Abstract.
Let K⊂ℝd (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX i (t) = ∑ j a(j−i) (X j (t) −X i (t))dt + σ (X i (t))dB i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a S (i) = a(i) + a(−i) is recurrent, then each component X i (∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a S on Abelian groups Λ.
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Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000
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Swart, J. Clustering of linearly interacting diffusions and universality of their long-time limit distribution. Probab Theory Relat Fields 118, 574–594 (2000). https://doi.org/10.1007/PL00008755
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DOI: https://doi.org/10.1007/PL00008755