On the distribution of planar Brownian motion at stopping times

Abstract

A simple extension is given of the well-known conformal invariance of harmonic measure in the plane. This equivalence depends on the interpretation of harmonic measure as an exit distribution of planar Brownian motion, and extends conformal invariance to analytic functions which are not injective, as well as allowing for stopping times more general than exit times. This generalization allow considerations of homotopy and reflection to be applied in order to compute new expressions for exit distributions of various domains, as well as the distribution of Brownian motion at certain other stopping times. An application of these methods is the derivation of a number of infinite sum identities, including the Leibniz formula for π and the values of the Riemann ζ function at even integers.2010 Mathematics subject classification: 60J65, 30A99.

Many proofs that $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{π^2}{6}$ can be found in the conformal invariance of planar Brownian motion

“…Suppose τ is a stopping time such that B τ ∈ γ a.s., and let ρ a τ (w)ds be the density of B τ on γ, when it exists, with ds denoting the arclength element. Then we have the following result from [9]. Theorem 2.…”

“…As a final disclaimer before beginning the proofs, it should be mentioned that very few, if any, of the ideas presented below are truly original to [10], [11], [9]; many of the same methods have surfaced over the years in various contexts, albeit often packaged somewhat differently. For example, the idea of wrapping the real axis around the circle as a mechanism for deducing summation identities (Proof 2 below) is essentially the same as is used in the Poisson summation formula (see for example [20]), and has been used also to deduce Jacobi's theta function identity from properties of one-dimensional Brownian motion wound around the circle ( [3]).…”

A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion

“…As another application of Theorem 4.6 (applied to the conformal map f (z) = ( 1−z 1+z ) θ π ), let us consider the following result (which can also be deduced from the explicit formula for the distribution of τ (W ) given in [19]). Proposition 4.9.…”

A note on invariance of the Cauchy and related distributions

“…We note that both of the preceding results can be obtained using the conformal invariance of Brownian motion and harmonic measure (see [Mar18]), and Theorem 1 can also be deduced by a direct calculation, using the Poisson kernel, or by properties of stable distributions; see for instance [Dur84, Sec. 1.9] or [Fel08, Ch.…”