Class Of Stochastic Delay Equations Research Articles

Asymptotic behaviours of a stochastic delay equation driven by an fBm in Hilbert space

ABSTRACTIn this paper, we consider a class of stochastic delay equations in Hilbert spaces driven by fractional Brownian motion with Hurst parameter . We obtain a sufficient condition for controllability of the systems and prove that their mild solutions are exponentially stable in pth moment with by using adequately the characteristic of fractional Brownian motion and the fractional power of a linear operator A with . In particular, when the mild solutions are exponentially stable in mean square.

Vector-valued stochastic delay equations—A weak solution and its Markovian representation

A class of stochastic delay equations in a type 2 umd Banach space E driven by the cylindrical Wiener process is studied. We investigate two concepts of solutions: weak and generalised strong, and give conditions under which they are equivalent. We present an evolution equation approach in a Banach space Ep:=E×Lp(−1,0;E) proving that the solutions can be reformulated as Ep-valued Markov processes. Based on the Markovian representation we prove the existence and continuity of the solutions. The results are applied to stochastic reaction–diffusion equations with delays.

Unstable invariant distributions for a class of stochastic delay equations

In our article [8] we examined asymptotic mean square stability for linear retarded f.d.e.'s which are perturbed by white noise. It is shown in [8] and [10] that if the deterministic linear retarded f.d.e. is asymptotically stable, then so is the perturbed stochastic f.d.e.