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We develop a practical approach to establish the stability, that is, the recurrence in a given set, of a large class of controlled Markov chains. These processes arise in various areas of applied science and encompass important numerical…
This paper is concerned with the development of rigorous approximations to various expectations associated with Markov chains and processes having non-stationary transition probabilities. Such non-stationary models arise naturally in…
Granular materials are characterized by large collections of discrete particles of sizes larger than one micron, where the particle-particle interactions are significantly more important than the particle-fluid interactions. These flows can…
Big networks express various large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big…
The parametrization of coarse-grained (CG) simulation models for molecular systems often aims at reproducing static properties alone. The reduced molecular friction of the CG representation usually results in faster, albeit inconsistent,…
We derive a thermodynamic uncertainty relation (TUR) for first-passage times (FPTs) on continuous time Markov chains. The TUR utilizes the entropy production coming from bidirectional transitions, and the net flux coming from unidirectional…
Metastability is a physical phenomenon ubiquitous in first order phase transitions. A fruitful mathematical way to approach this phenomenon is the study of rare transitions Markov chains. For Metropolis chains associated with Statistical…
We complete the existing literature on the kinetic theory of systems with long-range interactions. Starting from the BBGKY hierarchy, or using projection operator technics or a quasilinear theory, a general kinetic equation can be derived…
It is considered the integrated process $X(t)= x + \int _0^t Y(s) ds ,$ where $Y(t)$ is a Gauss-Markov process starting from $y.$ The first-passage time (FPT) of $X$ through a constant boundary and the first-exit time of $X$ from an…
We have developed a steady state theory of complex transport networks used to model the flow of commodity, information, viruses, opinions, or traffic. Our approach is based on the use of the Markov chains defined on the graph…
Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Inspired by the theory of random products of matrices, it has been shown that these Markov processes admit…
Mathematical models of biological neural networks are associated to a rich and complex class of stochastic processes. In this paper, we consider a simple {\em plastic} neural network whose {\em connectivity/synaptic strength} $(W(t))$…
The transition mechanism of jump processes between two different subsets in state space reveals important dynamical information of the processes and therefore has attracted considerable attention in the past years. In this paper, we study…
We give a short overview of recent results on a specific class of Markov process: the Piecewise Deterministic Markov Processes (PDMPs). We first recall the definition of these processes and give some general results. On more specific cases…
The number of observable degrees of freedom is typically limited in experiments. Here, we consider discrete Markov networks in which an observer has access to a few visible transitions and the waiting times between these transitions.…
Up to now, the nonparametric analysis of multidimensional continuous-time Markov processes has focussed strongly on specific model choices, mostly related to symmetry of the semigroup. While this approach allows to study the performance of…
Dynamic networks exhibit temporal patterns that vary across different time scales, all of which can potentially affect processes that take place on the network. However, most data-driven approaches used to model time-varying networks…
Many transport processes in ecology, physics and biochemistry can be described by the average time to first find a site or exit a region, starting from an initial position. Typical mathematical treatments are based on formulations that…
In this chapter, we consider the problem of a non-Markovian random walker (displaying memory effects) searching for a target. We review an approach that links the first passage statistics to the properties of trajectories followed by the…
Transition Path Theory (TPT) provides a rigorous statistical characterization of the ensemble of trajectories connecting directly, i.e., without detours, two disconnected (sets of) states in a Markov chain, a stochastic process that…