From Reflected Brownian Motion to the Critical Line: A Boundary Value Problem Analogy for the Riemann Hypothesis

Abstract

This essay explores a speculative analogy between the stationary distribution of reflected Brownian motion in the half-plane, as analyzed by Flin using Riemann boundary value problems and Sokhotski-Plemelj formulas, and the complex analytic theory of the Riemann zeta function.

Overview

This essay mines a paper by Jules Flin on the stationary distribution of reflecting Brownian motion (RBM) in the half-plane for structural patterns relevant to the Riemann Hypothesis (RH). Flin's work reduces the characterization of the stationary measure to solving a discontinuous Riemann boundary value problem (BVP) on the real line, using the Sokhotski-Plemelj formulas to derive explicit contour integral representations.

We propose a speculative bridge to RH by viewing the Riemann zeta functional equation ξ(s) = ξ(1−s) as a boundary value problem across the critical line Re(s) = 1/2. The analogy maps Flin's sectionally analytic Laplace transforms ϕ± to the zeta function on either side of the critical line, and the multiplicative jump function G(t) to the functional equation factor χ(1/2 + it). Both frameworks employ contour integrals and Tauberian theorems to connect boundary asymptotics to global behavior.

The correspondence is rated as a formal analogy: the mathematical structure of the BVP (multiplicative jump on the real line reduced to additive form via logarithms) is identical, though the specific kernels differ (algebraic vs. transcendental). The essay explores what would need to be true for this analogy to yield insights into the distribution of zeta zeros, and proposes numerical experiments comparing the Cauchy integral representations.

We conclude with an honest assessment of the analogy's limitations, noting the absence of an Euler product structure in the probabilistic model and the differing nature of the asymptotic results (power laws vs. oscillatory sums).

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.