A dynamical theory for singular stochastic delay differential equations II: Nonlinear equations and invariant manifolds

A dynamical theory for singular stochastic delay differential equations I: Linear equations and a Multiplicative Ergodic Theorem on fields of Banach spaces

We show that singular stochastic delay differential equations (SDDEs) induce cocycle maps on a field of Banach spaces. A general Multiplicative Ergodic Theorem on fields of Banach spaces is proved and applied to linear SDDEs. In Part II of this article, we use our results to prove a stable manifold theorem for non-linear singular SDDEs.

抛物型随机时滞偏微分方程的不变流形

Invariant manifolds for stochastic delayed partial differential equations of parabolic type

本文针对抛物型随机时滞偏微分方程，通过共轭变换、Lyapunov-Perron方法及谱间隙条件分析，证明了其稳定与不稳定不变流形的存在性与光滑性，并将结果应用于随机单物种种群模型。

Well-Posedness and Stability for a Class of Stochastic Delay Differential Equations with Singular Drift

In this paper we prove well-posedness and stabibility of a class of stochastic delay differential equations with singular drift. Moreover, we show local well-posedness under localized assumptions.

Stochastic delay differential equations with jumps in differentiable manifolds

In this article we propose a model for stochastic delay differential equation with jumps (SDDEJ) in a differentiable manifold $M$ endowed with a connection $\nabla$. In our model, the continuous part is driven by vector fields with a fixed delay and the jumps are assumed to come from a distinct source of (càdlàg) noise, without delay. The jumps occur along adopted differentiable curves with some dynamical relevance (with fictitious time) which allow to take parallel transport along them. Using a geometrical approach, in the last section, we show that the horizontal lift of the solution of an S…

On the Strong Feller Property of Stochastic Delay Differential Equations with Singular Drift

In this paper, we prove the strong Feller property for stochastic delay (or functional) differential equations with singular drift. We extend an approach of Maslowski and Seidler to derive the strong Feller property of those equations. The argumentation is based on the well-posedness and the strong Feller property of the equations' drift-free version. To this aim, we investigate a certain convergence of random variables in topological spaces in order to deal with discontinuous drift coefficients.

Exponential growth rate for a singular linear stochastic delay differential equation

We establish the existence of a deterministic exponential growth rate for the norm (on an appropriate function space) of the solution of the linear scalar stochastic delay equation dX(t) = X(t-1) dW(t) which does not depend on the initial condition as long as it is not identically zero. Due to the singular nature of the equation this property does not follow from available results on stochastic delay differential equations. The key technique is to establish existence and uniqueness of an invariant measure of the projection of the solution onto the unit sphere in the chosen function space via a…