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A Brownian motor for a Myosin II head sliding along an actin filament is designed. In the present work, without any specific assumption on the form of the potential, we obtain results on some of the most relevant features of a Brownian motor in terms of averages of certain deterministic functions of a random vector (X,S). Our approach appears to be of interest especially for numerical and simulation purposes. For instance, differently from other papers where ad hoc techniques are provided to determine the total flux, by our approach it suffices to simulate trajectories of the Brownian particle and to record its position X and chemical state S after a time that is long enough to enter the stationary state.

This paper focuses on the outline of some computational methods for the approximate solution of the integral equations for the neuronal firing probability density and an algorithm for the generation of sample-paths in order to construct histograms estimating the firing densities. Our results originate from the study of non-Markov stationary Gaussian neuronal models with the aim to determine the neuron's firing probability density function. A parallel algorithm has been implemented in order to simulate large numbers of sample paths of Gaussian processes characterized by damped oscillatory covariances in the presence of time dependent boundaries. The analysis based on the simulation procedure provides an alternative research tool when closed-form results or analytic evaluation of the neuronal firing densities are not available., 10 pages, 3 figures, to be published in Scientiae Mathematicae Japonicae

Making use of a Rice-like series expansion, for a class of stationary Gaussian processes the asymptotic behavior of the first passage time probability density function through certain time-varying boundaries, including periodic boundaries, is determined. Sufficient conditions are then given such that the density asymptotically exhibits an exponential behavior when the boundary is either asymptotically constant or asymptotically periodic.

The main results of our research related to first passage time (FPT) problems for stationary Gaussian processes are synthetically outlined. The vectorized and parallel algorithm, efficiently implemented on CRAY-T3E in FORTRN90-MPI, allows to simulat a large number of sample paths of Gaussian stochastic processes in order to obtain reliable estimates of probability density functions (pdf) of first passage times through pre-assigned boundaries. The class of Gaussian processes characterized by damped oscillatory covariance functions and by Butterworth-type covariances have been extensively analyzed in the presence of constant and/or periodic boundaries. The analysis based on our simulation procedure has been particularly profitable as it has proved to provide an efficient research tool in all cases of interest to us when closed-form results or analytic evaluation were not available. Last but not least, in some cases it has allowed us to conjecture certain general features of FPT densities that successively have been rigorously proved.

The first passage time problem through a monotone boundary is considered for stationary Gaussian processes possessing rational spectral densities. Using a parallel procedure to simulate sample paths, estimates of the upcrossing probability density function of the first passage time are constructed. A comparison is provided with the numerical results obtained by Kostyukov via numerical solution of a Volterra integral equation. Finally, the effect of covariance's oscillatory components and of the behaviour of the boundary on the shape of the upcrossing FPT densities are pinpointed.