Generalized retarded integral inequalities

We prove some new retarded integral inequalities. The results generalize those in [J. Math. Anal. Appl. 301 (2005), no. 2, 265--275].Comment: Changes suggested by the referee don

Stochastic nonlinear Schrodinger equations driven by a fractional noise - Well posedness, large deviations and support

We consider stochastic nonlinear Schrodinger equations driven by an additive noise. The noise is fractional in time with Hurst parameter H in (0,1). It is also colored in space and the space correlation operator is assumed to be nuclear. We study the local well-posedness of the equation. Under adequate assumptions on the initial data, the space correlations of the noise and for some saturated nonlinearities, we prove a sample path large deviations principle and a support result. These results are stated in a space of exploding paths which are Holder continuous in time until blow-up. We treat the case of Kerr nonlinearities when H > 1/2

On the almost sure running maxima of solutions of affine stochastic functional differential equations

This paper studies the large fluctuations of solutions of scalar and finite-dimensional affine stochastic functional differential equations with finite memory as well as related nonlinear equations. We find conditions under which the exact almost sure growth rate of the running maximum of each component of the system can be determined, both for affine and nonlinear equations. The proofs exploit the fact that an exponentially decaying fundamental solution of the underlying deterministic equation is sufficient to ensure that the solution of the affine equation converges to a stationary Gaussian process

Some New Delay Integral Inequalities in Two Independent Variables on Time Scales

Some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales are established, which can be used as a handy tool in the research of boundedness of solutions of delay dynamic equations on time scales. Some of the established results are 2D extensions of several known results in the literature, while some results unify existing continuous and discrete analysis