H. Bessaih - Academia.edu (original) (raw)

Papers by H. Bessaih

We formulate and prove a new criterion for stability of e-processes. It says that any e-process w... more We formulate and prove a new criterion for stability of e-processes. It says that any e-process which is averagely bounded and concentrating is asymptotically stable. In the second part, we show how this general result applies to some shell models (the Goy and the Sabra model). Indeed, we manage to prove that the processes corresponding to these models satisfy the

In this paper a non autonomous dynamical system is considered, a stochastic one that is obtained ... more In this paper a non autonomous dynamical system is considered, a stochastic one that is obtained from the dissipative Euler equation subject to a stochastic perturbation, an additive noise. Absorbing sets have been defined as sets that depend on time and attract from −∞. A stochastic weak attractor is constructed in phase space with respect to two metrics and is compact in the lower one.

We formulate and prove a new criterion for stability of e-processes. In particular, we prove that... more We formulate and prove a new criterion for stability of e-processes. In particular, we prove that any e-process which is averagely bounded and concentrating is asymptotically stable. This general result is applied to some stochastic shell models driven by an additive noise. For these equations, the e-process property holds even for degenerate noises. As a consequence, we also obtain the uniqueness of the invariant measure when the noise is affecting only finitely many modes.

Mathematical Models and Methods in Applied Sciences, 2004

Electronic Journal of Probability, 2009

A LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient ν conver... more A LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient ν converges to 0 and the noise intensity is multiplied by √ ν, we prove that some shell models of turbulence with a multiplicative stochastic perturbation driven by a H-valued Brownian motion satisfy a LDP in C([0, T ], V ) for the topology of uniform convergence on [0, T ], but where V is endowed with a topology weaker than the natural one. The initial condition has to belong to V and the proof is based on the weak convergence of a family of stochastic control equations. The rate function is described in terms of the solution to the inviscid equation.

Discrete and Continuous Dynamical Systems, 2014

In this paper we study the long-time dynamics of mild solutions to retarded stochastic evolution ... more In this paper we study the long-time dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. As a preparation for this purpose we have to show the existence and uniqueness of a cocycle solution of such an equation. We do not assume that the noise is given in additive form or that it is a very simple multiplicative noise. However, we need some smoothing property for the coefficient in front of the noise. The main idea of this paper consists of expressing the stochastic integral in terms of non-stochastic integrals and the noisy path by using an integration by parts. This latter term causes that in a first moment only a local mild solution can be obtained, since in order to apply the Banach fixed point theorem it is crucial to have the Hölder norm of the noisy path to be sufficiently small. Later, by using appropriate stopping times, we shall derive the existence and uniqueness of a global mild solution. Furthermore, the asymptotic behavior is investigated by using the Random Dynamical Systems theory. In particular, we shall show that the global mild solution generates a random dynamical system that, under an appropriate smallness condition for the time lag, have associated a random attractor.

We formulate and prove a new criterion for stability of e-processes. It says that any e-process w... more We formulate and prove a new criterion for stability of e-processes. It says that any e-process which is averagely bounded and concentrating is asymptotically stable. In the second part, we show how this general result applies to some shell models (the Goy and the Sabra model). Indeed, we manage to prove that the processes corresponding to these models satisfy the

In this paper a non autonomous dynamical system is considered, a stochastic one that is obtained ... more In this paper a non autonomous dynamical system is considered, a stochastic one that is obtained from the dissipative Euler equation subject to a stochastic perturbation, an additive noise. Absorbing sets have been defined as sets that depend on time and attract from −∞. A stochastic weak attractor is constructed in phase space with respect to two metrics and is compact in the lower one.

We formulate and prove a new criterion for stability of e-processes. In particular, we prove that... more We formulate and prove a new criterion for stability of e-processes. In particular, we prove that any e-process which is averagely bounded and concentrating is asymptotically stable. This general result is applied to some stochastic shell models driven by an additive noise. For these equations, the e-process property holds even for degenerate noises. As a consequence, we also obtain the uniqueness of the invariant measure when the noise is affecting only finitely many modes.

Mathematical Models and Methods in Applied Sciences, 2004

Electronic Journal of Probability, 2009

A LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient ν conver... more A LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient ν converges to 0 and the noise intensity is multiplied by √ ν, we prove that some shell models of turbulence with a multiplicative stochastic perturbation driven by a H-valued Brownian motion satisfy a LDP in C([0, T ], V ) for the topology of uniform convergence on [0, T ], but where V is endowed with a topology weaker than the natural one. The initial condition has to belong to V and the proof is based on the weak convergence of a family of stochastic control equations. The rate function is described in terms of the solution to the inviscid equation.

Discrete and Continuous Dynamical Systems, 2014

In this paper we study the long-time dynamics of mild solutions to retarded stochastic evolution ... more In this paper we study the long-time dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. As a preparation for this purpose we have to show the existence and uniqueness of a cocycle solution of such an equation. We do not assume that the noise is given in additive form or that it is a very simple multiplicative noise. However, we need some smoothing property for the coefficient in front of the noise. The main idea of this paper consists of expressing the stochastic integral in terms of non-stochastic integrals and the noisy path by using an integration by parts. This latter term causes that in a first moment only a local mild solution can be obtained, since in order to apply the Banach fixed point theorem it is crucial to have the Hölder norm of the noisy path to be sufficiently small. Later, by using appropriate stopping times, we shall derive the existence and uniqueness of a global mild solution. Furthermore, the asymptotic behavior is investigated by using the Random Dynamical Systems theory. In particular, we shall show that the global mild solution generates a random dynamical system that, under an appropriate smallness condition for the time lag, have associated a random attractor.