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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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