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作者：Campi, Luciano; Fischer, Markus
作者单位：University of London; London School Economics & Political Science; University of Padua
摘要：We introduce a simple class of mean-field games with absorbing boundary over a finite time horizon. In the corresponding N-player games, the evolution of players' states is described by a system of weakly interacting Ito equations with absorption on first exit from a bounded open set. Once a player exits, her/his contribution is removed from the empirical measure of the system. Players thus interact through a renormalized empirical measure. In the definition of solution to the mean-field game,...
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作者：Mangoubi, Oren; Smith, Aaron
作者单位：Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; University of Ottawa
摘要：We introduce a Markov chain for sampling from the uniform distribution on a Riemannian manifold M, which we call the geodesic walk. We prove that the mixing time of this walk on any manifold with positive sectional curvature C-x(u, v) bounded both above and below by 0 < m(2) <= C-x(u, v) <= M-2 <( )infinity is O*(M-2/m(2)). In particular, this bound on the mixing time does not depend explicitly on the dimension of the manifold. In the special case that M is the boundary of a convex body, we gi...
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作者：Benaim, Michel; Cloez, Bertrand; Panloup, Fabien
作者单位：Institut Agro; Montpellier SupAgro; Universite de Montpellier; INRAE; Universite d'Angers
摘要：As a continuation of a recent paper, dealing with finite Markov chains, this paper proposes and analyzes a recursive algorithm for the approximation of the quasi-stationary distribution of a general Markov chain living on a compact metric space killed in finite time. The idea is to run the process until extinction and then to bring it back to life at a position randomly chosen according to the (possibly weighted) empirical occupation measure of its past positions. General conditions are given ...
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作者：Duchamps, Jean-Jil; Lambert, Amaury
作者单位：Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Institut National de la Sante et de la Recherche Medicale (Inserm); Centre National de la Recherche Scientifique (CNRS); Universite PSL; College de France
摘要：Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure (infinitely-many-allele model) on the skeleton of the tree. The partition of the boundary into distinct alleles is the so-called allelic partition. In this paper, we are interested in the infinite trees generated by supercritical, possibly time-inhomogeneous, binary branching pr...
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作者：Leon, Jose R.; Samson, Adeline
作者单位：Universidad de la Republica, Uruguay; Communaute Universite Grenoble Alpes; Institut National Polytechnique de Grenoble; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS); Inria; University of Central Venezuela
摘要：The FitzHugh-Nagumo is a well-known neuronal model that describes the generation of spikes at the intracellular level. We study a stochastic version of the model from a probabilistic point of view. The hypoellipticity is proved, as well as the existence and uniqueness of the stationary distribution. The bi-dimensional stochastic process is beta-mixing. The stationary density can be estimated with an adaptive non-parametric estimator. Then we focus on the distribution of the length between succ...
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作者：Davis, Erik; Sethuraman, Sunder
作者单位：University of Arizona
摘要：Given a graph, the popular modularity clustering method specifies a partition of the vertex set as the solution of a certain optimization problem. In this paper, we discuss scaling limits of this method with respect to random geometric graphs constructed from i.i.d. points X-n = {X-1, X-2,..., X-n}, distributed according to a probability measure nu supported on a bounded domain D subset of R-d. Among other results, we show, via a Gamma convergence framework, a geometric form of consistency: Wh...
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作者：Litvak, Alexander E.; Tikhomirov, Konstantin
作者单位：University of Alberta; University of Alberta; Princeton University
摘要：Let X be an n-dimensional random centered Gaussian vector with independent but not identically distributed coordinates and let T be an orthogonal transformation of R-n. We show that the random vector Y = T (X) satisfies E Sigma j=1(k) j-min (i)(<= n) X-i(2 )<= CE Sigma j=1(k) j-min(i)(<= n) Y-i(2 ) for all k <= n, where j-min denotes the jth smallest component of corresponding vector and C >0 is a universal constant. This resolves (up to a multiplicative constant) an old question of S. Mallat ...
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作者：Bjornberg, Jakob E.; Ueltschi, Daniel
作者单位：Chalmers University of Technology; University of Gothenburg; University of Warwick
摘要：We give estimates of the critical parameter for random loop models that are related to quantum spin systems. A special case of the model that we consider is the interchange- or random-stirring process. We consider here the model defined on regular trees of large degrees, which are expected to approximate high spatial dimensions. We find a critical parameter that indeed shares similarity with existing numerical results for the cubic lattice. In the case of the interchange process, our results i...
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作者：Deligiannidis, George; Lee, Anthony
作者单位：University of Oxford; University of Warwick; University of Bristol; Alan Turing Institute
摘要：We show that the class of L-2 functions for which ergodic averages of a reversible Markov chain have finite asymptotic variance is determined by the class of L-2 functions for which ergodic averages of its associated jump chain have finite asymptotic variance. This allows us to characterize completely which ergodic averages have finite asymptotic variance when the Markov chain is an independence sampler. From a practical perspective, the most important result identifies a simple sufficient con...
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作者：Lionnet, Arnaud; dos Reis, Goncalo; Szpruch, Lukasz
作者单位：Institut Polytechnique de Paris; Ecole Nationale des Ponts et Chaussees; University of Edinburgh
摘要：The theory of Forward-Backward Stochastic Differential Equations (FBSDEs) paves a way to probabilistic numerical methods for nonlinear parabolic PDEs. The majority of the results on the numerical methods for FBSDEs relies on the global Lipschitz assumption, which is not satisfied for a number of important cases such as the Fisher-KPP or the FitzHugh- Nagumo equations. Furthermore, it has been shown in [Ann. Appl. Probab. 25 (2015) 2563-2625] that for BSDEs with monotone drivers having polynomi...
