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Self-interacting diffusions. II: Convergence in law. (English) Zbl 1064.60191
This paper is a continuation of the article [Probab. Theory Relat. Fields 122, 1–41 (2002; Zbl 1042.60060)] by the present authors with M. Ledoux. A self-interacting diffusion on a compact Riemannian manifold \(M\) is a family of probability measures \(\{P_{x,r,\mu}\mid x\in M, r> 0, \mu\) a probability measure on \(M\}\) each of which is the law of a diffusion process \(X_t\) solving an SDE of the form \[ dX_t = \sum_{i=1}^N F_i(X_t)\circ dB_t^i - \nabla V_{\mu_t(r,\mu)}(X_t)\,dt, \quad X_0 = x \] where \(B_t^i\) are independent Brownian motions, \(F_i\) are vector fields s.t. \(\sum_i F_i^2 = \Delta\), \(V_\bullet\) is a potential function and \(\mu_t\) is the empirical occupation measure with initial weight \(r\) and initial measure \(\mu\). Existence of such diffusions was established in the above mentioned paper.
The present paper studies convergence in law of self-interacting diffusions. The main results are: \(\lim_{t\to\infty}\text{dist}(P_{X_t,r+t,\mu_t(r,\mu)},P_{X_t,\mu^*}) = 0\) \(P_{x,r,\mu}\)-a.s. on \(\widetilde\Omega := \{\omega \mid \lim_{t\to\infty}\mu_t(r,\mu,\omega) = \mu^*(\omega)\}\) where \(\mu^*\) is a measure-valued r.v. Furthermore, \(\text{law}(X_{t+s(t)}|\mathcal B_t) \to \mu^*\) a.s. weakly as \(t\to\infty\) whenever \(s(t)\to\infty\) with the right rate. A similar result is established for \(X_{t+u}\).

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
58J65 Diffusion processes and stochastic analysis on manifolds
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60F05 Central limit and other weak theorems
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