Lyapunov exponents and stationary solutions for affine stochastic delay equations

A certain class of affine delay equations is considered. Two cases for the forcingfunction M are treated: M locally integrable deterministic, and M a random process with stationaryincrements. The Lyapunov spectrum of the homogeneous equation is used to decompose the state spaceinto finite-dimensional and finite-codimensional subspaces. Using a suitable variation of constants representation, formulas for the projection of the trajectories onto the above subspaces are obtained. If the homogeneous equation is hyperbolic and M has stationary increments, existence and uniqueness of a stationary solution for the affine stochastic delay equation is proved. The existence of Lyapunov exponents for the affine equation and their dependence on initial conditions is als studied.     

The Smoothness of Laws of Random Flags and Oseledets Spaces of Linear Stochastic Differential Equations

The Oseledets spaces of a random dynamical system generated by a linear stochastic differential equation are obtained as intersections of the corresponding nested invariant spaces of a forward and a backward flag, described as the stationary states of flows on corresponding flag manifolds. We study smoothness of their laws and conditional laws by applying Malliavin's calculus. If the Lie algebras induced by the actions of the matrices generating the system on the manifolds span the tangent spaces at any point, laws and conditional laws are seen to be C-smooth. As an application we find that the semimartingale property is well preserved if the Wiener filtration is enlarged by the information present in the flag or Oseledets spaces.     

The substitution theorem for semilinear stochastic partial differential equations

In this article we establish a substitution theorem for semilinear stochastic evolution equations (see's) depending on the initial condition as an infinite-dimensional parameter. Due to the infinite-dimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techniques together with new estimates on the underlying stochastic semiflow. Applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations (spde's) with anticipating initial conditions and non-ergodic stationary solutions. In particular, our result gives a new existence theorem for solutions of semilinear Stratonovich spde's with anticipating initial conditions.     

Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations

In this paper, based on a multidimensional Riemann theta function, a lucid and straightforward generalization of the Hirota-Riemann method is presented to explicitly construct multiperiodic Riemann theta functions periodic wave solutions for nonlinear equations such as the Caudrey-Dodd-Gibbon-Sawada-Kotera equation and (2+1)-dimensional breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional so that they have two independent spatial periods in two independent horizontal directions. A limiting procedure is presented to analyze in detail, asymptotic behavior of the multiperiodic waves and the relations between the periodic wave solutions and soliton solutions are rigorously established. This generalized Hirota-Riemann method can also be demonstrated on a class variety of nonlinear difference equations such as Toeplitz lattice equation.