arXiv subjects

Kening Lu

Publications and source records attributed to Kening Lu.

26 records · Page 2Linked to original sources

arXiv2015

Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in (1/3,1/2]$

We consider the stochastic evolution equation $ du=Audt+G(u)dω,\quad u(0)=u_0 $ in a separable Hilbert--space $V$. Here $G$ is supposed to be three times Fréchet--differentiable and $ω$ is a trace class fractional Brownian--motion with Hurst parameter $H\in (1/3,1/2]$. We prove the existence of a global solution where exceptional sets are independent of the initial state $u_0\in V$. In addition, we show that the above equation generates a random dynamical system.

math.DS↗

arXiv2014

Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$

In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a Hölder continuous function with Hölder exponent in $(1/3,1/2)$. Our stochastic integral is a generalization of the well-known Young integral. To be more precise, the integral is defined by using a fractional integration by parts formula and it involves a tensor for which we need to formulate a new equation. From this it turns out that we have to solve a system consisting in a path and an area equations. In this paper we prove the existence of a unique local solution of the system of equations. The results can be applied to stochastic evolution equations with a non-linear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which is particular includes white noise.

math.AP↗

arXiv2014

Lévy-areas of Ornstein-Uhlenbeck processes in Hilbert-spaces

In this paper we investigate the existence and some useful properties of the Lévy areas of Ornstein-Uhlenbeck processes associated to Hilbert-space-valued fractional Brownian-motions with Hurst parameter $H\in (1/3,1/2]$. We prove that this stochastic area has a Hölder-continuous version with sufficiently large Hölder-exponent and that can be approximated by smooth areas. In addition, we prove the stationarity of this area.

math.DS↗

arXiv2013

Pathwise solutions to stochastic partial differential equations driven by fractional Brownian motions with Hurst parameters in $(1/3,1/2]$

Combining fractional calculus and the Rough Path Theory we study the existence and uniqueness of mild solutions to evolutions equations driven by a Hölder continuous function with Hölder exponent in $(1/3,1/2)$. Our stochastic integral is in some sense a generalization of the well-known Young integral and can be defined independently of the initial condition. Similar to the Rough Path Theory we establish a second variable which is given, roughly speaking, by a tensor product. It is then necessary to formulate a second equation for this new variable, and we do in a mild sense. The crucial point in order to get this new equation is to construct a tensor depending on the noise path but also on the semigroup. We then prove the existence of a unique Hölder continuous solution of the system of equations, consisting of the path and the area components, if the nonlinear term and the initial condition are sufficiently smooth. Once the abstract theory is developed, we can present a pathwise nonlinear SPDE driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$.

math.AP↗

arXiv2009

Invariant manifolds for random and stochastic partial differential equations

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable and pseudo-unstable manifolds for a class of \emph{random} partial differential equations and \emph{stochastic} partial differential equations is shown. Unlike the invariant manifold theory for stochastic \emph{ordinary} differential equations, random norms are not used. The result is then applied to a nonlinear stochastic partial differential equation with linear multiplicative noise.

math.DS↗

arXiv2006

Rotation Numbers for Random Dynamical Systems on the Circle

In this paper, we study rotation numbers of random dynamical systems on the circle. We prove the existence of rotation numbers and the continuous dependence of rotation numbers on the systems. As an application, we prove a theorem on analytic conjugacy to a circle rotation.

math.DS↗

arXiv2004

Invariant manifolds for stochastic partial differential equations

Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invariant manifolds for infinite dimensional {\em random} dynamical systems generated by {\em stochastic} partial differential equations. We first introduce a random graph transform and a fixed point theorem for non-autonomous systems. Then we show the existence of generalized fixed points which give the desired invariant manifolds.

math.DS↗

arXiv2004

Smooth stable and unstable manifolds for stochastic partial differential equations

Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron's method. Then, we prove the smoothness of these invariant manifolds.

math.DS↗