Stability of a class of transformations of distribution-valued processes and stochastic evolution equations
type: Item
instance of: scholarly article
title: Stability of a class of transformations of distribution-valued processes and stochastic evolution equations
zbMATH Open document ID: 0762.60053
DOI: 10.1007/BF01058724
published in: Journal of Theoretical Probability
publication date: 1993-01-17T00:00:00Z
review text: Since 1976, many results on convergence of the processes taking values in distribution spaces have been proven. The processes and their limits are often solutions of stochastic evolution equations involving accompanying semimartingales and therefore the question can be posed as a problem of convergence of solutions to such equations and relationships between convergence of solutions and convergence of accompanying semimartingales. This problem belongs to a more general question of stability of a class of linear transformations of processes. This general question is studied and the results are translated into theorems on stochastic evolution equations. Two general results (Theorems 4 and 5) on convergence of solutions of stochastic evolution equations in distribution spaces which cover a large class of applications are given. In these results the distribution spaces are duals of Fréchet nuclear spaces. One of the results, for the case of continuous limits, considerably simplifies the tightness problem due to a recent result of Aldous (Theorem 5).
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 96936
zbMATH Keywords: stochastic evolution equations
zbMATH Keywords: relationships between convergence of solutions and convergence of accompanying semimartingales
zbMATH Keywords: stability of a class of linear transformations of processes
zbMATH Keywords: duals of Fréchet nuclear spaces
zbMATH Keywords: tightness problem
author: Begoña Fernández Fernández
author: Luis G. Gorostiza
reviewed by: T. Sasagawa
MaRDI profile type: Publication
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cites work: Stopping times and tightness. II
cites work: Passage to the Limit in Itô Stochastic Equations
cites work: Q5560061
cites work: Comparison of stochastic and deterministic models of a linear chemical reaction with diffusion
cites work: Langevin equations for S'-valued Gaussian processes and fluctuation limits of infinite particle systems
cites work: Inhomogenous infinite dimensional langevin equations
cites work: Gaussian and Non–Gaussian Distribution–Valued Ornstein–Uhlenbeck Processes
cites work: Convergence of \({\mathcal S}'\)-valued processes and space-time random fields
cites work: Stochastic integration for inhomogeneous Wiener process in the dual of a nuclear space
cites work: A fluctuations limit for scaled age distributions and weighted Sobolev spaces
cites work: Generalized jump-type Ornstein-Uhlenbeck processes as limits of infinite- particle systems
cites work: On weak convergence of integral functionals of stochastic processes with applications to processes taking paths in \(L^ E_ p\)
cites work: Stable hydrodynamic limit fluctuations of a critical branching particle system in a random medium
cites work: Generalized solutions of a class of nuclear-space-valued stochastic evolution equations
cites work: Spatially homogeneous random evolutions
cites work: Invariance principle and empirical mean large deviations of the critical Ornstein-Uhlenbeck process
cites work: Limit Theorems for Branching Diffusions in Hydrodynamical Rescaling
cites work: A stochastic particle system: Fluctuations around a nonlinear reaction- diffusion equation
cites work: Q3994935
cites work: Markov properties of the fluctuation limit of a particle system
cites work: Markov properties of the fluctuation limit of a particle system with death
cites work: A Criterion of Convergence of Generalized Processes and an Application to a Supercritical Branching Particle System
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cites work: Non-equilibrium fluctuations for a zero range process
cites work: La convergence en loi pour les processus à valeurs dans un espace nucléaire
cites work: High density limit theorems for infinite systems of unscaled branching Brownian motions
cites work: The demographic variation process of branching random fields
cites work: The demographic variation process of multitype branching random fields
cites work: Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions
cites work: Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions
cites work: Q3339049
cites work: On the Skorokhod topology
cites work: Q1099492
cites work: Weak convergence of sequences of semimartingales with applications to multitype branching processes
cites work: Stochastic evolution equations driven by nuclear-space-valued martingales
cites work: Infinite dimensional stochastic differential equation models for spatially distributed neurons
cites work: Linear parabolic differential equations as limits of space-time jump Markov processes
cites work: Law of large numbers and central limit theorem for linear chemical reactions with diffusion
cites work: Fluctuations near homogeneous states of chemical reactions with diffusion
cites work: Weak limit theorems for stochastic integrals and stochastic differential equations
cites work: Fluctuation limits of multitype branching random fields
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cites work: Convergence faible et principe d'invariance pour des martingales à valeurs dans des espaces de Sobolev
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cites work: Generalized Ornstein-Uhlenbeck process having a characteristic operator with polynomial coefficients
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cites work: On the existence, uniqueness, convergence and explosions of solutions of systems of stochastic integral equations
cites work: On existence and stability of weak solutions of multidimensional stochastic differential equations with measurable coefficients
cites work: Stability of strong solutions of stochastic differential equations
cites work: Equilibrium fluctuations for interacting Brownian particles
cites work: A mean field limit of the contact process with large range
cites work: Central limit theorem for a simple diffusion model of interacting particles
cites work: Fluctuations of Markovian systems in Kac's caricature of a Maxwellian gas
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cites work: Some applications of stochastic integration in infinite dimensions
full work available at URL: https://doi.org/10.1007/bf01058724
OpenAlex ID: W2001547248