Diffusion approximation of multi-class Hawkes processes: Theoretical and numerical analysis

Oscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced by Ditlevsen and Löcherbach (Stoch. Process. Appl., 2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. In this paper, first, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with mean-square order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity, and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.

Wigner ensemble Monte Carlo simulation without splitting error of a GaAs resonant tunneling diode

A Monte Carlo technique for the solution of the Wigner transport equation has been developed, based on the generation and annihilation of signed particles (Nedjalkov et al. in Phys Rev B 70:115319, 2004). A stochastic algorithm without time discretization error has been recently introduced (Muscato and Wagner in Kinet Relat Models 12(1):59–77, 2019). Its derivation is based on the theory of piecewise deterministic Markov processes. Numerical experiments are performed in the case of a GaAs resonant tunneling diode. Convergence of the time-splitting scheme to the no-splitting algorithm is demonstrated. The no-splitting algorithm is shown to be more efficient in terms of computational effort.

Time triggered stochastic hybrid system with nonlinear continuous dynamics

Time triggered stochastic hybrid systems (TTSHS) constitute a class of piecewise-deterministic Markov processes (PDMP), where continuous-time evolution of the state space is interspersed with discrete stochastic events. Whenever a stochastic event occurs, the state space is reset based on a random map. Prior work on this topic has focused on the continuous-time evolution being modeled as a linear time- invariant system, and in this contribution, we generalize these results to consider nonlinear continuous dynamics. Our approach relies on approximating the nonlinear dynamics between two successive events as a linear time-varying system and using this approximation to derive analytical solutions for the state space’s statistical moments. The TTSHS framework is used to model continuous growth in an individual cell’s size and its subsequent division into daughters. It is well known that exponential growth in cell size, together with a size- independent division rate, leads to an unbounded variance in cell size. Motivated by recent experimental findings, we consider nonlinear growth in cell size based on a Michaelis- Menten function and show that this leads to size homeostasis in the sense that the variance in cell size remains bounded. Moreover, we provide a closed-form expression for the variance in cell size as a function of model parameters and validate it by performing exact Monte Carlo simulations. In summary, our work provides an analytical approach for characterizing moments of a nonlinear stochastic dynamical system that can have broad applicability in studying random phenomena in both engineering and biology.

Fluid Approximation–based Analysis for Mode-switching Population Dynamics

Fluid approximation results provide powerful methods for scalable analysis of models of population dynamics with large numbers of discrete states and have seen wide-ranging applications in modelling biological and computer-based systems and model checking. However, the applicability of these methods relies on assumptions that are not easily met in a number of modelling scenarios. This article focuses on one particular class of scenarios in which rapid information propagation in the system is considered. In particular, we study the case where changes in population dynamics are induced by information about the environment being communicated between components of the population via broadcast communication. We see how existing hybrid fluid limit results, resulting in piecewise deterministic Markov processes, can be adapted to such models. Finally, we propose heuristic constructions for extracting the mean behaviour from the resulting approximations without the need to simulate individual trajectories.