McKean-Vlasov Equations Research Articles

Superposition and mimicking theorems for conditional McKean–Vlasov equations

We consider conditional McKean–Vlasov stochastic differential equations (SDEs), as the ones arising in the large-system limit of mean field games and particle systems with mean field interactions when common noise is present. The conditional time-marginals of the solutions to these SDEs are governed by non-linear stochastic partial differential equations (SPDEs) of the second order, whereas their laws satisfy Fokker–Planck equations on the space of probability measures. Our paper establishes two superposition principles: The first asserts that any solution of the SPDE can be lifted to a solution of the conditional McKean–Vlasov SDE, and the second guarantees that any solution of the Fokker–Planck equation on the space of probability measures can be lifted to a solution of the SPDE. We use these results to obtain a mimicking theorem which shows that the conditional time-marginals of an Itô process can be emulated by those of a solution to a conditional McKean–Vlasov SDE with Markovian coefficients. This yields, in particular, a tool for converting open-loop controls into Markovian ones in the context of controlled McKean–Vlasov dynamics.

Well-posedness and numerical schemes for one-dimensional McKean–Vlasov equations and interacting particle systems with discontinuous drift

In this paper, we first establish well-posedness results for one-dimensional McKean–Vlasov stochastic differential equations (SDEs) and related particle systems with a measure-dependent drift coefficient that is discontinuous in the spatial component, and a diffusion coefficient which is a Lipschitz function of the state only. We only require a fairly mild condition on the diffusion coefficient, namely to be non-zero in a point of discontinuity of the drift, while we need to impose certain structural assumptions on the measure-dependence of the drift. Second, we study Euler–Maruyama type schemes for the particle system to approximate the solution of the one-dimensional McKean–Vlasov SDE. Here, we will prove strong convergence results in terms of the number of time-steps and number of particles. Due to the discontinuity of the drift, the convergence analysis is non-standard and the usual strong convergence order 1/2 known for the Lipschitz case cannot be recovered for all presented schemes.

Well-posedness and propagation of chaos for McKean–Vlasov equations with jumps and locally Lipschitz coefficients

We study McKean–Vlasov equations where the coefficients are locally Lipschitz continuous. We prove the strong well-posedness and a propagation of chaos property. These questions are classical under the assumptions that the coefficients are Lipschitz continuous. In the locally Lipschitz case, we use truncation arguments and Osgood’s lemma instead of Grönwall’s lemma. Technical difficulties appear in the proofs, in particular for the existence of solution of the McKean–Vlasov equations. This proof relies on a Picard iteration scheme that is not guaranteed to converge in an L1−sense. However, we prove its convergence in distribution, and the (strong) well-posedness of the equation.

Large Deviations and Exit-times for reflected McKean–Vlasov equations with self-stabilising terms and superlinear drifts

We study reflected McKean–Vlasov diffusions over a convex, non-bounded domain with self-stabilising coefficients that do not satisfy the classical Wasserstein Lipschitz condition. We establish existence and uniqueness results for this class and address the propagation of chaos. Our results are of wider interest: without the McKean–Vlasov component they extend reflected SDE theory, and without the reflective term they extend the McKean–Vlasov theory.We prove a Freidlin–Wentzell type Large Deviations Principle and an Eyring–Kramer’s law for the exit-time from subdomains contained in the interior of the reflecting domain. Our characterisation of the rate function for the exit-time distribution is explicit.

Gradient flow formulation of diffusion equations in the Wasserstein space over a Metric graph

<p style='text-indent:20px;'>This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou–Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport. Secondly, in the spirit of Jordan–Kinderlehrer–Otto, we show that McKean–Vlasov equations can be formulated as gradient flow of the free energy in the Wasserstein space of probability measures. The proofs of these results are based on careful regularisation arguments to circumvent some of the difficulties arising in metric graphs, namely, branching of geodesics and the failure of semi-convexity of entropy functionals in the Wasserstein space.</p>

McKean-Vlasov equations with positive feedback through elastic stopping times

We prove existence and uniqueness of physical and minimal solutions to McKean-Vlasov equations with positive feedback through elastic stopping times. We do this by establishing a relationship between this problem and a problem with absorbing stopping times. We show convergence of a particle system to the McKean-Vlasov equation. Moreover, we establish convergence of the elastic McKean-Vlasov problem to the problem with absorbing stopping times and to a reflecting Brownian motion as the elastic parameter goes to infinity or zero respectively.

Long time behavior of stochastic Mckean–Vlasov equations

In this short paper, we consider the long time behavior of stochastic Mckean–Vlasov equations. The exponential, polynomial and logarithmic decay for stochastic equations are considered. Sufficient conditions are given to obtain these exponents.

Local Stability of McKean-Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies.

A family of heterogeneous mean-field systems with jumps is analyzed. These systems are constructed as a Gibbs measure on block graphs. When the total number of particles goes to infinity, the law of large numbers is shown to hold in a multi-class context, resulting in the weak convergence of the empirical vector towards the solution of a McKean–Vlasov system of equations. We then investigate the local stability of the limiting McKean–Vlasov system through the construction of a local Lyapunov function. We first compute the limit of adequately scaled relative entropy functions associated with the explicit stationary distribution of the N-particles system. Using a Laplace principle for empirical vectors, we show that the limit takes an explicit form. Then we demonstrate that this limit satisfies a descent property, which, combined with some mild assumptions shows that it is indeed a local Lyapunov function.

Central Limit Theorem and Moderate Deviation Principle for McKean-Vlasov SDEs

Under a Lipschitz condition on distribution dependent coefficients, the central limit theorem and the moderate deviation principle are obtained for solutions of McKean-Vlasov type stochastic differential equations, which generalize the corresponding results for classical stochastic differential equations to the distribution dependent setting.

Long-Time Behaviors of Mean-Field Interacting Particle Systems Related to McKean–Vlasov Equations

In this paper, we investigate gradient estimate of the Poisson equation and the exponential convergence in the Wasserstein metric $W_{1,d_{l^1}}$, uniform in the number of particles, and uniform-in-time propagation of chaos for the mean-field weakly interacting particle system related to McKean-Vlasov equation. By means of the known approximate componentwise reflection coupling and with the help of some new cost function, we obtain explicit estimates for those three problems, avoiding the technical conditions in the known results. Our results apply when the confinement potential $V$ has many wells, the interaction potential $W$ has bounded second mixed derivative $\nabla^2_{xy}W$ which should be not too big so that there is no phase transition. As an application, we obtain the concentration inequality of the mean-field interacting particle system with explicit and sharp constants, uniform in time. Several examples are provided to illustrate the results.

Rough nonlocal diffusions

We consider a nonlinear Fokker–Planck equation driven by a deterministic rough path which describes the conditional probability of a McKean–Vlasov diffusion with “common” noise. To study the equation we build a self-contained framework of non-linear rough integration theory which we use to study McKean–Vlasov equations perturbed by rough paths. We construct an appropriate notion of solution of the corresponding Fokker–Planck equation and prove well-posedness.

Strong Averaging Principle for Two-Time-Scale Stochastic McKean-Vlasov Equations

Strong Averaging Principle for Two-Time-Scale Stochastic McKean-Vlasov Equations

Solutions for nonlinear Fokker–Planck equations with measures as initial data and McKean-Vlasov equations

One proves the existence and uniqueness of a generalized (mild) solution for the nonlinear Fokker–Planck equation (FPE)ut−Δ(β(u))+div(D(x)b(u)u)=0,t≥0,x∈Rd,d≠2,u(0,⋅)=u0,in Rd, where u0∈L1(Rd), β∈C2(R) is a nondecreasing function, b∈C1, bounded, b≥0, D∈L∞(Rd;Rd) with divD∈(L2+L∞)(Rd), and (divD)−∈L∞(Rd), β strictly increasing, if b is not constant. Moreover, t→u(t,u0) is a semigroup of contractions in L1(Rd), which leaves invariant the set of probability density functions in Rd. If divD≥0, β′(r)≥a|r|α−1, and |β(r)|≤Crα, α≥1, d≥3, then |u(t)|L∞≤Ct−dd+(α−1)d|u0|22+(m−1)d, t>0, and if D∈L2(Rd;Rd) the existence extends to initial data u0 in the space Mb of bounded measures in Rd. As a consequence for arbitrary initial laws, we obtain weak solutions to a class of McKean-Vlasov SDEs with coefficients which have singular dependence on the time marginal laws.

First-order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems.

In this paper, we derive fully implementable first-order time-stepping schemes for McKean-Vlasov stochastic differential equations, allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretized interacting particle system associated with the McKean-Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with finite second-order moments. In addition, numerical examples are presented which support our theoretical findings.

Euler–Maruyama Approximations for Stochastic McKean–Vlasov Equations with Non-Lipschitz Coefficients

In this paper, we study a type of stochastic McKean–Vlasov equations with non-Lipschitz coefficients. Firstly, by an Euler–Maruyama approximation the existence of its weak solutions is proved. Then we observe the pathwise uniqueness of its weak solutions. Finally, it is shown that the Euler–Maruyama approximation has an optimal strong convergence rate.

Convergence in Monge-Wasserstein Distance of Mean Field Systems with Locally Lipschitz Coefficients

This paper focuses on stochastic systems of weakly interacting particles whose dynamics depend on the empirical measures of the whole populations. The drift and diffusion coefficients of the dynamical systems are assumed to be locally Lipschitz continuous and satisfy global linear growth condition. The limits of such systems as the number of particles tends to infinity are studied, and the rate of convergence of the sequences of empirical measures to their limits in terms of pth Monge-Wasserstein distance is established. We also investigate the existence, uniqueness, and boundedness, and continuity of solutions of the limiting McKean-Vlasov equations associated to the systems.

On mean field systems with multi-classes

This work focuses on stochastic systems of weakly interacting particles containing different populations represented by multi-classes. The dynamics of each particle depends not only on the empirical measure of the whole population but also on those of different populations. The limits of such systems as the number of particles tends to infinity are investigated. We establish the existence, uniqueness, and basic properties of solutions to the limiting McKean-Vlasov equations of these systems and then obtain the rate of convergence of the sequences of empirical measures associated with the systems to their limits in terms of the \begin{document}$ p^{\text{th}} $\end{document} Monge-Wasserstein distance.

Freidlin–Wentzell LDP in path space for McKean–Vlasov equations and the functional iterated logarithm law

We show two Freidlin–Wentzell-type Large Deviations Principles (LDP) in path space topologies (uniform and Hölder) for the solution process of McKean–Vlasov Stochastic Differential Equations (MV-SDEs) using techniques which directly address the presence of the law in the coefficients and altogether avoiding decoupling arguments or limits of particle systems. We provide existence and uniqueness results along with several properties for a class of MV-SDEs having random coefficients and drifts of superlinear growth. As an application of our results, we establish a functional Strassen-type result (law of iterated logarithm) for the solution process of a MV-SDE.

State and Parameter Estimation from Observed Signal Increments.

The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean–Vlasov equations as the starting point to derive ensemble Kalman–Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems.

On a strong form of propagation of chaos for McKean-Vlasov equations

This note shows how to considerably strengthen the usual mode of convergence of an $n$-particle system to its McKean-Vlasov limit, often known as propagation of chaos, when the volatility coefficient is nondegenerate and involves no interaction term. Notably, the empirical measure converges in a much stronger topology than weak convergence, and any fixed $k$ particles converge in total variation to their limit law as $n\rightarrow \infty $. This requires minimal continuity for the drift in both the space and measure variables. The proofs are purely probabilistic and rather short, relying on Girsanov’s and Sanov’s theorems. Along the way, some modest new existence and uniqueness results for McKean-Vlasov equations are derived.