Dirichlet's and Thomson's principles for non-selfadjoint elliptic operators with application to non-reversible metastable diffusion processes

We present two variational formulae for the capacity in the context of non-selfadjoint elliptic operators. The minimizers of these variational problems are expressed as solutions of boundary-value elliptic equations. We use these principles to provide a sharp estimate for the transition times between two different wells for non-reversible diffusion processes. This estimate permits to describe the metastable behavior of the system

Measure rigidity for leafwise weakly rigid actions

In this paper, given a Borel action $G\curvearrowright X$, we introduce a new approach to obtain classification of conditional measures along a $G$-invariant foliation along which $G$ has a controlled behavior. Given a Borel action $G\curvearrowright X$ over a Lebesgue space $X$ we show that if $G\curvearrowright X$ preserves an invariant system of metrics along a Borel lamination $\mathcal F$, which satisfy a good packing estimative hypothesis, then the ergodic measures preserved by the action are rigid in the sense that the system of conditional measures with respect to the partition $\mathcal F$ are the Hausdorff measures given by the metric system or are supported in a countable number of boundaries of balls. The argument we employ does not require any structure on $G$ other then second-countability and no hyperbolicity on the action as well. Our main result is interesting on its own, but to exemplify its strength and usefulness we show some applications in the context of cocycles over hyperbolic maps and to certain partially hyperbolic maps

Some particular self-interacting diffusions: Ergodic behaviour and almost sure convergence

This paper deals with some self-interacting diffusions $(X_t,t\geq 0)$ living on $\mathbb{R}^d$. These diffusions are solutions to stochastic differential equations: $$\mathrm{d}X_t=\mathrm{d}B_t-g(t)\nabla V(X_t-\bar{\mu}_t)\,\mathrm{d}t,$$ where $\bar{\mu}_t$ is the empirical mean of the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is a given function. We study the ergodic behaviour of $X$ and prove that it is strongly related to $g$. Actually, we show that $X$ is ergodic (in the limit quotient sense) if and only if $\bar{\mu}_t$ converges a.s. We also give some conditions (on $g$ and $V$) for the almost sure convergence of $X$.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ310 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm