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作者：Kozakova, Iva; Meester, Ronald; Nanda, Seema
作者单位：Vrije Universiteit Amsterdam; Tata Institute of Fundamental Research (TIFR); University of Tennessee System; University of Tennessee Knoxville
摘要：We study the size of connected components of random nearest-neighbor graphs with vertex set the points of a homogeneous Poisson point process in R-d. The connectivity function is shown to decay superexponentially, and we identify the exact exponent. From this we also obtain the decay rate of the maximal number of points of a path through the origin. We define the generation number of a point in a component and establish its asymptotic distribution as the dimension d tends to infinity.
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作者：Lototsky, S. V.; Rozovskii, B. L.
作者单位：University of Southern California
摘要：A new method is described for constructing a generalized solution of a stochastic evolution equation. Existence, uniqueness, regularity and a probabilistic representation of this Wiener Chaos solution are established for a large class of equations. As an application of the general theory, new results are obtained for several types of the passive scalar equation.
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作者：Adamczak, Radoslaw
作者单位：Polish Academy of Sciences; Institute of Mathematics of the Polish Academy of Sciences
摘要：We present moment inequalities for completely degenerate Banach space valued (generalized) U-statistics of arbitrary order. The estimates involve suprema of empirical processes which, in the real-valued case, can be replaced by simpler norms of the kernel matrix (i.e., norms of some multilinear operators associated with the kernel matrix). As a corollary, we derive tail inequalities for U-statistics with bounded kernels and for some multiple stochastic integrals.
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作者：Gallardo, Leonard; Yor, Marc
作者单位：Universite de Tours; Centre National de la Recherche Scientifique (CNRS); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite
摘要：Dunkl processes are martingales as well as cadlag homogeneous Markov processes taking values in R-d and they are naturally associated with a root system. In this paper we study the jumps of these processes, we describe precisely their martingale decompositions into continuous and purely discontinuous parts and we obtain a Wiener chaos decomposition of the corresponding L-2 spaces of these processes in terms of adequate mixed multiple stochastic integrals.
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作者：Nourdin, Ivan; Simon, Thomas
作者单位：Universite de Lorraine; Universite Paris Saclay
摘要：We consider the problem of absolute continuity for the one-dimensional SDE [GRAPHICS] where Z is a real Levy process without Brownian part and a a function of class C-1 with bounded derivative. Using an elementary stratification method, we show that if the drift a is monotonous at the initial point x, then X-t is absolutely continuous for every t > 0 if and only if Z jumps infinitely often. This means that the drift term has a regularizing effect, since Z(t) itself may not have a density. We a...
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作者：Graham, B. T.; Grimmett, G. R.
作者单位：University of Cambridge
摘要：The influence theorem for product measures on the discrete space to, {0, 1}(N) may be extended to probability measures with the property of monotonicity (which is equivalent to strong positive association). Corresponding results are valid for probability measures on the cube [0, 1](N) that are absolutely continuous with respect to Lebesgue measure. These results lead to a sharp-threshold theorem for measures of random-cluster type, and this may be applied to box crossings in the two-dimensiona...
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作者：Mytnik, Leonid; Perkins, Edwin; Sturm, Anja
作者单位：Technion Israel Institute of Technology; University of British Columbia; University of Delaware
摘要：We consider the existence and pathwise uniqueness of the stochastic heat equation with a multiplicative colored noise term on R-d for d >= 1. We focus on the case of non-Lipschitz noise coefficients and singular spatial noise correlations. In the course of the proof a new result on Holder continuity of the solutions near zero is established.
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作者：Cator, Eric; Groeneboom, Piet
作者单位：Delft University of Technology; Vrije Universiteit Amsterdam
摘要：We show that, for a stationary version of Hammersley's process, with Poisson sources on the positive x-axis and Poisson sinks on the positive y-axis, the variance of the length of a longest weakly North-East path L(t, t) from (0, 0) to (t, t) is equal to 2E(t - X(t))+, where X(t) is the location of a second class particle at time t. This implies that both E(t - X(t))+ and the variance of L (t, t) are of order t(2/3). Proofs are based on the relation between the flux and the path of a second cl...
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作者：Chan, HP; Lai, TL
作者单位：National University of Singapore; Stanford University
摘要：Several classical results on boundary crossing probabilities of Brownian motion and random walks are extended to asymptotically Gaussian random fields, which include sums of i.i.d. random variables with multidimensional indices, multivariate empirical processes, and scan statistics in change-point and signal detection as special cases. Some key ingredients in these extensions are moderate deviation approximations to marginal tail probabilities and weak convergence of the conditional distributi...
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作者：Quastel, Jeremy
作者单位：University of Toronto
摘要：We consider a system of random walks in a random environment interacting via exclusion. The model is reversible with respect to a family of disordered Bernoulli measures. Assuming some weak mixing conditions, it is shown that, under diffusive scaling, the system has a deterministic hydrodynamic limit which holds for almost every realization of the environment. The limit is a nonlinear diffusion equation with diffusion coefficient given by a variational formula. The model is nongradient and the...
