Patchwork sampling of stochastic differential equations

Abstract

We propose a method to sample stationary properties of solutions of stochastic differential equations, which is accurate and efficient if there are rarely visited regions or rare transitions between distinct regions of the state space. The method is based on a complete, nonoverlapping partition of the state space into patches on which the stochastic process is ergodic. On each of these patches we run simulations of the process strictly truncated to the corresponding patch, which allows effective simulations also in rarely visited regions. The correct weight for each patch is obtained by counting the attempted transitions between all different patches. The results are patchworked to cover the whole state space. We extend the concept of truncated Markov chains which is originally formulated for processes which obey detailed balance to processes not fulfilling detailed balance. The method is illustrated by three examples, describing the one-dimensional diffusion of an overdamped particle in a double-well potential, a system of many globally coupled overdamped particles in double-well potentials subject to additive Gaussian white noise, and the overdamped motion of a particle on the circle in a periodic potential subject to a deterministic drift and additive noise. In an appendix we explain how other well-known Markov chain Monte Carlo algorithms can be related to truncated Markov chains.

Random walks exhibiting anomalous diffusion: elephants, urns and the limits of normality

“…As well as providing a unifying perspective, the approximations used here to study the   T limit are controlled precisely and the analysis is consequently tighter than the Fokker-Planck approach. Similar findings are reported in different settings relating to random walks with slightly modified memory rules [4,11], bond percolation on random recursive trees [22], two-component urn processes [27,31,32] and voting models with two types of voter behaviour [37].…”

“…Minor modifications to the original formulation lead to sub-diffusion as well as super-diffusion [20,21], and various other models turn out to have a close connection to the ERW as well, see e.g. [22][23][24][25].…”

Discontinuous transitions in globally coupled potential systems with additive noise

“…The third property follows from direct calculation. From the properties (21,22) it follows by mean value theorem that there exists a critical value a c such that…”

“…As a standard simulation needs to much time to reach the stationary distribution we use the method of patchwork sampling [21] where the state space is cut into many patches that are then simulated separately. Eventually the results from the simulations of each piece are used to obtain the stationary distribution of the original model.…”

“…We further sample the stationary center of mass distribution for systems of finite size in the region of coexistence using patchwork sampling [21]. We find that the two limits of infinite observation time (or stationarity) and infinite system size do not commute.…”