Stochastics and Dynamics

Let T be the Pascal-adic transformation. For any measurable function g, we consider the corrections to the ergodic theorem

A nonlinear online Kalman type filter is proposed for the estimation of unknown function S(t) with the known smoothness β for the diffusion observed process with small, of the order ε², diffusion coefficient. Assuming that the drift coefficient of the observed process depends on an unknown function S(t), we propose an approach to the analysis of this estimator based on the Lyapunov's functions method. The best possible rate of convergence of risks to 0, as ε → 0, is proven for β ≤ 2.

We prove that the Wick–Itô–Skorohod integral with respect to a fractional Brownian motion is injective when it is restricted to a class of adapted integrands that satisfy an appropriate integrability condition.

The stochastic wave equation in one spatial dimension driven by a class of fractional noises or, alternately, by a class of smooth noises with arbitrary temporal covariance is studied. In either case, the wave equation is explicitly solved and the upper and lower bounds on both the large and small deviations of several sup norms associated with the solution are given. Finally the energy of a system governed by such an equation is calculated and its expected value is found.

We give a condition for nonperiodic rank-one systems to have non-exponential asymptotic distribution (equal to 1_[1,∞[) of return times along subsequences of cylinders. Applying this result to the staircase transformation, we derive mixing dynamical systems with non-exponential asymptotics. Moreover, we show for two columns rank-one systems unique asymptotic along full sequences of cylinders.

A discrete time model of large gene expression regulatory networks is considered. Transcriptional degrees of regulations and activation thresholds are shuffled randomly that helps to study the highly reproducible dynamical patterns of regulatory processes in a lack of empirical data concerning genetic switches. The multistationarity and multiperiodicity of oscillations exhibited by the system relay upon the feedback circuits. Statistics of their appearance depends upon the relative number of negative regulations between the genes of network, the number of cycles in the maximal graph, and their lengths.

The model defined on the scalable graphs demonstrates the high persistence in oscillations and the high error tolerance.

When a system of ordinary differential equations on ℜ^d have an attractor given as the omega-limit set of a compact set , then provided that D contains necessarily , i.e. the boundary of the attractor is the omega-limit set of the boundary of D. In general only a weaker result, , can be shown to hold for random attractors arising from systems of stochastic ordinary differential equations, where now D is any compact set such that is contained in D with a positive probability. However, if in addition is empty with positive probability, then in fact .

In many physical applications it is important to predict particle trajectories in a turbulent flow using a numerical model of the flow. One of the major reasons for a poor prediction is the limited resolution in space. We discuss the problem of predictability limits in the framework of a stochastic flow modeling inertial particles in turbulence.

Two stochastic flows called the "model" and "true" are considered. The "model" flow is the "true" flow smoothed in space. The principal measure of discrepancy of the flows is the distance between the centers of mass of two initially identical particle clusters driven by the "true" and "model" flows respectively. The predictability limit is defined as a moment when this distance exceeds the space correlation scale of the true flow. The main result is a simple formula relating the predictability limit with the smoothing scale. This relation is well supported by stochastic simulations and could be a helpful tool in planning experiments in numerical models of turbulent flows. Also the difference between particle scattering in the "true" and "model" flows is studied.