We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form $-ϵ\Delta +\nabla F(·)\nabla$ on ${ℝ}^d$ or subsets of ${ℝ}^d$, where $F$ is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum...

We study the law of functionals whose prototype is ∫0+∞ eBs(ν) dWs(μ),where B(ν) and W(μ) are independent Brownian motions with drift. These functionals appear naturally in risk theory as well as in the study of in variant diffusions on the hyperbolic half-plane. Emphasis is put on the fact that the results are obtained in two independent, very different fashions (invariant diffusions on the hyperbolic half-plane and Bessel processes).

Dans cet article, nous étudions les résultats de grandes déviations associés au couple $(X^{\epsilon },{\nu }^{\epsilon })$, solution de l’E.D.S. interprétée au sens d’Itô :$$d{X}_t^{\epsilon }=\sqrt{\epsilon }{\sigma }_{{\nu }^{\epsilon }\left(t\right)}\left(X_t^{\epsilon }\right)d{W}_t+b_{{\nu }^{\epsilon }\left(t\right)}\left(X_t^{\epsilon }\right)dt;X_0^{\epsilon }=x\in {ℝ}^d$$avec des conditions assez générales sur les coefficients et dans les deux cas suivants :Premier cas : ${\nu }^{\epsilon }$ est indépendant du mouvement brownien $W$ et satisfait à un principe de grandes déviations ;Deuxième cas : ${\nu }^{\epsilon }$ est un processus markovien avec un nombre fini d’états $\{1,...,n\}$ vérifiant$$\mathbb{P}\{{\nu }^{\epsilon }(t+\Delta )=j/{\nu }^{\epsilon }\left(t\right)=i,X^{\epsilon }\left(t\right)=x\}=d_{ij}\left(x\right)\Delta +o\left(\Delta \right)$$uniformément dans ${ℝ}^d$ pourvu que $\Delta \downarrow 0,1\le i,j\le n,i\ne j$.Ces résultats sont des extensions de ceux de Bezuidenhout...

If ξ(t) is the solution of homogeneous SDE in R m, and T ∃ is the first exit moment of the process from a small domain D ∃, then the total expansion for the following functional showing independence of the exit time and exit place is $$Eexp(-\lambda T_{\epsilon })f\left(\frac{\xi \left(T_{\epsilon }\right)}{\epsilon }\right)-Eexp(-\lambda T_{\epsilon })Ef\left(\frac{\xi \left(T_{\epsilon }\right)}{\epsilon }\right),\epsilon \searrow 0,\lambda >0.$$

We study the Hausdorff dimension of measures whose weight distribution satisfies a Markov non-homogeneous property. We prove, in particular, that the Hausdorff dimensions of this kind of measures coincide with their lower Rényi dimensions (entropy). Moreover, we show that the packing dimensions equal the upper Rényi dimensions. As an application we get a continuity property of the Hausdorff dimension of the measures, when viewed as a function of the distributed weights under the ${\ell }^{\infty }$ norm.