Titlebook: Random Walks, Brownian Motion, and Interacting Particle Systems; A Festschrift in Hon Rick Durrett,Harry Kesten Book 1991 Springer Science+

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Some Theorems Concerning 2-Dimensional Brownian Motionplane brownian motion. §1 is concerned with the process .(.)=|.(.)|, denoting the distance of the 2-dimensional separable Bachelier-Wiener process .(.) =.(.)+.(.) from the origin. We shall derive a law of the so-called strong type concerning the frequency of small values of .(.). This theorem dispro

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Recurrent Random Walk and Logarithmic Potentialses this fact has been studied intensely. Thus Hunt [9] bases a general potential theory on transient continuous parameter processes, Doob [7] and Hunt [10] use the theory to construct boundaries for discrete parameter processes, It? and McKean [11] solve the problem of characterizing the recurrent

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Interaction of Markov Processesplicity let . be a countable (or finite) set, and consider a countable (or finite if . is finite) collection of independent Markov processes on .,with common transition function ...,., ., .. It is natural to assume constant invariant measure, i.e.,

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Asymptotic Laplace-Transforms.d. with short tails. In the discrete case it is assumed that .(.) is concentrated on ? and not on any proper subgroup of ?. Here one is interested in the weights.for . in some neighborhood of .. In the continuous case it is assumed that,C(.), the law of ., satisfies Cramér’s condition. Using techni

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Higher Order Hydrodynamic Equations for a System of Independent Random Walksneous random walks. The main result states that, under some conditions on the increments of the random walk, the solution of the hydrodynamic equations of order n approximates the density of the original particle dynamics uniformly on time intervals [0, ∈.] for any γ > 0. Here . is the small paramet

evaculate

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Relations Between Solutions to a Discrete and Continuous Dirichlet Probleme line segment . in .. Let ? be a piecewise continuous function on . which vanishes on .. U .. and let .(?,?) be the harmonic function on . with boundary values ?. Thus △? = O. The discrete analogue, or the analogue for a simple random walk with step size ., is a function .. (?, ?) on .. which sati