On conformally invariant subsets of the planar Brownian curve

Vincent Beffara (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We define and study a family of generalized non-intersection exponents for planar Brownian motions that is indexed by subsets of the complex plane: For each $A\subset\CC$, we define an exponent $\xi(A)$ that describes the decay of certain non-intersection probabilities. To each of these exponents, we associate a conformally invariant subset of the planar Brownian path, of Hausdorff dimension $2-\xi(A)$. A consequence of this and continuity of $\xi(A)$ as a function of $A$ is the almost sure existence of pivoting points of any sufficiently small angle on a planar Brownian path.

Domaines: Probabilités [math.PR]

Vincent Beffara. On conformally invariant subsets of the planar Brownian curve. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2003, 39 (5), pp.793 - 821. ⟨10.1016/S0246-0203(03)00030-X⟩. ⟨hal-01693163⟩