A. Yu. Veretennikov’s research while affiliated with Institute for Information Transmission Problems, Russian Academy of Sciences and other places

... These nonlinear systems arise naturally as the limit of large finite mean-field particle systems through the so-called "propagation of chaos" [8,[13][14][15][16][17]32,33,37,49,52]. Several works have been devoted to the development of nonlinear Markov chain Monte Carlo methods [3,9,22,39,45,53], but in contrast with the classical (linear) Markov chain framework, quantitative results on convergence to stationarity are very limited and difficult to obtain. ...

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Nonlinear Dynamics for the Ising Model

... since the expression inside the infimum is a continuous function of x, y and it is taken on a compact set. Theorem 1 of [58] yields the claim. ...

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PERSISTENCE AND NEUTRALITY IN INTERACTING REPLICATOR DYNAMICS

... The system is ergodic when the diffusivity is nondegenerate [2,3,6,28,29,39]; therefore, the law will converge to a unique stationary distribution. Since we are interested in long time horizons during which systems recover from failure, as reflected in the long-term pay-offs (2.3), we may neglect the initial transient (and the initial conditions) and focus exclusively on the steady-state solution of (3.1). ...

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Optimizing Noisy Complex Systems Liable to Failure

... The lemmas is mainly about Coupling Markov in [14,15]. First, it's necessary to introduce the basic construction for next lemmas. ...

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Estimation and Application of the Convergence Bounds for Nonlinear Markov Chains

... Existence results were recently established by the third author of this paper in [40] under stronger conditions, specifically requiring the coefficients to be bounded (excluding the prototype MKV-Langevin equation (1.2)), the coefficient b 0 to be continuous with respect to all variables except t, and the diffusion matrix to be symmetric. The main tools used in the proof are Krylov's bounds [18], Skorokhod's technique of weak convergence [33], and Nisio's approach to SDEs in [26] and [39]. In addition to extending the original arguments of [40], in Proposition 2.5 we clarify certain technical aspects of the construction of the so-called ε-net, a crucial tool in the proof of Theorem 1.6: see, in particular, Remark 2.6. ...

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Existence and uniqueness results for strongly degenerate McKean-Vlasov equations with rough coefficients

... [25] proved existence and uniqueness for MV-SDEs with superlinear growth in both drift and diffusion coefficients. Further studies on well-posedness can be found in [5,11,34,41,12] and related references. ...

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Dimension-independent convergence rate of propagation of chaos and numerical analysis for McKean-Vlasov stochastic differential equations with coefficients nonlinearly dependent on measure

... We are looking for sufficient conditions for positive recurrence of the strong Markov process (X t , Z t ). Such a problem was considered in [2] for the exponentially recurrent case; for other references see [1,5,8,11], and the references therein. Under weak ergodic and transient conditions the setting was earlier investigated in [15] for the case of the constant intensities λ 0 , λ 1 (i.e., not depending on x). ...

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Positive recurrence of a solution of an SDE with variable switching intensities

... Moreover, since ∇V and ∇W are Lipschitz in θ by Assumption 3.1(ii) and polynomially bounded in x, using the uniform boundedness of the moments and the ergodic theorem, and due to the differentiability of the invariant measure with respect to θ [9], we have that there exists a constant L > 0, independent of T , such that ...

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Linearization of ergodic McKean SDEs and applications

... [1], Chapter I, Section 2). A more general situation concerns additive functionals of Markov processes (here we recall [26] as a reference with results based on the Gärtner Ellis Theorem); however, for simplicity, we refer to the case of Markov additive processes (see e.g. [1], Chapter III, Section 4; actually the presentation in that reference concerns the case h = 1). ...

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Large deviations for a class of tempered subordinators and their inverse processes

... Recently, the authors in [14] showed that the Borovkov-Sakhanenko bound is asymptotically better than the Van Trees bound, and asymptotically optimal in a certain class of bounds. The authors in [15] compared some Bayesian bounds from the point of view of asymptotic efficiency. ...

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Improvement of Bobrovsky–Mayor–Wolf–Zakai Bound