Cox JT, Greven A (1994)
Publication Language: English
Publication Type: Journal article, Original article
Publication year: 1994
Publisher: Institute of Mathematical Statistics (IMS)
Book Volume: 22
Pages Range: 833-853
Journal Issue: 2
URI: https://projecteuclid.org/euclid.aop/1176988732
Let x(t)={xi(t),i∈Zd}" role="presentation">x(t)={xi(t),i∈ℤd} be the solution of the system of stochastic differential equations dxi(t)=(∑j∈Zda(i,j)xj(t)−xi(t))dt+2g(xi(t))dwi(t),i∈Zd." role="presentation">dxi(t)=(∑j∈ℤda(i,j)xj(t)−xi(t))dt+2g(xi(t))‾‾‾‾‾‾‾‾√dwi(t),i∈ℤd. Here g:[0,1]→R+" role="presentation">g:[0,1]→ℝ+ satisfies g>0" role="presentation">g>0 on (0, 1), g(0)=g(1)=0,g" role="presentation">g(0)=g(1)=0,g is Lipschitz, a(i,j)" role="presentation">a(i,j) is an irreducible random walk kernel on Zd" role="presentation">ℤd and {wi(t),i∈Zd}" role="presentation">{wi(t),i∈ℤd} is a family of standard, independent Brownian motions on R;x(t)" role="presentation">ℝ;x(t) is a Markov process on X=[0,1]Zd" role="presentation">X=[0,1]ℤd. This class of processes was studied by Notohara and Shiga; the special case g(v)=v(1−v)" role="presentation">g(v)=v(1−v) has been studied extensively by Shiga. We show that the long term behavior of x(t)" role="presentation">x(t) depends only on a^(i,j)=(a(i,j)+a(j,i))/2" role="presentation">â (i,j)=(a(i,j)+a(j,i))/2 and is universal for the entire class of g" role="presentation">g considered. If a^(i,j)" role="presentation">â (i,j) is transient, then there exists a family {νθ,θ∈[0,1]}" role="presentation">{νθ,θ∈[0,1]} of extremal, translation invariant equilibria. Each νθ" role="presentation">νθ is mixing and has density θ=∫x0dνθ" role="presentation">θ=∫x0dνθ. If a^(i,j)" role="presentation">â (i,j), is recurrent, then the set of extremal translation invariant equilibria consists of the point masses {δ0,δ1}" role="presentation">{δ0,δ1}. The process starting in a translation invariant, shift ergodic measure μ" role="presentation">μ on X" role="presentation">X with ∫x0dμ=θ" role="presentation">∫x0dμ=θ converges weakly as t→∞" role="presentation">t→∞ to νθ" role="presentation">νθ if a^(i,j)" role="presentation">â (i,j) is transient, and to (1−θ)δ0+θδ1" role="presentation">(1−θ)δ0+θδ1 if a^(i,j)" role="presentation">â (i,j) is recurrent. (Our results in the recurrent case remove a mild assumption on g" role="presentation">g imposed by Notohara and Shiga.) For the case a^(i,j)" role="presentation">â (i,j) transient we use methods developed for infinite particle systems by Liggett and Spitzer. For the case a^(i,j)" role="presentation">â (i,j), recurrent we use a duality comparison argument.
APA:
Cox, J.T., & Greven, A. (1994). Ergodic theorems for systems of locally interacting diffusions. Annals of Probability, 22(2), 833-853. https://dx.doi.org/10.1214/aop/1176988732
MLA:
Cox, J. Theodore, and Andreas Greven. "Ergodic theorems for systems of locally interacting diffusions." Annals of Probability 22.2 (1994): 833-853.
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