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A dynamical system in a Hilbert space with a weakly attractive nonstationary point

Ivo Vrkoč (1993)

Mathematica Bohemica

A differential equation is a Hilbert space with all solutions bounded but with so finite nontrivial invariant measure is constructed. In fact, it is shown that all solutions to this equation converge weakly to the origin, nonetheless, there is no stationary point. Moreover, so solution has a non-empty Ω -set.

A note on γ-radonifying and summing operators

Zdzisław Brzeźniak, Hongwei Long (2015)

Banach Center Publications

In this note, we discuss certain generalizations of γ-radonifying operators and their applications to the regularity for linear stochastic evolution equations on some special Banach spaces. Furthermore, we also consider a more general class of operators, namely the so-called summing operators and discuss the application to the compactness of the heat semi-group between weighted L p -spaces.

An averaging principle for stochastic evolution equations. II.

Bohdan Maslowski, Jan Seidler, Ivo Vrkoč (1991)

Mathematica Bohemica

In the present paper integral continuity theorems for solutions of stochastic evolution equations of parabolic type on unbounded time intervals are established. For this purpose, the asymptotic stability of stochastic partial differential equations is investigated, the results obtained being of independent interest. Stochastic evolution equations are treated as equations in Hilbert spaces within the framework of the semigroup approach.

Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations

Renaud Marty (2005)

ESAIM: Probability and Statistics

We consider a differential equation with a random rapidly varying coefficient. The random coefficient is a gaussian process with slowly decaying correlations and compete with a periodic component. In the asymptotic framework corresponding to the separation of scales present in the problem, we prove that the solution of the differential equation converges in distribution to the solution of a stochastic differential equation driven by a classical brownian motion in some cases, by a fractional brownian...

Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations

Renaud Marty (2010)

ESAIM: Probability and Statistics

We consider a differential equation with a random rapidly varying coefficient. The random coefficient is a Gaussian process with slowly decaying correlations and compete with a periodic component. In the asymptotic framework corresponding to the separation of scales present in the problem, we prove that the solution of the differential equation converges in distribution to the solution of a stochastic differential equation driven by a classical Brownian motion in some cases, by a fractional Brownian motion...

Asymptotic behaviour of stochastic quasi dissipative systems

Giuseppe Da Prato (2002)

ESAIM: Control, Optimisation and Calculus of Variations

We prove uniqueness of the invariant measure and the exponential convergence to equilibrium for a stochastic dissipative system whose drift is perturbed by a bounded function.

Asymptotic behaviour of stochastic quasi dissipative systems

Giuseppe Da Prato (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We prove uniqueness of the invariant measure and the exponential convergence to equilibrium for a stochastic dissipative system whose drift is perturbed by a bounded function.

Asymptotic normality of eigenvalues of random ordinary differential operators

Martin Hála (1991)

Applications of Mathematics

Boundary value problems for ordinary differential equations with random coefficients are dealt with. The coefficients are assumed to be Gaussian vectorial stationary processes multiplied by intensity functions and converging to the white noise process. A theorem on the limit distribution of the random eigenvalues is presented together with applications in mechanics and dynamics.

Asymptotic stability condition for stochastic Markovian systems of differential equations

Efraim Shmerling (2010)

Mathematica Bohemica

Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by d X ( t ) = A ( ξ ( t ) ) X ( t ) d t + H ( ξ ( t ) ) X ( t ) d w ( t ) , where ξ ( t ) is a finite-valued Markov process and w(t) is a standard Wiener process, is considered. It is proved that the existence of a unique positive solution of the system of coupled Lyapunov matrix equations derived in the paper is a necessary asymptotic stability condition.

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