Boundary crossing identities for Brownian motion and some nonlinear ode's

TL;DR

This paper introduces a nonlinear involution operator linking Sturm-Liouville solutions to nonlinear ODEs, and demonstrates how it connects Brownian motion first passage times to a family of curves through algebraic and analytical methods.

Contribution

It presents a novel nonlinear involution operator and a composition method that relates Brownian first passage times to solution families of nonlinear ODEs, with multiple proofs including symmetry analysis.

Findings

Connection between Brownian first passage times and nonlinear ODE solutions

Introduction of a unique two-parameter family of solution operators

Multiple proofs including symmetry and harmonic transform methods

Abstract

We start by introducing a nonlinear involution operator which maps the space of solutions of Sturm-Liouville equations into the space of solutions of the associated equations which turn out to be nonlinear ordinary differential equations. We study some algebraic and analytical properties of this involution operator as well as some properties of a two-parameter family of operators describing the set of solutions of Sturm-Liouville equations. Next, we show how a specific composition of these mappings allows to connect, by means of a simple analytical expression, the law of the first passage time of a Brownian motion over a curve to a two-parameter family of curves. We offer three different proofs of this fact which may be of independent interests. In particular, one is based on the construction of parametric time-space harmonic transforms of the law of some Gauss-Markov processes. Another one, which is of algebraic nature, relies on the Lie group symmetry methods applied to the heat equation and reveals that our two-parameter transformation is the unique non-trivial one.