Mean-Square Numerical Methods for Stochastic Differential Equations with Small Noises

A new approach to the construction of mean-square numerical methods for the solution of stochastic differential equations with small noises is proposed. The approach is based on expanding the exact solution of the system with small noises in powers of time increment and small parameter. The theorem on the mean-square estimate of method errors is proved. Various efficient numerical schemes are derived for a general system with small noises and for systems with small additive and small colored noises. The proposed methods are tested by calculation of Lyapunov exponents and simulation of a laser Langevin equation with multiplicative noises

Mean-square approximation for stochastic differential equations with small noises

New approach to construction of mean-square numerical methods for solution of stochastic differential equations with small noises is proposed. The approach is based on expanding of the exact solution of the system with small noises by powers of time increment and regrouping of expansion terms according to powers of time increment and small parameter. The theorem on mean-square estimate of method errors is proved. Various efficient numerical schemes are derived for a general system with small noises and for systems with small additive and small colored noises. The proposed methods are tested by calculation of Lyapunov exponents and simulation of a laser Langevin equation with multiplicative noises

Complex noise in diffusion-limited reactions of replicating and competing species

We derive exact Langevin-type equations governing quasispecies dynamics. The inherent multiplicative noise has both real and imaginary parts. The numerical simulation of the underlying complex stochastic partial differential equations is carried out employing the Cholesky decomposition for the noise covariance matrix. This noise produces unavoidable spatio-temporal density fluctuations about the mean field value. In two dimensions, the fluctuations are suppressed only when the diffusion time scale is much smaller than the amplification time scale for the master species.Comment: 10 pages, 2 composite figure

Positive-P phase space method simulation in superradiant emission from a cascade atomic ensemble

The superradiant emission properties from an atomic ensemble with cascade level configuration is numerically simulated. The correlated spontaneous emissions (signal then idler fields) are purely stochastic processes which are initiated by quantum fluctuations. We utilize the positive-P phase space method to investigate the dynamics of the atoms and counter-propagating emissions. The light field intensities are calculated, and the signal-idler correlation function is studied for different optical depths of the atomic ensemble. Shorter correlation time scale for a denser atomic ensemble implies a broader spectral window needed to store or retrieve the idler pulse.Comment: To be published in Phys. Rev.