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作者：Alt, Johannes; Ducatez, Raphael; Knowles, Antti
作者单位：University of Geneva
摘要：We complete the analysis of the extremal eigenvalues of the adjacency matrix A of the Erdos-Renyi graph G(N, d/N) in the critical regime d asymptotic to log N of the transition uncovered in (Ann. Inst. Henri Poincare Probab. Stat. 56 (2020) 2141-2161; Ann. Probab. 47 (2019) 1653-1676), where the regimes d >> log N and d << log N were studied. We establish a one-to-one correspondence between vertices of degree at least 2d and nontrivial (excluding the trivial top eigenvalue) eigenvalues of A/ro...
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作者：Roeckner, Michael; Xie, Longjie
作者单位：University of Bielefeld; Jiangsu Normal University; Jiangsu Normal University
摘要：We consider a Poisson equation in R-d for the elliptic operator corresponding to an ergodic diffusion process. Optimal regularity and smoothness with respect to the parameter are obtained under mild conditions on the coefficients. The result is then applied to establish a general diffusion approximation for fully coupled multitime scales stochastic differential equations with only Holder continuous coefficients. Four different averaged equations as well as rates of convergence are obtained. Mo...
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作者：Livshyts, Galyna, V; Tikhomirov, Konstantin; Vershynin, Roman
作者单位：University System of Georgia; Georgia Institute of Technology; University of California System; University of California Irvine
摘要：We show that, for an n x n random matrix A with independent uniformly anticoncentrated entries such that E parallel to A parallel to(2)(HS) <= Kn(2), the smallest singular value sigma(n)(A) of A satisfies P{sigma(n)(A) <= epsilon/root n} <= C epsilon + 2e(-cn), epsilon >= 0. This extends earlier results (Adv. Math. 218 (2008) 600-633; Israel J. Math. 227 (2018) 507-544) by removing the assumption of mean zero and identical distribution of the entries across the matrix as well as the recent res...
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作者：Bakhtin, Yuri; Chen, Hong-Bin
作者单位：New York University
摘要：We consider exit problems for small, white noise perturbations of a dynamical system generated by a vector field and a domain containing a critical point with all positive eigenvalues of linearization. We prove that, in the vanishing noise limit, the probability of exit through a generic set on the boundary is asymptotically polynomial in the noise strength with exponent depending on the mutual position of the set and the flag of the invariant manifolds associated with the top eigenvalues. Fur...
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作者：Linker, Amitai; Mitsche, Dieter; Schapira, Bruno; Valesin, Daniel
作者单位：Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Aix-Marseille Universite; University of Groningen
摘要：We consider the contact process on the model of hyperbolic random graph, in the regime when the degree distribution obeys a power law with exponent x is an element of(1, 2) (so that the degree distribution has finite mean and infinite second moment). We show that the probability of nonextinction as the rate of infection goes to zero decays as a power law with an exponent that only depends on. and which is the same as in the configuration model, suggesting some universality of this critical exp...
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作者：Banerjee, Sayan; Burdzy, Krzysztof
作者单位：University of North Carolina; University of North Carolina Chapel Hill; University of Washington; University of Washington Seattle
摘要：We analyze the local and global smoothing rates of the smoothing process and obtain convergence rates to stationarity for the dual process known as the potlatch process. For general finite graphs we connect the smoothing and convergence rates to the spectral gap of the associated Markov chain. We perform a more detailed analysis of these processes on the torus. Polynomial corrections to the smoothing rates are obtained. They show that local smoothing happens faster than global smoothing. These...
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作者：Vanneuville, Hugo
作者单位：Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet
摘要：In this paper, we introduce and study the annealed spectral sample of Voronoi percolation, which is a continuous and finite point process in R-2 whose definition is mostly inspired by the spectral sample of Bernoulli percolation introduced in (Acta Math. 205 (2010) 19-104) by Garban, Pete and Schramm. We show a clustering effect as well as estimates on the full lower tail of this spectral object. Our main motivation is the study of two models of dynamical critical Voronoi percolation in the pl...
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作者：Priola, Enrico
作者单位：University of Pavia
摘要：We show uniqueness in law for the critical SPDE dX(t) = AX(t) dt + (-A)(1/2) F(X(t)) dt + dW(t), X-0 = x is an element of H, where A : dom(A) subset of H -> H is a negative definite self-adjoint operator on a separable Hilbert space H having A(-1) of trace class and W is a cylindrical Wiener process on H. Here, F : H -> H can be continuous with, at most, linear growth (some functions F which grow more than linearly can also be considered). This leads to new uniqueness results for generalized s...
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作者：Tolomeo, Leonardo
作者单位：University of Bonn
摘要：We study the two-dimensional wave equation with cubic nonlinearity posed on R-2 with space-time white noise forcing. After a suitable renormalisation of the nonlinearity, we prove global well-posedness for this equation for initial data in H-s, s > 4/5.
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作者：Feldheim, Naomi; Feldheim, Ohad; Nitzan, Shahaf
作者单位：Bar Ilan University; Hebrew University of Jerusalem; University System of Georgia; Georgia Institute of Technology
摘要：We study the persistence probability of a centered stationary Gaussian process on Z or R, that is, its probability to remain positive for a long time. We describe the delicate interplay between this probability and the behavior of the spectral measure of the process near zero and infinity.
