On conformally invariant subsets of the planar Brownian curve
Abstract
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.