NDLI: N/V-limit for stochastic dynamics in continuous particle systems

Content Provider	Springer Nature Link
Author	Kondratiev, Yuri G. Röckner, Michael Grothaus, Martin
Copyright Year	2006
Abstract	We provide an N/V-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on ℝ d ,d≥1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ⊂ℝ d with finite volume (Lebesgue measure) V=\|Λ\|<∞. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above N-particle dynamic in Λ as N→∞ and V→∞ such that N/V→ρ, where ρ is the particle density. First we derive an improved Ruelle bound for the canonical correlation functions under an appropriate relation between N and V. Then tightness is shown by using the Lyons–Zheng decomposition. The equilibrium measures of the accumulation points are identified as infinite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle, infinite volume stochastic dynamics via the associated martingale problem. Assuming a property closely related to Markov uniqueness and weaker than essential self-adjointness, via Mosco convergence techniques we can identify the accumulation points as Markov processes and show uniqueness. I.e., all accumulation corresponding to one invariant canonical Gibbs measure coincide. The proofs work for general repulsive interaction potentials ϕ of Ruelle type and all temperatures, densities, and dimensions d≥1, respectively. ϕ may have a nontrivial negative part and infinite range as e.g. the Lennard–Jones potential. Additionally, our result provides as a by-product an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary condition.
Ending Page	160
Page Count	40
Starting Page	121
File Format	PDF
ISSN	01788051
e-ISSN	14322064
Journal	Probability Theory and Related Fields
Issue Number	1
Volume Number	137
Language	English
Publisher	Springer-Verlag
Publisher Date	2006-04-24
Publisher Place	Berlin/Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Limit theorems for vector-valued random variables (infinite-dimensional case) Statistics for Business/Economics/Mathematical Finance/Insurance Interacting particle systems Diffusion processes Operations Research/Decision Theory Mathematical and Computational Physics Stochastic ordinary differential equations Interacting random processes; statistical mechanics type models; percolation theory Probability Theory and Stochastic Processes Limit theorems Mathematical Biology in General Quantitative Finance
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Analysis Statistics, Probability and Uncertainty

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