Barriers of the McKean–Vlasov energy via a mountain pass theorem in the space of probability measures

Gvalani, Rishabh S.; Schlichting, André

doi:10.1016/j.jfa.2020.108720

Cited by 11 publications

(14 citation statements)

References 29 publications

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“…(25), we have X r = x r 1 e. Hence ( 24) hold. Assume that s ̸ = k. Then, by (25), X s = x s 1 e. Hence the equalities (24) also hold.…”

Section: Basic Concepts and Classical Resultsmentioning

confidence: 85%

“…The research tools of the paper include the functional analysis, discrete mathematics, probability [17,19,20,[22][23][24][25][26][27], convex geometry [28], linear algebra [2], inequality and the mean value [15,16,18,27,[29][30][31][32][33][34] theories, especially the mean value theory. The research methods of the paper are based on the mathematical induction [2,17,18,32], reorder method [2] and the dimension reduction method [32].…”

Section: Introductionmentioning

confidence: 99%

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Chebyshev-Jensen-type inequalities and their applications in probability theory with surround system

Wen

Han

Yuan

2023

Preprint

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This paper will establish Chebyshev-Jensen-type inequalities which involving the Chebyshev products as follows:\[\langle X^{1}, X^{2}, \ldots, X^{m}\rangle_{\chi}\geq \left[ X^{1}, X^{2}, \ldots, X^{m}\right]_{\chi}\geq \chi\left( \overline{X^{1}}, \overline{X^{2}}, \ldots, \overline{X^{m}}\right)\]and\[\langle f_{1}, f_{2}, \ldots, f_{m}\rangle_{\chi}\geq \left[f_{1}, f_{2}, \ldots,f_{m}\right]_{\chi}\geq \chi\left( \overline{f_{1}}, \overline{f_{2}}, \ldots, \overline{f_{m}}\right),\]as well as display their applications in probability theory and surround system, especially, the observation angles inequalities are obtained as follows:\[1\geq \frac{1}{n}\sum_{i=1}^{n}P\left(R_{i}\right)\geq \frac{1}{n^{m}}\sum_{1\leq i_{1},\ldots,i_{m}\leq n}P(R_{i_{1},\ldots,i_{m}})\geq\left(\frac{2}{n}\right)^{m+|\alpha|}.\]The research methods of the paper are based on the mathematical induction, reorder method and the dimension reduction method. 2010 MSC: 26D15, 26E60, 51K05, 52A40

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“…(25), we have X r = x r 1 e. Hence ( 24) hold. Assume that s ̸ = k. Then, by (25), X s = x s 1 e. Hence the equalities (24) also hold.…”

Section: Basic Concepts and Classical Resultsmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Chebyshev-Jensen-type inequalities and their applications in probability theory with surround system

Wen

Han

Yuan

2023

Preprint

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“…Consider the case in which β < β ♯ but E M F has more than one critical point one of which is ρ ∞ . Furthermore, we know from [33,Lemma 5.6] that ρ ∞ is a strict local minimum. If at least one of the other critical points is not a minimiser then clearly Property C is violated since the numerator can be chosen to be zero while the denominator remains positive.…”

Section: Resultsmentioning

confidence: 99%

“…Assume it is not, that is all other critical points are minimisers. Then, we can apply the mountain pass theorem in P(Ω) (using the fact that ρ ∞ is a strict local minimum) [33,Theorem 1.1] to construct a new non-minimising critical point thus obtaining a contradiction.…”

Section: Resultsmentioning

confidence: 99%

Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions

Delgadino¹,

Gvalani²,

Pavliotis³

et al. 2021

Preprint

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In this article, we study the mean field limit of weakly interacting diffusions for confining and interaction potentials that are not necessarily convex. We explore the relationship between the large N limit of the constant in the logarithmic Sobolev inequality (LSI) for the N -particle system and the presence or absence of phase transitions for the mean field limit. The non-degeneracy of the LSI constant is shown to have far reaching consequences, especially in the context of uniform-in-time propagation of chaos and the behaviour of equilibrium fluctuations. Our results extend previous results related to unbounded spin systems and recent results on propagation of chaos using novel coupling methods. As incidentals, we provide concise and, to our knowledge, new proofs of a generalised form of Talagrand's inequality and of quantitative propagation of chaos by employing techniques from the theory of gradient flows, specifically the Riemannian calculus on the space of probability measures.1. Introduction. Interacting particle systems have attracted a lot of attention in recent years since they appear in diverse areas ranging from plasma physics and galactic dynamics to machine learning and optimization. For systems of identical (or exchangeable) particles in which the pair-wise interactions scale like the inverse of the number of particles, it is possible to pass to the mean field limit and obtain a coarse-grained description of the system via a nonlinear nonlocal PDE that governs the evolution of the one-particle density. In this paper, we consider systems of weakly interacting diffusions driven by pair-wise interactions, confinement and independent Brownian motions (see (2.1) ). In this case the mean field PDE is the so-called McKean-Vlasov equation.A natural problem that one would like to address is how to obtain sharp quantitative estimates on the rate at which the empirical measure of the particle system converges to the mean field limit, as the number of particles N goes to infinity. When considering arbitrarily long time scales, this problem is intimately connected to the rate of convergence to steady states as time t goes to infinity. For the study of such quantitative results, a crucial role is played by the Poincaré (PI) and logarithmic Sobolev (LSI) inequalities. Our focus in this paper is to elucidate the connection between the validity of the LSI for the N -particle Gibbs measure uniformly in the number of particles N and the properties of the mean field limit. We establish connections with uniform-in-time propagation of chaos, (non-)uniqueness of steady states of the mean field equation, exponential convergence to equilibrium, and the behaviour

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“…[55] and the references therein. Thus we view this paper as the first step towards a new method of designing importance sampling schemes for studying properties of the empirical measure which are of interest to the greater scientific community, such as the free-energy differences for chemical and biological molecular systems [3,28,59,71,76] or exit times of the empirical measure from domains of attraction in models related to social dynamics and consensus convergence [6,31,47,48,57].…”

Section: Introductionmentioning

confidence: 99%

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

Zachary¹,

Heldman²

2022

Preprint

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We construct an importance sampling method for computing statistics related to rare events for weakly interacting diffusions. Standard Monte Carlo methods behave exponentially poorly with the number of particles in the system for such problems. Our scheme is based on subsolutions of a Hamilton-Jacobi-Bellman (HJB) Equation on Wasserstein Space which arises in the theory of mean-field (McKean-Vlasov) control. We identify conditions under which such a scheme is asymptotically optimal. In the process, we make connections between the large deviations principle for the empirical measure of weakly interacting diffusions, mean-field control, and the HJB Equation on Wasserstein Space. We also provide evidence, both analytical and numerical, that with sufficient regularity of the HJB Equation, our scheme can have vanishingly small relative error in the many particle limit.

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Barriers of the McKean–Vlasov energy via a mountain pass theorem in the space of probability measures

Cited by 11 publications

References 29 publications

Chebyshev-Jensen-type inequalities and their applications in probability theory with surround system

Chebyshev-Jensen-type inequalities and their applications in probability theory with surround system

Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

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