Optimal random search, fractional dynamics and fractional calculus

What is the most efficient search strategy for the random located target sites subject to the physical and biological constraints? Previous results suggested the L\'evy flight is the best option to characterize this optimal problem, however, which ignores the understanding and learning abilities of the searcher agents. In the paper we propose the Continuous Time Random Walk (CTRW) optimal search framework and find the optimum for both of search length's and waiting time's distributions. Based on fractional calculus technique, we further derive its master equation to show the mechanism of such complex fractional dynamics. Numerous simulations are provided to illustrate the non-destructive and destructive cases.Comment: 12 pages, 7 figure

Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise

Little seems to be known about the invariant manifolds for stochastic partial differential equations (SPDEs) driven by nonlinear multiplicative noise. Here we contribute to this aspect and analyze the Lu-Schmalfu{\ss} conjecture [Garrido-Atienza, et al., J. Differential Equations, 248(7):1637--1667, 2010] on the existence of stable manifolds for a class of parabolic SPDEs driven by nonlinear mutiplicative fractional noise. We emphasize that stable manifolds for SPDEs are infinite-dimensional objects, and the classical Lyapunov-Perron method cannot be applied, since the Lyapunov-Perron operator does not give any information about the backward orbit. However, by means of interpolation theory, we construct a suitable function space in which the discretized Lyapunov-Perron-type operator has a unique fixed point. Based on this we further prove the existence and smoothness of local stable manifolds for such SPDEs.Comment: To appear in Journal of Functional Analysi

Lyapunov Techniques for Stochastic Differential Equations Driven by Fractional Brownian Motion

Little seems to be known about evaluating the stochastic stability of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) via stochastic Lyapunov technique. The objective of this paper is to work with stochastic stability criterions for such systems. By defining a new derivative operator and constructing some suitable stochastic Lyapunov function, we establish some sufficient conditions for two types of stability, that is, stability in probability and moment exponential stability of a class of nonlinear SDEs driven by fBm. We will also give an example to illustrate our theory. Specifically, the obtained results open a possible way to stochastic stabilization and destabilization problem associated with nonlinear SDEs driven by fBm

Smooth center-stable/unstable manifolds and foliations of stochastic evolution equations with non-dense domain

The current paper is devoted to the asymptotic behavior of a class of stochastic PDE. More precisely, with the help of the theory of integrated semigroups and a crucial estimate of the random Stieltjes convolution, we study the existence and smoothness of center-unstable invariant manifolds and center-stable foliations for a class of stochastic PDE with non-dense domain through the Lyapunov-Perron method. Finally, we give two examples about a stochastic age-structured model and a stochastic parabolic equation to illustrate our results.Comment: 46 page

Existence and Uniqueness of Solution for a Class of Stochastic Differential Equations

A class of stochastic differential equations given by dx(t)=f(x(t))dt+g(x(t))dW(t),  x(t0)=x0,  t0≤t≤T<+∞, are investigated. Upon making some suitable assumptions, the existence and uniqueness of solution for the equations are obtained. Moreover, the existence and uniqueness of solution for stochastic Lorenz system, which is illustrated by example, are in good agreement with the theoretical analysis

Stationary Wong-Zakai Approximation of Fractional Brownian Motion and Stochastic Differential Equations with Noise Perturbations

In this article, we introduce a Wong-Zakai type stationary approximation to the fractional Brownian motions and provide a sharp rate of convergence in L-p (Omega). Our stationary approximation is suitable for all values of H is an element of (0, 1). As an application, we consider stochastic differential equations driven by a fractional Brownian motion with H &gt; 1 / 2. We provide sharp rate of convergence in a certain fractional-type Sobolev space of the approximation, which in turn provides rate of convergence for the solution of the approximated equation. This generalises some existing results in the literature concerning approximation of the noise and the convergence of corresponding solutions

Well-posedness of the time-space fractional stochastic Navier-Stokes equations driven by fractional Brownian motion

The current paper is devoted to the time-space fractional Navier-Stokes equations driven by fractional Brownian motion. The spatial-temporal regularity of the nonlocal stochastic convolution is firstly established, and then the existence and uniqueness of mild solution are obtained by Banach Fixed Point theorem and Mittag-Leffler families operators

Variational Solutions and Random Dynamical Systems to SPDEs Perturbed by Fractional Gaussian Noise

This paper deals with the following type of stochastic partial differential equations (SPDEs) perturbed by an infinite dimensional fractional Brownian motion with a suitable volatility coefficient Φ: dX(t)=A(X(t))dt+Φ(t)dBH(t), where A is a nonlinear operator satisfying some monotonicity conditions. Using the variational approach, we prove the existence and uniqueness of variational solutions to such system. Moreover, we prove that this variational solution generates a random dynamical system. The main results are applied to a general type of nonlinear SPDEs and the stochastic generalized p-Laplacian equation