Abstract
AbstractModern methods of simulating molecular systems are based on the mathematical theory of Markov operators with a focus on autonomous equilibrated systems. However, non‐autonomous physical systems or non‐autonomous simulation processes are becoming more and more important. A representation of non‐autonomous Markov jump processes is presented as autonomous Markov chains on space‐time. Augmenting the spatial information of the embedded Markov chain by the temporal information of the associated jump times, the so‐called augmented jump chain is derived. The augmented jump chain inherits the sparseness of the infinitesimal generator of the original process and therefore provides a useful tool for studying time‐dependent dynamics even in high dimensions. Furthermore, possible generalizations and applications to the computation of committor functions and coherent sets in the non‐autonomous setting are discussed. After deriving the theoretical foundations, the concepts with a proof‐of‐concept Galerkin discretization of the transfer operator of the augmented jump chain applied to simple examples are illustrated.
Highlights
We will introduce the notation and recall some basic results needed for the subsequent sections.Let the set X = {xi}i=1,...,N denote a finite state space and {Xt}t∈T a time-continuous Markov chain on X with T = R+0 denoting the time domain
Where the jump chain and jump times are defined as in Definition 1. We call this the augmented jump chain since its state space is that of the original process Xt augmented by the time component
We extended the known representation of autonomous Markov jump processes as embedded Markov chain (Theorem 1) to the non-autonomous regime
Summary
We will introduce the notation and recall some basic results needed for the subsequent sections. Let the set X = {xi}i=1,...,N denote a finite state space and {Xt}t∈T a time-continuous Markov chain ( called Markov jump process) on X with T = R+0 denoting the time domain. It is well known[2] that this process can be described by means of its associated stochastic transition kernel k(x, s, y, t) = P(Xt = y|Xs = x). Let Xt be an autonomous Markov jump process with infinitesimal generator Q = (qij). Are independent exponential random variables with parameters qY0 , qY1 , ..., respectively Using this decomposition for sampling, that is, drawing the state from the Markov chain Yt and the exponentially distributed holding time Hn leads to the well known Gillespie (Stochastic Simulation) Algorithm[20] for sampling from Markov Jump chains
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
