Propagation of chaos: A review of models, methods and applications. Ⅱ. Applications

Chaintron, Louis-Pierre; Diez, Antoine

doi:10.3934/krm.2022018

Cited by 29 publications

(9 citation statements)

References 295 publications

(713 reference statements)

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“…Nonlinear overdamped/kinetic Langevin diffusions have also been studied in the context of non-convex learning [43,47]. For a broad, two-part survey on nonlinear processes, propagation of chaos, and applications, see [14,15].…”

Section: Introductionmentioning

confidence: 99%

Mixing time guarantees for unadjusted Hamiltonian Monte Carlo

Bou-Rabee

Eberle

2023

Bernoulli

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We present a nonlinear (in the sense of McKean) generalization of Hamiltonian Monte Carlo (HMC) termed nonlinear HMC (nHMC) capable of sampling from nonlinear probability measures of mean-field type. When the underlying confinement potential is K-strongly convex and L-gradient Lipschitz, and the underlying interaction potential is gradient Lipschitz, nHMC can produce an ε-accurate approximation of a d-dimensional nonlinear probability measure in L 1 -Wasserstein distance using O((L/K) log(1/ε)) steps. Owing to a uniform-in-steps propagation of chaos phenomenon, and without further regularity assumptions, unadjusted HMC with randomized time integration for the corresponding particle approximation can achieve ε-accuracy in L 1 -Wasserstein distance using O((L/K) 5/3 (d/K) 4/3 (1/ε) 8/3 log(1/ε)) gradient evaluations. These mixing/complexity upper bounds are a specific case of more general results developed in the paper for a larger class of non-logconcave, nonlinear probability measures of mean-field type.

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Section: Introductionmentioning

confidence: 99%

Mixing time guarantees for unadjusted Hamiltonian Monte Carlo

Bou-Rabee

Eberle

2023

Bernoulli

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“…On the other hand, in the slow switching limit one could apply DDFT for fixed N(t) = n and then consider switching between the n-dependent An alternative way of deriving a closed hPDE for an effective one-particle density is to use mean field theory. In the case of non-switching, weakly-interacting Brownian particles, there is an extensive mathematical literature on the mean field limit (or propagation of chaos), see for example [59][60][61][62]. More specifically, suppose that both the external and interaction potentials are independent of the environmental state, and take K = K 0 /N where K 0 is a smooth function.…”

Section: Interacting Brownian Particles and Mean Field Theorymentioning

confidence: 99%

Global density equations for a population of actively switching particles

Bressloff

2024

J. Phys. A: Math. Theor.

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There are many processes in cell biology that can be modelled in terms of an actively switching particle. The continuous degrees of freedom of the particle evolve according to a hybrid stochastic differential equation (hSDE) whose drift term depends on a discrete internal or environmental state that switches according to a continuous time Markov chain. Examples include Brownian motion in a randomly switching environment, membrane voltage fluctuations in neurons, protein synthesis in gene networks, bacterial run-and-tumble motion, and motor-driven intracellular transport. In this paper we derive generalized Dean-Kawasaki (DK) equations for a population of actively switching particles, either independently switching or subject to a common randomly switching environment. In the case of a random environment, we show that the global particle density evolves according to a hybrid DK equation. Averaging with respect to the Gaussian noise processes in the absence of particle interactions yields a hybrid partial differential equation for the one-particle density. We use this to show how a randomly switching environment induces statistical correlations between the particles. We also discuss methods for handling the moment closure problem for interacting particles, including dynamical density functional theory and mean field theory. We then develop the analogous constructions for independently switching particles. In order to derive a DK equation, we introduce a discrete set of global densities that are indexed by the single-particle internal states, and take expectations with respect to the switching process. However, the resulting DK equation is no longer closed when particle interactions are included. We conclude by deriving Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD) path integrals for the global density equations in the absence of interactions, and relate this to recent field theoretic studies of Brownian gases and run-and-tumble particles (RTPs).

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“…When b and σ are Lipschitz continuous in x and the unknown distribution µ with respect to the Euclidean distance and Wasserstein metric, it is by now standard to show the existence and uniqueness of strong solutions to DDSDE (1.42) (cf. [67] and [13]). Moreover, under some one-side Lipschitz assumptions, Wang [71] showed the strong well-posedness and some functional inequalities to DDSDE (1.42).…”

Section: Introductionmentioning

confidence: 98%

“…In statistical sense, it is assumed that µ N t weakly converges to some density f (t, x, v)dxdv, which is also called mean-field limit or propagation of chaos in Kac's sense (see [44], [35] and [13]), where the density f satisfies the following Vlasov-Poisson-Fokker-Planck equation (abbreviated as VPFP)…”

Section: Introductionmentioning

confidence: 99%

Second order fractional mean-field SDEs with singular kernels and measure initial data

Hao¹,

Röckner²,

Zhang³

2023

Preprint

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In this paper we establish the local and global well-posedness of weak and strong solutions to second order fractional mean-field SDEs with singular/distribution interaction kernels and measure initial value, where the kernel can be Newton or Coulomb potential, Riesz potential, Biot-Savart law, etc. Moreover, we also show the stability, smoothness and the short time singularity and large time decay estimates of the distribution density. Our results reveal a phenomenon that for nonlinear mean-field equations, the regularity of the initial distribution could balance the singularity of the kernel. The precise relationship between the singularity of kernels and the regularity of initial value are calculated, which belongs to the subcritical regime in scaling sense. In particular, our results provide microscopic probability explanation and establish a unified treatment for many physical models such as fractional Vlasov-Poisson-Fokker-Planck system, the vorticity formulation of 2D-fractal Navier-Stokes equations, surface quasi-geostrophic models, fractional porous medium equation with viscosity, etc.

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Propagation of chaos: A review of models, methods and applications. Ⅱ. Applications

Cited by 29 publications

References 295 publications

Mixing time guarantees for unadjusted Hamiltonian Monte Carlo

Mixing time guarantees for unadjusted Hamiltonian Monte Carlo

Global density equations for a population of actively switching particles

Second order fractional mean-field SDEs with singular kernels and measure initial data

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