Weak Poincaré inequalities on domains defined by Brownian rough paths. (English) Zbl 1067.60029

The aim of this article is to prove the weak Poincaré inequality (WPI) on some infinite-dimensional sets of Brownian paths. More precisely, the set of paths \(w\) under consideration are described as \(\{w;F(w)>0\}\), for a functional \(F\) which is for instance deduced from a stochastic differential equation. This functional is generally not continuous for the usual topology of the Wiener space, so the set is not open. Thus the idea is to replace classical Brownian paths by rough paths. Then continuity properties of \(F\) can be applied, and under some assumptions, the author can prove the WPI. For the proof, several results concerning rough paths are recalled, and useful complements are given.


60H07 Stochastic calculus of variations and the Malliavin calculus
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
58J65 Diffusion processes and stochastic analysis on manifolds
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