Non-Markovian process driven by quadratic noise: Kramers-Moyal expansion and Fokker-Planck modeling

Spatial and temporal memory effects in the Nagel-Schreckenberg model for crowdsourced traffic property determination

We investigate the spatial and temporal memory effects of traffic density and velocity in the Nagel-Schreckenberg cellular automaton model. We show that the two-point correlation function of vehicle occupancy provides access to spatial memory effects, such as headway, and the velocity autocovariance function to temporal memory effects such as traffic relaxation time and traffic compressibility. We develop stochasticity-density plots that permit determination of traffic density and stochasticity from the isotherms of first- and second-order velocity statistics of a randomly selected vehicle. Specifically, provided ergodicity and stationarity, these stochasticity-density plots permit a direct determination of traffic properties from crowdsourced measurements of velocities of vehicles. We illustrate the predictive prowess of the approach for crowdsourced vehicle speed data collected by anonymous smartphone measurements for the state of Massachusetts, USA, as a powerful alternative to classical traffic property estimates from spatially distributed user counts.

Chemical Langevin equation: A path-integral view of Gillespie's derivation

In 2000, Gillespie rehabilitated the chemical Langevin equation (CLE) by describing two conditions that must be satisfied for it to yield a valid approximation of the chemical master equation (CME). In this work, we construct an original path-integral description of the CME and show how applying Gillespie's two conditions to it directly leads to a path-integral equivalent to the CLE. We compare this approach to the path-integral equivalent of a large system size derivation and show that they are qualitatively different. In particular, both approaches involve converting many sums into many integrals, and the difference between the two methods is essentially the difference between using the Euler-Maclaurin formula and using Riemann sums. Our results shed light on how path integrals can be used to conceptualize coarse-graining biochemical systems and are readily generalizable.

Stochastic resonance in a linear system: An exact solution

Multistable systems can exhibit stochastic resonance which is characterized by the amplification of small periodic signals by additive noise. Here we consider a nonmultistable linear system with a multiplicative noise forced by an external periodic signal. The noise is the sum of a colored noise of mean value zero and a noise with a definite sign. We show that the system exhibits stochastic resonance through the numerical study of an exact analytical expression for the mean value obtained by functional integral techniques. This is proof of the effect for a very general kind of noise which can even have a definite sign.

Fokker-Planck perspective on stochastic delay systems: exact solutions and data analysis of biological systems

Stochastic delay systems with additive noise are examined from the perspective of Fokker-Planck equations. For a linear system, the exact stationary probability density is derived by means of a delay Fokker-Planck equation. We show how to determine the delay equation of the linear system from experimental data, and corroborate a fundamental result previously obtained by Küchler and Mensch. We also propose a method to derive delay equations of nonlinear systems from experimental data. To this end, the theory of multivariate Fokker-Planck equations is used. The applicability of this method is demonstrated for stochastic models describing tracking and pointing movements of humans.