Diffusions conditioned on occupation measures

TL;DR

This paper constructs a driven Markov process for diffusions conditioned on spending atypical fractions of time in certain regions, generalizing known problems like the Brownian meander and applicable to various stochastic systems.

Contribution

It introduces a method to explicitly construct the effective process for diffusions conditioned on occupation measures, extending the theory to new classes of conditioned stochastic processes.

Findings

Constructed driven processes for diffusions with specified occupation measures

Generalized the Brownian meander to broader regions and conditions

Discussed applications to metastability, chemical reactions, and queues

Abstract

A Markov process fluctuating away from its typical behavior can be represented in the long-time limit by another Markov process, called the effective or driven process, having the same stationary states as the original process conditioned on the fluctuation observed. We construct here this driven process for diffusions spending an atypical fraction of their evolution in some region of state space, corresponding mathematically to stochastic differential equations conditioned on occupation measures. As an illustration, we consider the Langevin equation conditioned on staying for a fraction of time in different intervals of the real line, including the positive half-line which leads to a generalization of the Brownian meander problem. Other applications related to quasi-stationary distributions, metastable states, noisy chemical reactions, queues, and random walks are discussed.