Modeling and analysis of random and stochastic input flows in the chemostat model

In this paper we study a new way to model noisy input flows in the chemostat model, based on the Ornstein-Uhlenbeck process. We introduce a parameter β as drift in the Langevin equation, that allows to bridge a gap between a pure Wiener process, which is a common way to model random disturbances, and no noise at all. The value of the parameter β is related to the amplitude of the deviations observed on the realizations. We show that this modeling approach is well suited to represent noise on an input variable that has to take non-negative values for almost any time.European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Ministerio de Economía y Competitividad (MINECO). EspañaJunta de Andalucí

On fractional Brownian motions and random dynamical systems

In this paper we consider a class of nonlinear stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion with the Hurst parameter bigger than 1/2. We show that these SPDEs generate random dynamical systems

Stochastic stabilization of differential systems with general decay rate

Some sufficient conditions concerning stability of solutions of stochastic differential evolution equations with general decay rate are first proved. Then, these results are interpreted as suitable stabilization ones for deterministic and stochastic systems. Also, they permit us to construct appropriate linear stabilizers in some particular situations

Existence and uniqueness of solutions for delay stochastic evolution equations

Some results on the existence and uniqueness of solutions for stochastic evolution equations containing some hereditary characteristics are proved. In fact, our theory is developed from a variational point of view and in a general functional setting which permit us to deal with several kinds of delay terms in a unified formulation

Stability of Nonlinear Functional Stochastic Evolution Equations of Second Order in Time

Sufficient conditions for exponential mean square stability of solutions to delayed stochastic partial differential equations of second order in time are established. As a consequence of these results, the pathwise exponential stability of the system is also deduced. The stability results derived can be applied also to partial differential equations without hereditary characteristics. The results are illustrated with various examples

Asymptotic stability of nonlinear stochastic evolution equations

Some results on the pathwise asymptotic stability of solutions to stochastic partial differential equations are proved. Special attention is paid in proving sufficient conditions ensuring almost sure asymptotic stability with a nonexponential decay rate. The situation containing some hereditary characteristics is also treated. The results are illustred with several examples

The exponential behaviour of nonlinear stochastic functional equations of second order in time

Sufficient conditions for exponential mean square stability of solutions to delayed stochastic partial differential equations of second order in time are established. As a consequence of these results, some ones on the pathwise exponential stability of the system are proved. The stability results derived are applied also to partial differential equations without hereditary characteristics. The results are illustrated with several examples

Existence of exponentially attracting stationary solutions for delay evolution equations

We consider the exponential stability of semilinear stochastic evolution equations with delays when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution exponentially stable, for which we use a general random fixed point theorem for general cocycles. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly, by means of the theory of random dynamical systems and their conjugation properties

The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional brownian motion

In this paper we investigate the existence, uniqueness and exponential asymptotic behavior of mild solutions to stochastic delay evolution equations perturbed by a fractional Brownian motion BH Q (t): dX(t) = (AX(t) + f(t;Xt))dt + g(t)dBH Q (t); with Hurst parameter H 2 (1=2; 1). We also consider the existence of weak solutions

Navier-Stokes equations with delays on unbounded domains

Some results on the existence and uniqueness of solutions to Navier–Stokes equations when the domain is unbounded and the external force contains some hereditary characteristics are proved for both the evolutionary and the stationary cases. Exponential stability of the stationary solution is also established in dimension two