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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
Full Text: Numdam EuDML