## Topologie faible et meta-stabilite. (Weak topology and meta-stability).(French)Zbl 0633.60010

Sémin. probabilités XXI, Lect. Notes Math. 1247, 544-562 (1987).
[For the entire collection see Zbl 0606.00022.]
The problem regarded in this paper comes from the work of M. Cassandro, A. Galves, E. Olivieri and M. Vares [J. Stat. Phys. 35, 603-634 (1984; Zbl 0591.60080)] about the asymptotic behaviour of a randomly perturbed particle in a two-well potential, when the random perturbation goes to zero. Depending on the time length of observation in relation to the perturbation parameter, the local-time measure of the particle has different asymptotic limits.
Since this result cannot hold in the usual Skorokhod space, a special weak topology, that of convergence of pseudo-trajectories, is needed here: If $$\mu$$ ($$\cdot)$$ is a trajectory with values in the space $${\mathcal M}$$ of local-time measures, its pseudo-trajectory $${\hat \mu}$$ is the measure on $${\mathbb{R}}^+\times {\mathcal M}$$ given by $<{\hat \mu},f\otimes g>=\int^{\infty}_{0}e^{-t}f(t) g(\mu (t))dt,$ $$f\in L^{\infty}({\mathbb{R}}^+)$$, $$g\in L^{\infty}({\mathcal M})$$. The set of measures on $${\mathbb{R}}^+\times {\mathcal M}$$ is endowed with the usual weak topology.
The paper gives tightness criteria for pseudo-trajectory-valued processes, which apply to the situation regarded by Cassandro et al.
Reviewer: Th.Eisele

### MSC:

 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60K35 Interacting random processes; statistical mechanics type models; percolation theory

### Citations:

Zbl 0606.00022; Zbl 0591.60080
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