Generating stochastic trajectories with global dynamical constraints

TL;DR

This paper introduces a method to generate stochastic trajectories, like Brownian paths and random walks, that satisfy specific global constraints such as returning to the origin, fixed area, or occupation time, using effective Langevin equations and jump distributions.

Contribution

It provides an exact framework for generating constrained stochastic paths, including continuous and discrete processes, with applications to various global constraints.

Findings

Derived an exact effective Langevin equation for constrained Brownian paths.

Developed an approach for discrete-time random walks with arbitrary jump distributions.

Generalized the method to other constraints like occupation time and quadratic area.

Abstract

We propose a method to exactly generate Brownian paths $x_c(t)$ that are constrained to return to the origin at some future time $t_f$, with a given fixed area $A_f = \int_0^{t_f}dt\, x_c(t)$ under their trajectory. We derive an exact effective Langevin equation with an effective force that accounts for the constraint. In addition, we develop the corresponding approach for discrete-time random walks, with arbitrary jump distributions including L\'evy flights, for which we obtain an effective jump distribution that encodes the constraint. Finally, we generalise our method to other types of dynamical constraints such as a fixed occupation time on the positive axis $T_f=\int_0^{t_f}dt\, \Theta\left[x_c(t)\right]$ or a fixed generalised quadratic area $\mathcal{A}_f=\int_0^{t_f}dt \,x_c^2(t)$.