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S-asymptotically ω-periodic solutions in the p-th mean for a Stochastic Evolution Equation driven by Q-Brownian motion
International audienceIn this paper, we study the existence (uniqueness) and asymptotic stability of the p-th mean S-asymptotically ω-periodic solutions for some non-autonomous Stochastic Evolution Equations driven by a Q-Brownian motion. This is done using the Banach fixed point Theorem and a Gron-wall inequality
One and two-phase cell cycle models
In this review paper we present deterministic and stochastic one and two-phase models of the cell cycle. The deterministic models are given by partial differential equations of the first order with time delay and space variable retardation. The stochastic models are given by stochastic iterations or by piecewise deterministic Markov processes. We study asymptotic stability and sweeping of stochastic semigroups which describe the evolution of densities of these processes. We also present some results concerning chaotic behaviour of models and relations between different types of models
Stability of stochastic differential equations in infinite dimensions
In engineering, physics and economics, many dynamical systems involving with stochastic components and random noise are often modeled by stochastic models. The stochastic effects of these models are often used to describe the uncertainty about the operating systems. Motivated by the development of analysis and theory of stochastic processes, as well as the studies of natural sciences, the theory of stochastic differential equations in infinite dimensional spaces evolves gradually into a branch of modern analysis. Many qualitative properties of such systems have been studied in the past few decades, among which, investigation of stability of such systems is often regarded as the first characteristic of the dynamical systems or models. In general, this thesis is mainly concerned with the studies of the stability property of stochastic differential equations in Hilbert spaces. Chapter 1 is an introduction to a brief history of stochastic differential equations in infinite dimensions, together with an overview of the studies. Chapter 2 is a presentation of preliminaries to some basic stochastic analysis. In Chapter 3, we study the stability in distribution of mild solutions to stochastic delay differential equations with Poisson jumps. Firstly, we use approximation of strong solutions to pass on the stability of strong solutions to the mild ones. Then, by constructing a suitable metric between the transition probability functions of mild solutions, we obtain the desired stability result under some suitable conditions. In Chapter 4, we investigate the stochastic partial delay differential equations with Markovian switching and Poisson jumps. By estimating the coefficients of energy equality, both the exponential stability and almost sure exponential stability of energy solutions to the equations are obtained. In Chapter 5, we study the relationship among strong, weak and mild solutions to the stochastic functional differential equations of neutral type. Finally, in Chapter 6, we study the asymptotic stability of two types of equations, impulsive stochastic delay differential equations with Poisson jumps and stochastic evolution equations with Poisson jumps. By employing the fixed point theorem, we derive the desired stability results under some criteria
Trait evolution in two-sex populations
We present an individual-based model of phenotypic trait evolution in two-sex
populations, which includes semi-random mating of individuals of the opposite
sex, natural death and intra-specific competition. By passing the number of
individuals to infinity, we derive the macroscopic system of nonlinear
differential equations describing the evolution of trait distributions in male
and female subpopulations. We study solutions, give criteria for persistence or
extinction, and state theorem on asymptotic stability, which we apply later to
particular examples of trait inheritance
Stochastic collective dynamics of charged--particle beams in the stability regime
We introduce a description of the collective transverse dynamics of charged
(proton) beams in the stability regime by suitable classical stochastic
fluctuations. In this scheme, the collective beam dynamics is described by
time--reversal invariant diffusion processes deduced by stochastic variational
principles (Nelson processes). By general arguments, we show that the diffusion
coefficient, expressed in units of length, is given by ,
where is the number of particles in the beam and the Compton
wavelength of a single constituent. This diffusion coefficient represents an
effective unit of beam emittance. The hydrodynamic equations of the stochastic
dynamics can be easily recast in the form of a Schr\"odinger equation, with the
unit of emittance replacing the Planck action constant. This fact provides a
natural connection to the so--called ``quantum--like approaches'' to beam
dynamics. The transition probabilities associated to Nelson processes can be
exploited to model evolutions suitable to control the transverse beam dynamics.
In particular we show how to control, in the quadrupole approximation to the
beam--field interaction, both the focusing and the transverse oscillations of
the beam, either together or independently.Comment: 15 pages, 9 figure
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