Usually the time evolution is discussed from the present to the future or from the past, precisely, from some fixed initial time in the past, to the present or to the future. But we sometimes consider the time evolution from the remote past to the remote future as in the theory of stationary stochastic processes. In the present paper we consider time evolutions governed by noise driven automorp...

The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking values in a Hilbert space H, is the sum of a local martingale and a suitable orthogonal process. The new concept is shown to be useful in several contexts and d...

Semilinear stochastic evolution equations with multiplicative L'evy noise are considered‎. ‎The drift term is assumed to be monotone nonlinear and with linear growth‎. ‎Unlike other similar works‎, ‎we do not impose coercivity conditions on coefficients‎. ‎We establish the continuous dependence of the mild solution with respect to initial conditions and also on coefficients. ‎As corollaries of ...

Numerical studies of the May–Leonard model for cyclically competing species exhibit spontaneous spatial structures in the form of spirals. It is desirable to obtain a simple coarse-grained evolution equation describing spatio-temporal pattern formation in such spatially extended stochastic population dynamics models. Extending earlier work on the corresponding deterministic system, we derive th...

A model of genome evolution is proposed. Based on three assumptions the evolutionary theory of a genome is formulated. The general law on the direction of genome evolution is given. Both the deterministic classical equation and the stochastic quantum equation are proposed. It is proved that the classical equation can be put in a form of the least action principle and the latter can be used for ...

The dynamics of the two{dimensional turbulence is studied by a stochastic simulation method. The latter is based on a representation of the random vorticity eld and stream function by a multivariate stochastic process de-ned by a discrete master equation. It is demonstrated that in the continuum limit the complete hierarchy of coupled moment equations for the statistical formulation of the 2D N...

Quantum systems of interest are typically coupled to several quantum channels (more generally environments). In this paper, we develop an exact stochastic Schrödinger equation for an open quantum system coupled to a hybrid environment containing both bosonic and fermionic particles. Such a stochastic differential equation may be obtained directly from a microscopic model through employing a cla...

We consider a nonlinear controlled stochastic evolution equation in a Hilbert space, with a Wiener process affecting the control, assuming Lipschitz conditions on the coefficients. We take a cost functional quadratic in the control term, but otherwise with general coefficients that may even take infinite values. Under a mild finiteness condition, and after appropriate formulation, we prove exis...

We develop the general formalism for the study of neutrino propagation in presence of stochastic media. This formalism allows the systematic derivation of evolution equations for averaged quantities as survival probabilities and higher order distribution moments. The formalism applies equally to any finite dimensional Schroedinger equation in presence of a stochastic external field. New integro...