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作者：Cerrai, Sandra; Freidlin, Mark
作者单位：University System of Maryland; University of Maryland College Park
摘要：We consider a class of stochastic reaction-diffusion equations also having a stochastic perturbation on the boundary and we show that when the diffusion rate is much larger than the rate of reaction, it is possible to replace the SPDE by a suitable one-dimensional stochastic differential equation. This replacement is possible under the assumption of spectral gap for the diffusion and is a result of averaging in the fast spatial transport. We also study the fluctuations around the averaged moti...
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作者：Blath, Jochen; Doering, Leif; Etheridge, Alison
作者单位：Technical University of Berlin; University of Oxford
摘要：In this paper we introduce a critical curve separating the asymptotic behavior of the moments of the symbiotic branching model, introduced by Etheridge and Fleischmann [Stochastic Process. Appl. 114 (2004) 127-160] into two regimes. Using arguments based on two different dualities and a classical result of Spitzer [Trans. Amer Math. Soc. 87 (1958) 187-197] on the exit-time of a planar Brownian motion from a wedge, we prove that the parameter governing the model provides regimes of bounded and ...
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作者：Hu, Yaozhong; Nualart, David; Song, Jian
作者单位：University of Kansas
摘要：We establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the Feynman-Kac formula is a weak solution of the stochastic heat equation. From the Feynman-Kac formula, we establish the smoothness of the density of the solution and the Holder regularity in the space and time variables. We also derive a Feynman-Kac formula for the...
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作者：Sznitman, Alain-Sol
作者单位：Swiss Federal Institutes of Technology Domain; ETH Zurich
摘要：The vacant set of random interlacements on Z(d), d >= 3, has nontrivial percolative properties. It is known from Sznitman [Ann. Math. 171 (2010) 2039-2087], Sidoravicius and Sznitman [Comm. Pure Appl. Math. 62 (2009) 831-858] that there is a nondegenerate critical value u(*) such that the vacant set at level u percolates when u < u(*) and does not percolate when u > u(*). We derive here an asymptotic upper bound on u(*), as d goes to infinity, which complements the lower bound from Sznitman [P...
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作者：Biskup, Marek; Spohn, Herbert
作者单位：University of California System; University of California Los Angeles; University of South Bohemia Ceske Budejovice; Technical University of Munich
摘要：We consider gradient fields (phi(x) : x is an element of Z(d)) whose law takes the Gibbs-Boltzmann form Z(-1) exp{-Sigma(< x,y >) V(phi(y) - phi(x))}, where the sum runs over nearest neighbors. We assume that the potential V admits the representation V(eta) := -log integral rho(dk)exp[-1/2 kappa eta(2)], where rho is a positive measure with compact support in (0, infinity). Hence, the potential V is symmetric, but nonconvex in general. While for strictly convex V's, the translation-invariant, ...
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作者：Lalley, Steven P.; Zheng, Xinghua
作者单位：University of Chicago; Hong Kong University of Science & Technology
摘要：Consider a critical nearest-neighbor branching random walk on the d-dimensional integer lattice initiated by a single particle at the origin. Let G(n) be the event that the branching random walk survives to generation n. We obtain the following limit theorems, conditional on the event G(n), for a variety of occupation statistics: ( I) Let V-n be the maximal number of particles at a single site at time n. If the offspring distribution has finite alpha th moment for some integer alpha >= 2, then...
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作者：Le Gall, Jean-Francois; Miermont, Gregory
作者单位：Universite Paris Saclay
摘要：We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index alpha is an element of (1, 2). When the number n of vertices of the map tends to infinity, the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process, which can be constructed from a centered stable process with no negative ju...
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作者：Ben Arous, Gerard; Corwin, Ivan
作者单位：New York University
摘要：We consider the family of two-sided Bernoulli initial conditions for TASEP which, as the left and right densities (rho-, rho+) are varied, give rise to shock waves and rarefaction fans-the two phenomena which are typical to TASEP. We provide a proof of Conjecture 7.1 of [Progr. Probab. 51 (2002) 185-204] which characterizes the order of and scaling functions for the fluctuations of the height function of two-sided TASEP in terms of the two densities rho-, rho+ and the speed y around which the ...
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作者：Jara, Milton; Landim, Claudio; Teixeira, Augusto
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite PSL; Universite Paris-Dauphine; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Rouen Normandie; Swiss Federal Institutes of Technology Domain; ETH Zurich
摘要：In this paper, we study Bouchaud's trap model on the discrete d-dimensional torus T-n(d) = (Z/nZ)(d). In this process, a particle performs a symmetric simple random walk, which waits at the site x is an element of T-n(d), an exponential time with mean xi x, where {xi x, x is an element of T-n(d)} is a realization of an i.i.d. sequence of positive random variables with an alpha-stable law. Intuitively speaking, the value of xi x gives the depth of the trap at x. In dimension d = 1, we prove tha...
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作者：Lacoin, Hubert
作者单位：Roma Tre University
摘要：In this paper, we study a model of a Brownian polymer in R+ x R-d, introduced by Rovira and Tindel [J. Fund. Anal. 222 (2005) 178-201]. Our investigation focuses mainly on the effect of strong spatial correlation in the environment in that model in terms of free energy, fluctuation exponent and volume exponent. In particular, we prove that under some assumptions, very strong disorder and superdiffusivity hold at all temperatures when d >= 3 and provide a novel approach to Petermann's superdiff...
