Generating stochastic trajectories with global dynamical constraints – Archive ouverte HAL

Benjamin de Bruyne ¹ Satya N. Majumdar ¹ Henri Orland ² Gregory Schehr ³

Benjamin de Bruyne, Satya N. Majumdar, Henri Orland, Gregory Schehr. Generating stochastic trajectories with global dynamical constraints. Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2021, 12, ⟨10.1088/1742-5468/ac3e70⟩. ⟨hal-03534085⟩

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 $\mathcal{A}_f=\int_0^{t_f}dt \,x_c^2(t)$.

• 1. LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
• 2. IPHT – Institut de Physique Théorique – UMR CNRS 3681
• 3. LPTHE – Laboratoire de Physique Théorique et Hautes Energies