Generating stochastic trajectories with global dynamical constraints
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 ${A}_{f}={\int }_{0}^{{t}_{f}}\mathrm{d}t\enspace {x}_{c}(t)$ Af=∫0tfdtxc(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 discretetime random walks, with arbitrary jump distributions including Lévy 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 ${T}_{f}={\int }_{0}^{{t}_{f}}\mathrm{d}t\enspace {\Theta}\left[{x}_{c}(t)\right]$ Tf=∫0tfdtΘxc(t) or a fixed generalised quadratic area ${\mathcal{A}}_{f}={\int }_{0}^{{t}_{f}}\mathrm{d}t\enspace {x}_{c}^{2}(t)$ Af=∫0tfdtxc2(t) .
 Publication:

Journal of Statistical Mechanics: Theory and Experiment
 Pub Date:
 December 2021
 DOI:
 10.1088/17425468/ac3e70
 arXiv:
 arXiv:2110.07573
 Bibcode:
 2021JSMTE2021l3204D
 Keywords:

 Brownian motion;
 extreme value;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Probability
 EPrint:
 32 pages, 7 figures