On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion

The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen (2005) and subsequently used by others to study the L²$L^2$ and L³$L^3$ moduli of continuity of Brownian local time. A version of the DSLT for fractional Brownian motion (fBm) was introduced in Yan et al. (2008); however, the definition given there presents difficulties, since it is motivated by an incorrect application of the fractional Itô formula. To rectify this, we introduce a modified DSLT for fBm and prove existence using an explicit Wiener chaos expansion. We will then argue that our modification is the natural version of the DSLT by rigorously proving the corresponding Tanaka formula. This formula corrects a formal identity given in both Rosen (2005) and Yan et al. (2008). In the course of this endeavor we prove a Fubini theorem for integrals with respect to fBm. The Fubini theorem may be of independent interest, as it generalizes (to Hida distributions) similar results previously seen in the literature. As a further byproduct of our investigation, we also provide a small correction to an important technical second-moment bound for fBm which has appeared in the literature many times.     

Lower classes for fractional Brownian motion

We characterize the lower classes of fractional Brownian motion by an integral test.Work partially supported by an NSF grant. Equipe d'Analyse, Tour 46, U.A. at C.N.R.S. n^o 754, Université Paris VI, 4 place Jussieu, 75230 Paris Cedex 05, and Department of Mathematics, 231 West 18th Avenue, Columbus, Ohio 43210.     

The local time of iterated Brownian motion

We define and study the local time process {L ^*(x,t);x¹,t0} of the iterated Brownian motion (IBM) {H(t):=W ₁(|W ₂ (t)|); t0}, whereW ₁(·) andW ₂(·) are independent Wiener processes.Research supported by Hungarian National Foundation for Scientific Research, Grant No. T 016384.Research supported by an NSERC Canada Grant at Carleton University, Ottawa.Research supported by a PSC CUNY Grant, No. 6-66364.     

Self-intersection local times and collision local times of bifractional Brownian motions

In this paper, we consider the local time and the self-intersection local time for a bifractional Brownian motion, and the collision local time for two independent bifractional Brownian motions. We mainly prove the existence and smoothness of the self-intersection local time and the collision local time, through the strong local nondeterminism of bifractional Brownian motion, L2 convergence and Chaos expansion.     

Stochastic delay fractional evolution equations driven by fractional Brownian motion

In this paper, we consider a class of stochastic delay fractional evolution equations driven by fractional Brownian motion in a Hilbert space. Sufficient conditions for the existence and uniqueness of mild solutions are obtained. An application to the stochastic fractional heat equation is presented to illustrate the theory. Copyright © 2014 John Wiley & Sons, Ltd.     

Packing measure of the sample paths of fractional Brownian motion

Let be a fractional Brownian motion of index in If , then there exists a positive finite constant such that with probability 1,

where and - is the -packing measure of .

     

Uniqueness and explosion time of solutions of stochastic differential equations driven by fractional Brownian motion

In this paper, we first study the existence and uniqueness of solutions to the stochastic differential equations driven by fractional Brownian motion with non-Lipschitz coefficients. Then we investigate the explosion time in stochastic differential equations driven by fractional Browmian motion with respect to Hurst parameter more than half with small diffusion.     

Smoothness of local times and self-intersection local times of Gaussian random fields

This paper is concerned with the smoothness (in the sense of Meyer- Watanabe) of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.     

Hilbert-valued self-intersection local times for planar Brownian motion

Dynkin's construction for self-intersection local time of a planar Wiener process is extended to Hilbert-valued weights. In Dynkin's construction, the weight is bounded and measurable. Since the weight function describes the properties of the medium in which the Brownian motion moves, relative to the external medium's properties, the weight function can be random and unbounded. In this article, we discuss the possibility to consider Hilbert-space-valued weights. It appears that the existence of Hilbert-valued Dynkin-renormalized self-intersection local time is equivalent to the embedding of the values of Hilbert-valued weight into a Hilbert–Schmidt brick. Using Dorogovtsev's sufficient condition for the embedding of compact sets into a Hilbert–Schmidt brick in terms of an isonormal process, we prove the existence of Hilbert-valued Dynkin-renormalized self-intersection local time. Also using Dynkin's construction we construct the self-intersection local time for the deterministic image of the planar Wiener process.     

Synchronization for fractional FitzHugh–Nagumo equations with fractional Brownian motion

This paper is devoted to dynamics of the Caputo-type fractional FitzHugh–Nagumo equations (FHN) driven by fractional Brownian motion (fBm). The existence and uniqueness of mild solution for of the Caputo-type fractional FHN are established, and the exponential synchronization and finite-time synchronization for the stochastic FHN are provided. Finally, the numerical simulation of the synchronization for time-fractional FHN perturbed by fBm is provided; the effects of the order of time fractional derivative and Hurst parameter $H$ on synchronization are also revealed.     

Large deviations for local time fractional Brownian motion and applications

Let be a fractional Brownian motion of Hurst index H∈(0,1) with values in R, and let be the local time process at zero of a strictly stable Lévy process of index 1<α?2 independent of WH. The α-stable local time fractional Brownian motion is defined by ZH(t)=WH(Lt). The process ZH is self-similar with self-similarity index and is related to the scaling limit of a continuous time random walk with heavy-tailed waiting times between jumps [P. Becker-Kern, M.M. Meerschaert, H.P. Scheffler, Limit theorems for coupled continuous time random walks, Ann. Probab. 32 (2004) 730-756; M.M. Meerschaert, H.P. Scheffler, Limit theorems for continuous time random walks with infinite mean waiting times, J. Appl. Probab. 41 (2004) 623-638]. However, ZH does not have stationary increments and is non-Gaussian. In this paper we establish large deviation results for the process ZH. As applications we derive upper bounds for the uniform modulus of continuity and the laws of the iterated logarithm for ZH.     

Smoothness for the collision local times of bifractional Brownian motions

Let BHi,Ki={BtHi,Ki,t≥0},i=1,2 be two independent bifractional Brownian motions with respective indices H i ∈(0,1) and K i ∈(0,1].One of the main motivations of this paper is to investigate the smoothness of the collision local time,introduced by Jiang and Wang in 2009,lT = integral(δ(BsH1,K1-BsH2,K2)ds) from n=0 to T,T > 0,where δ denotes the Dirac delta function.By an elementary method,we show that T is smooth in the sense of the Meyer-Watanabe if and only if min{H1K1,H2K2} <1/3.     

Collision local times of two independent fractional Brownian motions

In this paper, the collision local times for two independent fractional Brownian motions are considered as generalized white noise functionals. Moreover, the collision local times exist in L ² under mild conditions and chaos expansions are also given.     

Stochastic integration for tempered fractional Brownian motion

Tempered fractional Brownian motion is obtained when the power law kernel in the moving average representation of a fractional Brownian motion is multiplied by an exponential tempering factor. This paper develops the theory of stochastic integrals for tempered fractional Brownian motion. Along the way, we develop some basic results on tempered fractional calculus.     

Integral representation of renormalized self-intersection local times

In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1). As a consequence, we derive the existence of some exponential moments for this random variable.     

Fractional Brownian Density Process and Its Self-Intersection Local Time of Order k

The fractional Brownian density process is a continuous centered Gaussian ( ^d )-valued process which arises as a high-density fluctuation limit of a Poisson system of independent d-dimensional fractional Brownian motions with Hurst parameter H. ( ( ^d ) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure of the (initial) Poisson random measure on ^d is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order k 2 if and only if Hd < k/(k – 1). The latter is also the necessary and sufficient condition for existence of multiple points of order k for d-dimensional fractional Brownian motion, as proved by Talagrand¹². This result extends to a non-Markovian case the relationship known for (Markovian) symmetric -stable Lévy processes and their corresponding density processes. New methods are used in order to overcome the lack of Markov property. Other properties of the fractional Brownian density process are also given, in particular the non-semimartingale property in the case H 1/2, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes that we also prove.     

On $L_p$-solution of fractional heat equation driven by fractional Brownian motion

In this paper, we study the fractional stochastic heat equation driven by fractional Brownian motions of the form $$ du(t,x)=\left(-(-\Delta)^{\alpha/2}u(t,x)+f(t,x)\right)dt +\sum\limits^{\infty}_{k=1} g^k(t,x)\delta\beta^k_t $$ with $u(0,x)=u_0$, $t\in[0,T]$ and $x\in\mathbb{R}^d$, where $\beta^k=\{\beta^k_t,t\in[0,T]\},k\geq1$ is a sequence of i.i.d. fractional Brownian motions with the same Hurst index $H>1/2$ and the integral with respect to fractional Brownian motion is Skorohod integral. By adopting the framework given by Krylov, we prove the existence and uniqueness of $L_p$-solution to such equation.