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作者：Moreno Flores, Gregorio R.
作者单位：Pontificia Universidad Catolica de Chile
摘要：We give a new proof of the fact that the solutions of the stochastic heat equation, started with nonnegative initial conditions, are strictly positive at positive times. The proof uses concentration of measure arguments for discrete directed polymers in Gaussian environments, originated in M. Talagrand's work on spin glasses and brought to directed polymers by Ph. Carmona and Y. Hu. We also get slightly improved bounds on the lower tail of the solutions of the stochastic heat equation started ...
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作者：Chen, Xia
作者单位：University of Tennessee System; University of Tennessee Knoxville
摘要：In this paper, we study the long-term asymptotics for the quenched moment E-x exp{integral(t)(0) V (B-s) ds} consisting of a d-dimensional Brownian motion {B-s; s >= 0} and a generalized Gaussian field V. The major progress made in this paper includes: Solution to an open problem posted by Carmona and Molchanov [Probab. Theory Related Fields 102 (1995) 433-453], the quenched laws for Brownian motions in Newtonian-type potentials and in the potentials driven by white noise or by fractional whit...
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作者：Shao, Qi-Man; Zhou, Wen-Xin
作者单位：Chinese University of Hong Kong; Hong Kong University of Science & Technology
摘要：Let x(1),...,x(n) be a random sample from a p-dimensional population distribution, where p = p(n) -> infinity and log p = o(n(beta)) for some 0 < beta <= 1, and let L-n be the coherence of the sample correlation matrix. In this paper it is proved that root n/log pL(n) -> 2 in probability if and only if Ee(t0 vertical bar x11 vertical bar alpha) < infinity for some t(0) > 0, where alpha satisfies beta = alpha/(4 - alpha). Asymptotic distributions of L-n are also proved under the same sufficient...
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作者：Menz, Georg; Schlichting, Andre
作者单位：Stanford University; University of Bonn
摘要：We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian H : R-n -> R in the regime of low temperature epsilon. We proof the Eyring-Kramers formula for the optimal constant in the Poincare (PI) and logarithmic Sobolev inequality (LSI) for the associated generator L = epsilon Delta - del H . del of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald et al. [Ann. Inst. Henri Poincare Probab. Stat. 45 (2009) 302-351] a...
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作者：Nguyen, Hoi H.
作者单位：University System of Ohio; Ohio State University
摘要：Let X be a matrix sampled uniformly from the set of doubly stochastic matrices of size n x n. We show that the empirical spectral distribution of the normalized matrix root n(X - EX) converges almost surely to the circular law. This confirms a conjecture of Chatterjee, Diaconis and Sly.
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作者：Last, Guenter; Moerters, Peter; Thorisson, Hermann
作者单位：Helmholtz Association; Karlsruhe Institute of Technology; University of Bath; University of Iceland
摘要：Let B = (B-t)(t epsilon R) be a two-sided standard Brownian motion. An unbiased shift of B is a random time T, which is a measurable function of B, such that (BT+t - B-T)(t epsilon R) is a Brownian motion independent of B-T. We characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of B. For any probability distribution nu on R we construct a stopping time T >= 0 with the above properties such that B-T has distribution nu. We also study moment and minimalit...
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作者：Mueller, Carl; Mytnik, Leonid; Perkins, Edwin
作者单位：University of Rochester; Technion Israel Institute of Technology; University of British Columbia
摘要：Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) partial derivative u/partial derivative t = Delta/2 u(t, x) + vertical bar u(t, x)vertical bar(gamma) W(t, x), u(0, x) = 0 Here W is a space time white noise on R x R. More precisely, we show the above stochastic PDE has a nonzero solution for 0 < y < 3/4. Since u(t, x) = 0 solves the equation, it follows that solutions are neither unique in law nor ...
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作者：Garet, Olivier; Marchand, Regine
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Lorraine; Universite de Lorraine; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
摘要：The asymptotic shape theorem for the contact process in random environment gives the existence of a norm mu on R-d such that the hitting time t(x) is asymptotically equivalent to mu(x) when the contact process survives. We provide here exponential upper bounds for the probability of the event {t(x)/mu(x) is not an element of[1 - epsilon, 1 + epsilon]); these bounds are optimal for independent random environment. As a special case, this gives the large deviation inequality for the contact proce...
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作者：Alberts, Tom; Khanin, Konstantin; Quastel, Jeremy
作者单位：California Institute of Technology; University of Toronto
摘要：We introduce a new disorder regime for directed polymers in dimension 1 + 1 that sits between the weak and strong disorder regimes. We call it the intermediate disorder regime. It is accessed by scaling the inverse temperature parameter beta to zero as the polymer length n tends to infinity. The natural choice of scaling is beta(n) :=beta n(-1/4). We show that the polymer measure under this scaling has previously unseen behavior. While the fluctuation exponents of the polymer endpoint and the ...
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作者：Denis, Laurent; Matoussi, Anis; Zhang, Jing
作者单位：Universite Paris Saclay; Le Mans Universite; Institut Polytechnique de Paris; Ecole Polytechnique
摘要：We prove an existence and uniqueness result for quasilinear Stochastic PDEs with obstacle (OSPDE in short). Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair (u, v) where u is a predictable continuous process which takes values in a proper Sobolev space and v is a random regular measure satisfying the minimal Skohorod condition.
