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作者：Curien, Nicolas; Le Gall, Jean-Francois
作者单位：Universite Paris Saclay
摘要：We study properties of the harmonic measure of balls in typical large discrete trees. For a ball of radius n centered at the root, we prove that, although the size of the boundary is of order n, most of the harmonic measure is supported on a boundary set of size approximately equal to n(beta), where beta approximate to 0.78 is a universal constant. To derive such results, we interpret harmonic measure as the exit distribution of the ball by simple random walk on the tree, and we first deal wit...
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作者：Chen, Le; Cranston, Michael; Khoshnevisan, Davar; Kim, Kunwoo
作者单位：Utah System of Higher Education; University of Utah; University of California System; University of California Irvine
摘要：Given a field {B(x)}x is an element of Z(d) of independent standard Brownian motions, indexed by Z(d), the generator of a suitable Markov process on Z(d), G, and sufficiently nice function sigma : [0, infinity) (bar right arrow) [0, infinity), we consider the influence of the parameter lambda on the behavior of the system, du(t) (x) = (Gu(t))(x) dt + lambda sigma(u(t)(x)) dB(t)(x) [t > 0, x is an element of Z(d)], u(0)(x) = c(0)delta(0)(x). We show that for any lambda > 0 in dimensions one and...
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作者：Kendall, Wilfrid S.
作者单位：University of Warwick
摘要：Consider an improper Poisson line process, marked by positive speeds so as to satisfy a scale-invariance property (actually, scale-equivariance). The line process can be characterized by its intensity measure, which belongs to a one-parameter family if scale and Euclidean invariance are required. This paper investigates a proposal by Aldous, namely that the line process could be used to produce a scale-invariant random spatial network (SIRSN) by means of connecting up points using paths which ...
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作者：Ren, Yan-Xia; Song, Renming; Zhang, Rui
作者单位：Peking University; Peking University; University of Illinois System; University of Illinois Urbana-Champaign
摘要：In this paper, we establish a spatial central limit theorem for a large class of supercritical branching, not necessarily symmetric, Markov processes with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem generalizes and unifies all the central limit theorems obtained recently in Ren, Song and Zhang [J. Funct. Anal. 266 (2014) 1716-1756] for supercritical branching symmetric Markov processes. To prove our central limit theorem, we have to...
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作者：Joseph, Mathew; Khoshnevisan, Davar; Mueller, Carl
作者单位：University of Sheffield; Utah System of Higher Education; University of Utah; University of Rochester
摘要：We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation on the real line. As a consequence, we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities.
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作者：van der Hofstad, Remco; Holmes, Mark; Perkins, Edwin A.
作者单位：Eindhoven University of Technology; University of Auckland; University of British Columbia
摘要：We give a sufficient condition for tightness for convergence of rescaled critical spatial structures to the canonical measure of super-Brownian motion. This condition is formulated in terms of the r-point functions for r = 2,..., 5. The r-point functions describe the expected number of particles at given times and spatial locations, and have been investigated in the literature for many high-dimensional statistical physics models, such as oriented percolation and the contact process above 4 dim...
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作者：Barlow, M. T.; Croydon, D. A.; Kumagai, T.
作者单位：University of British Columbia; University of Warwick; Kyoto University
摘要：The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embedded into Euclidean space. Various properties of the intrinsic metrics, measures and embeddings of the subsequential limits in this space are obtained, with it being proved in particular that the Hausdorff dimension of any limit in its intrinsic metric is almost surely equal to 8/5. In addition, the tight...
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作者：Banuelos, Rodrigo; Burdzy, Krzysztof; Vares, Maria Eulalia
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作者：Evans, Steven N.; Gruebel, Rudolf; Wakolbinger, Anton
作者单位：University of California System; University of California Berkeley; Leibniz University Hannover; Goethe University Frankfurt
摘要：Remy's algorithm is a Markov chain that iteratively generates a sequence of random trees in such a way that the nth tree is uniformly distributed over the set of rooted, planar, binary trees with 2n + 1 vertices. We obtain a concrete characterization of the Doob-Martin boundary of this transient Markov chain and thereby delineate all the ways in which, loosely speaking, this process can be conditioned to go to infinity at large times. A (deterministic) sequence of finite rooted, planar, binary...
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作者：Osekowski, Adam
作者单位：University of Warsaw
摘要：We present a probabilistic study of the Hilbert operator, T f (x) = 1/pi integral(infinity)(0) f(y) dy/x+y, x >= 0, defined on integrable functions f on the positive halfline. Using appropriate novel estimates for orthogonal martingales satisfying the differential subordination, we establish sharp moment, weak-type and phi-inequalities for T. We also show similar estimates for higher dimensional analogues of the Hilbert operator, and by the further careful modification of martingale methods, w...
