Generalized Ornstein-Uhlenbeck Processes and Infinite Particle Branching Brownian Motions

Holley, Richard A.; Stroock, Daniel W.

doi:10.2977/prims/1195188837

Cited by 185 publications

(85 citation statements)

References 13 publications

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“…The proof employs two rather separate techniques (i) the martingale approach to the convergence of stochastic processes worked out in a form applicable to our problem by R. A. Holley and D. W. Stroock [20] and (ii) low fugacity cluster expansion as is well known from Statistical Mechanics [21, Chap. IV].…”

Section: Strategy Of the Proofmentioning

confidence: 99%

Equilibrium fluctuations for interacting Brownian particles

Spohn

1986

Commun.Math. Phys.

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Abstract. We consider an infinitely extended system of Brownian particles interacting by a pair force-grad V. Their initial distribution is stationary and given by the Gibbs measure associated with the potential V with fugacity z. We assume that V is symmetric, finite range, three times continuously differentiable, superstable, and positive and that the fugacity is small in the sense that 0 < z < 0.28/eSdq (1 -e-v(q)). In addition a certain essential self-adjointness property is assumed. We prove then that the time-dependent fluctuations in the density on a spatial scale of order e-1 and on a time scale of order e-2 converge as s ~ 0 to a Gaussian field with covariance xSdqg(q)(e (p/2x)a jt~f)(q) with p the density and X the compressibility.

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Section: Strategy Of the Proofmentioning

confidence: 99%

Equilibrium fluctuations for interacting Brownian particles

Spohn

1986

Commun.Math. Phys.

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“…with the boundary conditions, We denote by X t ,2(σ) and X t$z (σ) 9 the second term and the third term of the right hand side of the equation (2.7), respectively. When K -1, the fundamental solution will be denoted by q(t, σ, τ) instead of p (t, σ, τ).…”

Section: The Sequence {X N } Is Tight If It Satisfies These Two Condimentioning

confidence: 99%

Random motion of strings and related stochastic evolution equations

Funaki

1983

Nagoya Mathematical Journal

161

121

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In this paper, we shall investigate the random motion of an elastic string by using the theory of infinite dimensional stochastic differential equations. The paper consists of three main parts and appendices. In the first part (§2), we shall derive a basic equation which describes the random motion of a string. Several properties of this equation will be investigated in § 3, 4 and 5. In the third part (§ 6), we shall deal with a stochastic differential equation on a Hilbert space as a generalization of the equation of the string.

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“…Much of the recent work on this and related processes has been on limit theorems of various sorts (e.g. Dawson [1977], Holley and Stroock [1978], Iscoe [1986], Cox and Griffeath [1985]). We study a sample path property of the process, namely the Hausdorff measure of its support as time varies.…”

mentioning

confidence: 99%

“…Yt denotes a d-dimensional symmetric stable process of index a e [0,2], starting at x g Rd under the measure P£, and scaled so that The measure-valued diffusions that we study may be characterized as solutions of a martingale problem (see Holley and Stroock [1978], Dawson and Kurtz [1982], Roelly-Coppoletta [1986]). The last reference contains a detailed proof of Theorem 1.1.…”

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confidence: 99%

A Space-Time Property of a Class of Measure-Valued Branching Diffusions

Perkins¹

1988

Transactions of the American Mathematical Society

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Abstract. If d > a, it is shown that the ¿-dimensional branching diffusion of index a, studied by Dawson and others, distributes its mass over a random support in a uniform manner with respect to the Hausdorff "-measure, where "(x) = "loglogl/x. More surprisingly, it does so for all positive times simultaneously. Slightly less precise results are obtained in the critical case d = a. In particular, the process is singular at all positive times a.s. for d> a.

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Generalized Ornstein-Uhlenbeck Processes and Infinite Particle Branching Brownian Motions

Cited by 185 publications

References 13 publications

Equilibrium fluctuations for interacting Brownian particles

Equilibrium fluctuations for interacting Brownian particles

Random motion of strings and related stochastic evolution equations

A Space-Time Property of a Class of Measure-Valued Branching Diffusions

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