The reliability of processes with moving elastic and isotropic material containing initial cracks is considered in terms of fracture. The material is modelled as a moving plate which is simply supported from two of its sides and subjected to homogeneous tension acting in the travelling direction. For tension, two models are studied: i) tension is constant with respect to time, and ii) tension v...

An elementary approach is used to derive a Bayes-type formula, extending the Kallianpur–Striebel formula for the nonlinear filters associated with the Gaussian noise processes. In the particular cases of certain Gaussian processes, recent results of Kunita and of Le Breton on fractional Brownian motion are derived. We also use the classical approximation of the Brownian motion by the Ornstein–U...

The generalized Ornstein-Uhlenbeck process Vt = e −ξt ( V0 + ∫ t 0 edηs ) , t ≥ 0, driven by a bivariate Lévy process (ξt, ηt)t≥0 with starting random variable V0 independent of (ξ, η) fulfills the stochastic differential equation dVt = Vt−dUt + dLt for another bivariate Lévy process (Ut, Lt)t≥0, which is determined completely by (ξ, η). In particular it holds ξt = − log(E(U)t), t ≥ 0, where E(...

Abstract We study the long time behavior of an Ornstein–Uhlenbeck process under the influence of a periodic drift. We prove that, under the standard diffusive rescaling, the law of the particle position converges weakly to the law of a Brownian motion whose covariance can be expressed in terms of the solution of a Poisson equation. We also derive upper bounds on the convergence rate in several ...

for simultaneously diagonalizable matrices A,B ∈ C . The unbounded drift term is defined by a skew-symmetric matrix S ∈ R. Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain D(Ap) of the generator Ap belonging to the OrnsteinUhlenbeck semigroup coincides with the dom...

Denote by γ the Gauss measure on R and by L the Ornstein– Uhlenbeck operator. In this paper we introduce a Hardy space h(γ) of Goldberg type and show that for each u in R\{0} and r > 0 the operator (rI+L) is unbounded from h(γ) to L(γ). This result is in sharp contrast both with the fact that (rI+L) is bounded from H(γ) to L(γ), where H(γ) denotes the Hardy type space introduced in [MM], and wi...