Random motion of strings and related stochastic evolution equations

Funaki, Tadahisa

doi:10.1017/s0027763000020298

Cited by 161 publications

(124 citation statements)

References 17 publications

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“…In Section 4 we consider such nonlinear noise coefficients which may, in particular, be non-Lipschitz. Since in this case we need to show convergence of approximate densities directly to the solution of the limiting SPDE (rather than in a measure sense), it is convenient to start with the corresponding lattice systems instead of the particle model itself (see Funaki [16] and Gyöngy [18] for this approach applied to related systems). Thus, in Section 4, we first consider existence and uniqueness questions of the following system:…”

Section: Formulation Of the Main Resultsmentioning

confidence: 99%

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On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise

Sturm

2003

Electron. J. Probab.

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We consider the stochastic heat equation with a multiplicative colored noise term on R d for d ≥ 1. First, we prove convergence of a branching particle system in a random environment to this stochastic heat equation with linear noise coefficients. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems, which are infinite dimensional stochastic differential equations with correlated noise terms, provided that uniqueness of the limit is known. In the course of the proof, we establish existence and uniqueness of solutions to the lattice systems, as well as a new existence result for solutions to the stochastic heat equation. The latter are shown to be jointly continuous in time and space under some mild additional assumptions.

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Section: Formulation Of the Main Resultsmentioning

confidence: 99%

“…Thus we obtain, from (46) and (49), X n t (1) = X n 0 (1) + M b,n tn (1). By the same version of Burkholder's Inequality as above and for small enough as in (16), setting p = 1 + 2 , it now follows that…”

Section: Proof Of Theorem 22mentioning

confidence: 86%

On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise

Sturm

2003

Electron. J. Probab.

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“…whereẆ (x, t) is an R-valued space-time white noise, which is assumed to be adapted with respect to a filtered probability space (Ω, F, F t , P), where F is complete and the filtration {F t , t ≥ 0} is right continuous; see Funaki (1983) and Mueller and Tribe (2002) for more information.…”

Section: The Stochastic Heat Equationmentioning

confidence: 99%

Sample Path Properties of Anisotropic Gaussian Random Fields

Xiao¹

Lecture Notes in Mathematics

140

227

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Anisotropic Gaussian random fields arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operator-scaling Gaussian fields with stationary increments, and the solution to the stochastic heat equation.This paper is concerned with sample path properties of anisotropic Gaussian random fields in general. Let X = {X(t), t ∈ R N } be a Gaussian random field with values in R d and with parameters H 1 , . . . , H N . Our goal is to characterize the anisotropic nature of X in terms of its parameters explicitly.Under some general conditions, we establish results on the modulus of continuity, small ball probabilities, fractal dimensions, hitting probabilities and local times of anisotropic Gaussian random fields. An important tool for our study is the various forms of strong local nondeterminism.

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“…The regularities of the solution of an even order (larger than 2) stochastic partial differential equation driven by cylindrical Brownian motion with uniformly bounded Lipschitz drift coefficients and uniformly bounded constant diffusion coefficients were initially studied by T. Funaki [5,6] and then were restudied in [1]. Relative results will be generalized to all fractional order α > 1 in this paper.…”

Section: Introductionmentioning

confidence: 99%