Search ResultsShowing 1-12 of 12
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arXiv:2409.08882 (Published 2024-09-13)
Quantitative propagation of chaos for non-exchangeable diffusions via first-passage percolation
Categories: math.PRThis paper develops a non-asymptotic approach to mean field approximations for systems of $n$ diffusive particles interacting pairwise. The interaction strengths are not identical, making the particle system non-exchangeable. The marginal law of any subset of particles is compared to a suitably chosen product measure, and we find sharp relative entropy estimates between the two. Building upon prior work of the first author in the exchangeable setting, we use a generalized form of the BBGKY hierarchy to derive a hierarchy of differential inequalities for the relative entropies. Our analysis of this complicated hierarchy exploits an unexpected but crucial connection with first-passage percolation, which lets us bound the marginal entropies in terms of expectations of functionals of this percolation process.
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arXiv:2403.05454 (Published 2024-03-08)
Quantitative Propagation of Chaos for Singular Interacting Particle Systems Driven by Fractional Brownian Motion
We consider interacting systems particle driven by i.i.d. fractional Brownian motions, subject to irregular, possibly distributional, pairwise interactions. We show propagation of chaos and mean field convergence to the law of the associated McKean--Vlasov equation, as the number of particles $N\to\infty$, with quantitative sharp rates of order $N^{-1/2}$. Our results hold for a wide class of possibly time-dependent interactions, which are only assumed to satisfy a Besov-type regularity, related to the Hurst parameter $H\in (0,+\infty)\setminus \mathbb{N}$ of the driving noises. In particular, as $H$ decreases to $0$, interaction kernels of arbitrary singularity can be considered, a phenomenon frequently observed in regularization by noise results. Our proofs rely on a combinations of Sznitman's direct comparison argument with stochastic sewing techniques.
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Quantitative Propagation of Chaos in $L^η(η\in(0,1])$-Wasserstein distance for Mean Field Interacting Particle System
Comments: 17 pagesCategories: math.PRIn this paper, quantitative propagation of chaos in $L^\eta$($\eta\in(0,1]$)-Wasserstein distance for mean field interacting particle system is derived, where the diffusion coefficient is allowed to be interacting and the initial distribution of interacting particle system converges to that of the limit equation in $L^1$-Wasserstein distance. The non-degenerate and second order system are investigated respectively and the main tool relies on the gradient estimate of the decoupled SDEs.
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Coupling by Change of Measure for Conditional McKean-Vlasov SDEs and Applications
Comments: 21 pagesCategories: math.PRIn this paper, couplings by change of measure are constructed to derive log-Harnack inequalities for conditional McKean-Vlasov SDEs, where the diffusion coefficients corresponding to the common noise are distribution free or merely depend on the distribution variable and for the latter one, the stochastic Hamiltonian system is also considered. Moreover, the quantitative propagation of chaos in Wasserstein distance is obtained, which combined with the coupling by change of measure implies the quantitative propagation of chaos in the relative entropy and total variation distance and in the additive noise case, the initial distributions of interacting particle system and conditional McKean-Vlasov SDEs are allowed to be singular, which is new even in the McKean-Vlasov frame.
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arXiv:2207.12738 (Published 2022-07-26)
Quantitative propagation of chaos for mean field Markov decision process with common noise
We investigate propagation of chaos for mean field Markov Decision Process with common noise (CMKV-MDP), and when the optimization is performed over randomized open-loop controls on infinite horizon. We first state a rate of convergence of order $M_N^\gamma$, where $M_N$ is the mean rate of convergence in Wasserstein distance of the empirical measure, and $\gamma \in (0,1]$ is an explicit constant, in the limit of the value functions of $N$-agent control problem with asymmetric open-loop controls, towards the value function of CMKV-MDP. Furthermore, we show how to explicitly construct $(\epsilon+\mathcal{O}(M_N^\gamma))$-optimal policies for the $N$-agent model from $\epsilon$-optimal policies for the CMKV-MDP. Our approach relies on sharp comparison between the Bellman operators in the $N$-agent problem and the CMKV-MDP, and fine coupling of empirical measures.
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arXiv:2106.05255 (Published 2021-06-09)
Quantitative Propagation of Chaos for the Mixed-Sign Viscous Vortex Model on the Torus
Comments: 16 pagesWe derive a quantiative propagation of chaos result for a mixed-sign point vortex system on $\mathbb{T}^2$ with independent Brownian noise, at an optimal rate. We introduce a pairing between vortices of opposite sign, and using the vorticity formulation of 2D Navier-Stokes, we define an associated tensorized vorticity equation on $\mathbb{T}^2\times\mathbb{T}^2$ with the same well-posedness theory as the original equation. Solutions of the new PDE can be projected onto solutions of Navier-Stokes, and the tensorized equation allows us to exploit existing propagation of chaos theory for identical particles.
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arXiv:2105.02983 (Published 2021-05-06)
Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions
Categories: math.PRThis paper develops a non-asymptotic, local approach to quantitative propagation of chaos for a wide class of mean field diffusive dynamics. For a system of $n$ interacting particles, the relative entropy between the marginal law of $k$ particles and its limiting product measure is shown to be $O((k/n)^2)$ at each time, as long as the same is true at time zero. A simple Gaussian example shows that this rate is optimal. The main assumption is that the limiting measure obeys a certain functional inequality, which is shown to encompass many potentially irregular but not too singular finite-range interactions, as well as some infinite-range interactions. This unifies the previously disparate cases of Lipschitz versus bounded measurable interactions, improving the best prior bounds of $O(k/n)$ which were deduced from global estimates involving all $n$ particles. We also cover a class of models for which qualitative propagation of chaos and even well-posedness of the McKean-Vlasov equation were previously unknown. At the center of a new approach is a differential inequality, derived from a form of the BBGKY hierarchy, which bounds the $k$-particle entropy in terms of the $(k+1)$-particle entropy.
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arXiv:2007.06352 (Published 2020-07-13)
Quantitative Propagation of Chaos for SGD in Wide Neural Networks
In this paper, we investigate the limiting behavior of a continuous-time counterpart of the Stochastic Gradient Descent (SGD) algorithm applied to two-layer overparameterized neural networks, as the number or neurons (ie, the size of the hidden layer) $N \to +\infty$. Following a probabilistic approach, we show 'propagation of chaos' for the particle system defined by this continuous-time dynamics under different scenarios, indicating that the statistical interaction between the particles asymptotically vanishes. In particular, we establish quantitative convergence with respect to $N$ of any particle to a solution of a mean-field McKean-Vlasov equation in the metric space endowed with the Wasserstein distance. In comparison to previous works on the subject, we consider settings in which the sequence of stepsizes in SGD can potentially depend on the number of neurons and the iterations. We then identify two regimes under which different mean-field limits are obtained, one of them corresponding to an implicitly regularized version of the minimization problem at hand. We perform various experiments on real datasets to validate our theoretical results, assessing the existence of these two regimes on classification problems and illustrating our convergence results.
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arXiv:1906.01051 (Published 2019-06-03)
Quantitative Propagation of Chaos in the bimolecular chemical reaction-diffusion model
Comments: 37 pagesWe study a stochastic system of $N$ interacting particles which models bimolecular chemical reaction-diffusion. In this model, each particle $i$ carries two attributes: the spatial location $X_t^i\in \mathbb{T}^d$, and the type $\Xi_t^i\in \{1,\cdots,n\}$. While $X_t^i$ is a standard (independent) diffusion process, the evolution of the type $\Xi_t^i$ is described by pairwise interactions between different particles under a series of chemical reactions described by a chemical reaction network. We prove that in the large particle limit the stochastic dynamics converges to a mean field limit which is described by a nonlocal reaction-diffusion partial differential equation. In particular, we obtain a quantitative propagation of chaos result for the interacting particle system. Our proof is based on the relative entropy method used recently by Jabin and Wang \cite{JW18}. The key ingredient of the relative entropy method is a large deviation estimate for a special partition function, which was proved previously by technical combinatorial estimates. We give a simple probabilistic proof based on a novel martingale argument.
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Quantitative propagation of chaos for generalized Kac particle systems
Comments: 19 pagesCategories: math.PRWe study a class of one dimensional particle systems with binary interactions of Bird type, which includes Kac's model of the Boltzmann equation and nonlinear equations for the evolution of wealth distribution arising in kinetic economic models. We obtain explicit rates of convergence for the Wasserstein distance between the law of the particles and their limiting law, which are linear in time and depend in a mild polynomial manner on the number of particles. The proof is based on a novel coupling between the particle system and a suitable system of non-independent nonlinear processes, as well as on recent sharp estimates for empirical measures.
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Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules
Comments: 52 pages. Typos corrected, improvement on the presentationWe prove a quantitative propagation of chaos, uniformly in time, for the spatially homogeneous Landau equation in the case of Maxwellian molecules. We improve the results of Fontbona, Gu\'erin and M\'el\'eard \cite{FonGueMe} and Fournier \cite{Fournier} where the propagation of chaos is proved for finite time. Moreover, we prove a quantitative estimate on the rate of convergence to equilibrium uniformly in the number of particles.
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A new approach to quantitative propagation of chaos for drift, diffusion and jump processes
Comments: v2 (55 pages): many improvements on the presentation, v3: correction of a few typos, to appear In Probability Theory and Related FieldsKeywords: jump processes, quantitative propagation, inelastic boltzmann collision jump process, diffusion processes, stability estimatesTags: journal articleThis paper is devoted the the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles, as the number of particles goes to infinity. To this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting nonlinear equation (stability estimates). Then we apply this method to a Boltzmann collision jump process (for Maxwell molecules), to a McKean-Vlasov drift-diffusion process and to an inelastic Boltzmann collision jump process with (stochastic) thermal bath. To our knowledge, our approach yields the first such quantitative results for a combination of jump and diffusion processes.