent processes. Numerical experiments illustrate the competitive performance of our SGD-based method compared with the IPS benchmarks. This work offers a theoretical foundation for using the SGD method in the context of numerical approximation of MV-SDEs, and provides analytical tools to study its stability and convergence.

Quantitative Propagation of Chaos for 2D Viscous Vortex Model on the Whole Space

In this paper, we investigate a strong convergence of propagation of chaos for the Linear-Formation particle model. We obtain an explicit bound on the relative entropy between the joint law of the particle system and the tensorized law of the Linear-Formation kinetic model. This result implies the mean-field limit for the Linear-Formation particle system and propagation of chaos via strong convergence of all marginal distributions. The proof is based on the modified law of large numbers for Jabin and Wang’s relative entropy at an exponential or large deviation scale.

For a class of time inhomogenous distribution dependent birth-death processes, we derive the well-posedness, Wp-estimate, exponential ergodicity, and uniform in time propagation of chaos. These extend the corresponding results derived for distribution dependent stochastic differential equations and mean field particle systems. As preparation, a criterion on the well-posedness of inhomogenous jump process is presented in the end of the paper, which should be interesting by itself.

Abstract Genetic algorithms are high‐level heuristic optimization methods that enjoy great popularity thanks to their intuitive description, flexibility, and, of course, effectiveness. The optimization procedure is based on the evolution of possible solutions through three mechanisms: selection, mutation, and crossover. In this paper, we look at the algorithm as an interacting particle system and show that it is described by a Boltzmann‐type equation in the many‐particle limit. Specifically, we prove a propagation of chaos result using a technique that leverages the optimal transport formulation of the bounded Lipschitz norm and naturally incorporates the crossover mechanism into the analysis. The convergence admits a rate with respect to the number of particles, corresponding to the optimal rate in the Wasserstein‐1 distance.

Quantitative Propagation of Chaos in $$L^\eta (\eta \in [0,1])$$-Wasserstein Distance for Mean Field Interacting Particle System

In this paper, we consider a class of McKean–Vlasov stochastic differential equations with oblique reflection on non-smooth time-dependent domains. We establish existence and uniqueness results for this class, address the propagation of chaos, and prove a Freidlin–Wentzell type large deviation principle (LDP). The analysis of Lundström et al. [Stochastic and partial differential equations on non-smooth time-dependent domains, Stoch. Process. Appl. 129(4) (2019), pp. 1097–1131] and a pair of recently developed sufficient conditions for the weak convergence method by Liu et al. [Large and moderate deviation principles for McKean–Vlasov SDEs with Jumps. Potential Anal. 2023;59(3):1141–1190] play an important role.

Propagation of chaos in infinite horizon and numerical stability for stochastic McKean-Vlasov equations

Sharp local propagation of chaos for mean field particles with W−1,∞ kernels

Abstract We derive quantitative propagation of chaos in the sense of relative entropy for the 2D viscous vortex model with general circulations, approximating the vorticity formulation of the 2D Navier–Stokes equation on the whole Euclidean space. Our results work on the general setting that the vortices are positioned on the whole space R 2 and that the circulations are allowed to be in different magnitudes and orientations, which can be adapted to general unconfined realistic fluids with vorticity that may change sign. We provide explicit convergence rates which are optimal in N and optimal in t among existing literature. The key technical tools, which are our major novelty, are the sharp logarithmic growth estimates and a new ODE hierarchy and iterated integral estimates.

We consider a metapopulation made up of K demes, each containing N individuals bearing a heritable quantitative trait. Demes are connected by migration and undergo independent Moran processes with mutation and selection based on trait values. Mutation and migration rates are tuned so that each deme receives a migrant or a mutant in the same slow timescale and is thus essentially monomorphic at all times for the trait value (adaptive dynamics). In the timescale of mutation/migration, the metapopulation can then be seen as a giant spatial Moran model with size K that we characterize. As Krightarrow infty and physical space becomes continuous, the empirical distribution of the trait value (over the physical and trait spaces) evolves deterministically according to an integro-differential evolution equation. In this limit, the trait value of every migrant is drawn from this global distribution, so that conditional on its initial state, trait values from finitely many demes evolve independently (propagation of chaos). Under mean-field dispersal, the value X_t of the trait at time t and at any given location has a law denoted mu _t and a jump kernel with two terms: a mutation-fixation term and a migration-fixation term involving mu _{t-} (McKean–Vlasov equation). In the limit where mutations have small effects and migration is further slowed down accordingly, we obtain the convergence of X, in the new migration timescale, to the solution of a stochastic differential equation which can be referred to as a new, canonical jump-diffusion of adaptive dynamics. This equation includes an advection term representing selection, a diffusive term due to genetic drift, and a jump term, representing the effect of migration, to a state distributed according to its own law.

McKean-Vlasov stochastic variational inequalities with oblique subgradients and propagation of chaos

Regularity and propagation of chaos for conditional McKean–Vlasov equations

Dimension-independent L convergence rate of propagation of chaos and numerical analysis for McKean-Vlasov stochastic differential equations

In this paper, the reflected Mckean-Vlasov equations with regime-switching over a convex domain is studied for the first time. We first establish the well-posedness of stochastic differential equations with regime-switching by using fixed point Theorem. Then, by using coupling method, the propagation of chaos for the particle system is also obtained.

A model for the evolution of a large population interacting system is considered in which a marked Poisson processes influences their evolution, together with a Brownian motion. Mean field McKean–Vlasov limits of such system are formulated studying first both systems individually. By letting the population size growing to infinite, the weak convergence of the solutions of such systems is proved. In other words, propagation of chaos of such systems is obtained.

Well-Posedness and Propagation of Chaos for McKean–Vlasov Stochastic Variational Inequalities

Uniform-in-time propagation of chaos for mean field Langevin dynamics