Strongly constrained stochastic processes: the multi-ends Brownian bridge

TL;DR

This paper explores the independence of certain conditioned stochastic processes from external flow fields, introducing a new multi-ends Brownian bridge and demonstrating its properties through theoretical derivations and numerical simulations.

Contribution

It introduces a novel multi-ends Brownian bridge process and applies the Langevin approach to analyze flow-independent conditioned diffusions.

Findings

Distribution of first hitting times is flow-independent.

New multi-ends Brownian bridge with two different final points.

Numerical simulations confirm theoretical results.

Abstract

In a recent article, Krapivsky and Redner (J. Stat. Mech. 093208 (2018)) established that the distribution of the first hitting times for a diffusing particle subject to hitting an absorber is independent of the direction of the external flow field. In the present paper, we build upon this observation and investigate when the conditioning on the diffusion leads to a process that is totally independent of the flow field. For this purpose, we adopt the Langevin approach, or more formally the theory of conditioned stochastic differential equations. This technique allows us to derive a large variety of stochastic processes: in particular, we introduce a new kind of Brownian bridge ending at two different final points and calculate its fundamental probabilities. This method is also very well suited for generating statistically independent paths. Numerical simulations illustrate our findings.