$\alpha$-Time Fractional Brownian Motion: PDE Connections and Local Times

TL;DR

This paper explores $ ext{α}$-time fractional Brownian motion, revealing its connections to PDEs, scaling limits of random walks, and analyzing the existence and properties of its local times using strong local nondeterminism.

Contribution

It introduces and studies $ ext{α}$-time fractional Brownian motion, establishing its PDE connections, scaling limits, and detailed local time properties, which are novel contributions.

Findings

Processes have natural PDE connections.

Can arise as scaling limits of random walks.

Established existence and regularity of local times.

Abstract

For $0<\alpha \leq 2$ and $0<H<1$, an $\alpha$-time fractional Brownian motion is an iterated process $Z = \{Z(t)=W(Y(t)), t \ge 0\}$ obtained by taking a fractional Brownian motion $\{W(t), t\in \RR{R} \}$ with Hurst index $0<H<1$ and replacing the time parameter with a strictly $\alpha$-stable L\'evy process $\{Y(t), t\geq 0 \}$ in $\RR{R}$ independent of $\{W(t), t \in \R\}$. It is shown that such processes have natural connections to partial differential equations and, when $Y$ is a stable subordinator, can arise as scaling limit of randomly indexed random walks. The existence, joint continuity and sharp H\"older conditions in the set variable of the local times of a $d$-dimensional $\alpha$-time fractional Brownian motion $X = \{X(t), t \in \R_+$\} defined by $ X(t)=\big(X_{1}(t),..., X_{d}(t) \big), $ where $t\geq 0$ and $X_{1},..., X_{d}$ are independent copies of $Z$, are investigated. Our methods rely on the strong local ' nondeterminism of fractional Brownian motion.

Equations

$${\mathbb{E}}\big{(}W(t)W(s)\big{)}=\frac{1}{2}\big{(}|t|^{2H}+|s|^{2H}-|t-s|^{2H}\big{)}$$

$$Z(t)\equiv W(Y(t)),t\ge 0.$$

$$X(t)=\big{(}X_{1}(t),\cdots,X_{d}(t)\big{)},\ \ \ \ \ \ (t\geq 0),$$

$$\underset{t\to 0}{\lim }\frac{T(t)\phi (x)-\phi (x)}{t}$$

$$T(t)\phi (x)={\int }_{{\mathbb{R}}^d}f(t,x-y)\phi (y)dy$$

$$\begin{array}{rl}\frac{\partial }{\partial t}u(t,x)& =\frac{1}{2}\Delta u(t,x),t>0,x\in {\mathbb{R}}^d,\\ u(0,x)& =\phi (x),x\in {\mathbb{R}}^d.\end{array}$$

$$q(t,x)={\int }_0^{\infty }f(s,x)p_t(s)ds={\int }_0^{\infty }\frac{e^{-|x{|}^2/2s}}{(2\pi s{)}^{d/2}}{p}_t(s)ds,$$

$$u(t,x)={\mathbb{E}}_x[\psi (W(Y(t)))]={\int }_0^{\infty }[T(s)\psi (x)]p_t(s)ds$$

$$\displaystyle\frac{\partial}{\partial t}u(t,x)\$$

$$u(0,x)$$

$${\int }_{{\mathbb{R}}^d}{e}^{i⟨k,x⟩}[-(-\Delta {)}^{\beta /2}\psi (x)]dx=-|k{|}^{\beta }{\int }_{{\mathbb{R}}^d}{e}^{i⟨k,x⟩}\psi (x)dx,$$

$$\displaystyle\frac{\partial}{\partial t}u(t,x)\$$

$$u(0,x)$$

$$\frac{{\partial }^{\frac{1}{2}}}{\partial t^{\frac{1}{2}}}u(t,x)$$

$$u(0,x)$$

$$\frac{\partial^{\frac{1}{2}}}{\partial t^{\frac{1}{2}}}u(t,x)=\frac{1}{\sqrt{\pi}}\int_{0}^{t}\bigg{[}\frac{\partial u(s,x)}{\partial s}\bigg{]}\frac{ds}{(t-s)^{\frac{1}{2}}}=\frac{1}{\Gamma(1-\frac{1}{2})}\int_{0}^{t}\bigg{[}\frac{\partial u(s,x)}{\partial s}\bigg{]}\frac{ds}{(t-s)^{\frac{1}{2}}},$$

$$\displaystyle\frac{\partial^{2}}{\partial t^{2}}u(t,x)\$$

$$u(0,x)$$

$$q(t,x)=2{\int }_0^{\infty }\frac{e^{-\frac{x^2}{2s^{2H}}}}{\sqrt{2\pi s^{2H}}}\frac{e^{-\frac{s^2}{2t}}}{\sqrt{2\pi t}}ds$$

$$t\frac{\partial q(t,x)}{\partial t}=-\frac{H}{2}\frac{\partial }{\partial x}(xq(t,x)),t>0,x\in \mathbb{R}.$$

$$q(t,x)=2{\int }_0^{\infty }{f}^H(s,x)p_t(s)ds=2{\int }_0^{\infty }\frac{e^{-\frac{|x{|}^2}{2s^{2H}}}}{(2\pi s^{2H}{)}^{d/2}}\frac{t}{\pi (t^2+s^2)}ds$$

$$\begin{array}{rl}\frac{{\partial }^{2}q(t,x)}{\partial t^2}& =-\frac{2H{I}_{(0,1/2]}(H)}{\pi t}\Delta \delta (x)-H(2H-1)\Delta G_{(2H-2),t}q(t,x)\\ & -H^2{\Delta }^2{G}_{(4H-2),t}q(t,x),x\in {\mathbb{R}}^d,t>0,\end{array}$$

$$G_{\gamma ,t}q(t,x)=2{\int }_0^{\infty }{s}^{\gamma }{p}_t(s)f^H(s,x)ds,\gamma \ne 0,$$

$$p_t(s)=\frac{t}{\pi (t^2+s^2)},t\ge 0,s\in \mathbb{R}.$$

$$\frac{{\partial }^2}{\partial t^2}{p}_t(s)=\frac{-2t(3s^2-t^2)}{(t^2+s^2{)}^3}$$

$$2\int_{0}^{\infty}f^{H}(s,x)\bigg{|}\frac{\partial^{2}}{\partial t^{2}}p_{t}(s)\bigg{|}ds=2\int_{0}^{\infty}\frac{e^{-\frac{|x|^{2}}{2s^{2H}}}}{(2\pi s^{2H})^{d/2}}\bigg{|}\frac{-2t(3s^{2}-t^{2})}{(t^{2}+s^{2})^{3}}\bigg{|}ds<\infty,$$

$$\frac{{\partial }^2}{\partial t^2}q(t,x)=2{\int }_0^{\infty }{f}^H(s,x)\frac{{\partial }^2}{\partial t^2}{p}_t(s)ds.$$

$$\begin{split}&\bigg{(}\frac{\partial^{2}}{\partial s^{2}}+\frac{\partial^{2}}{\partial t^{2}}\bigg{)}p_{t}(s)=0;\\ &\frac{\partial}{\partial s}f^{H}(s,x)=Hs^{2H-1}\Delta f^{H}(s,x),\end{split}$$

$$\frac{{\partial }^2}{\partial t^2}q(t,x)$$

$$\begin{array}{rl}\underset{s\to 0}{\lim }{f}^H(s,x)\frac{\partial }{\partial s}{p}_t(s)& =0,\\ \underset{s\to \infty }{\lim }{f}^H(s,x)\frac{\partial }{\partial s}{p}_t(s)& =0,\\ \underset{s\to \infty }{\lim }{p}_t(s)\frac{\partial }{\partial s}{f}^H(s,x)& =0,\end{array}$$

$$\begin{split}\lim_{s\to 0}p_{t}(s)\frac{\partial}{\partial s}f^{H}(s,x)&=\lim_{s\to 0}\frac{H}{\pi t}s^{2H-1}\Delta f^{H}(s,x)\\ &=\left\{\begin{array}[]{c}0\quad\qquad\qquad\quad\mbox{ if }\,H>\frac{1}{2},\\ \frac{H}{\pi t}\Delta\delta(x)\quad\mbox{ if }\,0<H\leq\frac{1}{2}.\end{array}\right.\end{split}$$

$$q(t,x)=2{\int }_0^{\infty }\frac{e^{-\frac{x^2}{2s^{2H}}}}{\sqrt{2\pi s^{2H}}}\frac{t}{\pi (t^2+s^2)}ds$$

Full text

$\alpha$-Time

Fractional Brownian Motion: PDE Connections and Local Times

Erkan Nane

Auburn University

  

Dongsheng Wu

University of Alabama in Huntsville

  

Yimin Xiao

Michigan State University Research partially supported by NSF grant DMS-1006903.

For $0<\alpha\leq 2$ and $0<H<1$, an $\alpha$-time fractional Brownian motion is an iterated process $Z=\{Z(t)=W(Y(t)),t\geq 0\}$ obtained by taking a fractional Brownian motion $\{W(t),t\in\mathbb{R}\}$ with Hurst index $0<H<1$ and replacing the time parameter with a strictly $\alpha$-stable Lévy process $\{Y(t),t\geq 0\}$ in $\mathbb{R}$ independent of $\{W(t),t\in{\mathbb{R}}\}$. It is shown that such processes have natural connections to partial differential equations and, when $Y$ is a stable subordinator, can arise as scaling limit of randomly indexed random walks. The existence, joint continuity and sharp Hölder conditions in the set variable of the local times of a $d$-dimensional $\alpha$-time fractional Brownian motion $X=\{X(t),t\in{\mathbb{R}}_{+}$}* defined by X(t)=\big{(}X_{1}(t),\cdots,X_{d}(t)\big{)}, where $t\geq 0$ and $X_{1},\cdots,X_{d}$ are independent copies of $Z$, are investigated. Our methods rely on the strong local ’ nondeterminism of fractional Brownian motion. *

Key words: Fractional Brownian motion, strictly $\alpha$-stable Lévy process, $\alpha$-time Brownian motion, $\alpha$-time fractional Brownian motion, partial differential equation, local time, Hölder condition.

Mathematics Subject Classification (2000): 60G17, 60J65, 60K99.

1 Introduction

In recent years, iterated Brownian motion and related iterated processes have received much research interest. Such iterated processes are connected naturally with partial differential equations and have interesting probabilistic and statistical features such as self-similarity, non-Markovian dependence structure, non-Gaussian distributions; see [2, 10, 11, 12, 17, 18, 28, 29, 33, 42] and references therein for further information. Inspired by these results, we consider a new class of iterated processes called $\alpha$-time fractional Brownian motion (fBm) for $0<\alpha\leq 2$ and $0<H<1$. These are obtained by taking a fractional Brownian motion of index $H$ and replacing the time parameter with a strictly $\alpha$-stable Lévy process $Y$. More precisely, let $W=\{W(t),t\in{\mathbb{R}}\}$ be a fractional Brownian motion in $\mathbb{R}$ with index $H$, which is a centered, real-valued Gaussian process with covariance function

[TABLE]

and $W(0)=0$ a.s. Here and in the sequel, $|\cdot|$ denotes the Euclidean norm. Let $Y=\{Y(t),t\geq 0\}$ be a real-valued strictly $\alpha$-stable Lévy process, $0<\alpha\leq 2$, starting from [math]; see Section 3 for its definition and [9, 36] for further information. We assume that $W$ and $Y$ are independent. Then a real-valued $\alpha$-time fractional Brownian motion $Z=\{Z(t),t\geq 0\}$ is defined by

[TABLE]

For $\alpha=2$ and $H=1/2$, this is the iterated Brownian motion of Burdzy [10]. When $0<\alpha<2$ and $H=1/2$, $Z$ is called an $\alpha$-time Brownian motion by Nane [30]. Aurzada and Lifshits [3] and Linde and Shi [27] studied the small deviation problem for real-valued $\alpha$-time Brownian motion. Nane [32] studied laws of the iterated logarithm for a version of $Z$. Moreover, when $Y$ is symmetric, for $\alpha=1,\,2$ and $H=1/2$ these processes have connections with partial differential operators as described in [2, 31].

More generally, it is easy to verify that the process $Z$ has stationary increments and is a self-similar process of index $H/\alpha$. The latter means that, for every constant $c>0$, the processes $\{Z(t):\ t\geq 0\}$ and $\{c^{-H/\alpha}Z(c\,t):\ t\geq 0\}$ have the same finite-dimensional distributions. Gaussian and stable self-similar processes have been studied extensively in recent years; see Samorodnitsky and Taqqu [35], Embrechts and Maejima [20] for further information. The $\alpha$-time fractional Brownian motions form an important class of non-Markovian and non-stable self-similar processes, except in the special case when $H=1/2$ and $Y$ is a stable subordinator [In this case, $Z$ is a symmetric stable Lévy process]. As will be shown in this paper, they have natural connections to partial differential equations and can arise as scaling limit of randomly indexed random walks with dependent jumps. Hence they can serve as useful stochastic models in many scientific areas including physics, insurance risk theory and communication networks. Moreover, because they are non-Markovian and have non-stable distributions, new methods are often needed in order to study their properties.

When $\alpha<2$, the sample function of the $\alpha$-time fractional Brownian motion $Z$ is not continuous and its irregularity is closely related to those of $W$ and $Y$. One of our motivations of this paper is to characterize the irregularity of $Z$ in terms of the parameters $H$ and $\alpha$. We do this by studying the existence and regularity of the local times of $\alpha$-time fractional Brownian motion $X=\{X(t),t\geq 0\}$ with values in ${\mathbb{R}}^{d}$ defined by

[TABLE]

where $X_{j}=W_{j}(Y_{j}(t))$ ($j=1,\cdots,d$). We assume that $W_{1},\cdots,W_{d}$ are independent copies of $W$, $Y_{1},\cdots,Y_{d}$ are independent copies of $Y$, and $\{W_{j}\}$ and $\{Y_{j}\}$ are independent. We will call $X=\{X(t),t\geq 0\}$ a $d$-dimensional $\alpha$-time fractional Brownian motion. It is clear that $X$ is also self-similar of index $H/\alpha$ and has stationary increments.

The rest of this paper is organized as follows. In Section 2, we study the PDE connections of $\alpha$-time fractional Brownian motions, and prove that they can be obtained as scaling limit of randomly indexed random walks with dependent jumps. These results provide some analytic and physical interpretations for $\alpha$-time fractional Brownian motions.

In Sections 3 and 4 we investigate the existence, joint continuity and sharp Hölder conditions in the set variable of the local times of a $d$ dimensional $\alpha$-time fractional Brownian motion $X$. In the special case of $d=1$, $W$ is Brownian motion and $Y$ is a symmetric $\alpha$-stable Lévy process with $\alpha>1$, the existence and joint continuity of the local time of $Z$ have been proved by Nane [30]. The methods used in this paper differ from those of Nane [30]. The latter uses the existence of local times of Brownian motion in $\mathbb{R}$ as well as the existence of local time of symmetric stable Lévy processes, which does not exist whenever $\alpha\leq 1$ or $d>1$. Our Theorem 3.1 implies that, for the $\alpha$-time Brownian motion $X$ in ${\mathbb{R}}^{d}$, local times exist in the case $d=1$ for $\alpha>1/2$; in the case $d=2$ for $\alpha>1$ and in the case $d=3$ for $\alpha>3/2$. Moreover, Theorem 4.1 shows that these local times have jointly continuous versions.

The methods of Sections 3 and 4 rely on the Fourier analytic argument of Berman [7, 8] and a chaining argument in Ehm [19]. In order to derive crucial moment estimates for the local times, we make use of the strong local nondeterminism (SLND) of fractional Brownian motion proved by Pitt [34] as well as several nontrivial modifications of the arguments in Xiao [41, 42].

Finally, we provide some technique lemmas as an Appendix in Section 5.

Throughout the paper, we will use $K$ to denote an unspecified positive finite constant which may not necessarily be the same in each occurrence.

2 PDE connections and scaling limits of randomly

indexed random walks

In this section we show that $\alpha$-time fractional Brownian motions have natural connections to partial differential equations. They may also arise as scaling limit of randomly indexed random walks with dependent jumps. These results provide some analytic and physical interpretations of $\alpha$-time fractional Brownian motions. These results show that $\alpha$-time fBm can serve as useful stochastic model in various scientific areas.

2.1 PDE connections

The domain of the infinitesimall generator $\mathcal{A}$ of a semigroup $T(t)$ defined on a Banach space $\mathcal{H}$ is the set of all $\varphi\in\mathcal{H}$ such that the limit

[TABLE]

exists in the strong norm of $\mathcal{H}$.

Let $\Delta=\sum_{j=1}^{d}\frac{\partial^{2}}{\partial x_{j}^{2}}$ be the Laplacian operator, and let $\delta(x)$ be the Dirac-delta function. The density of Brownian motion $W$ in ${\mathbb{R}}^{d}$ is $f(t,x)=\frac{1}{(2\pi t)^{d/2}}e^{-|x|^{2}/2t}$. Let

[TABLE]

be the semigroup of Brownian motion on $L^{2}({\mathbb{R}^{d}})$. Then the generator of $T(t)$ is $\Delta$ with the domain $Dom(\Delta)=\{\varphi\in L^{2}({\mathbb{R}^{d}}):\nabla\varphi\in L^{2}({\mathbb{R}^{d}})\}$, where $\nabla\varphi$ is the weak derivative of $\varphi$. See Section 31 in Sato [36] for more details and semigroups on other Banach spaces. Let $\varphi$ be a function in the domain of the Laplacian. Then the function $u(t,x)=T(t)\varphi(x)$ is a solution of the heat equation

[TABLE]

In the case $H=\frac{1}{2}$ and $Y(t)$ is stable subordinator of index $\beta/2$, $0<\beta\leq 2$ with ${\mathbb{E}}(e^{-sY(t)})=e^{-ts^{\beta/2}}$, $W(Y(t))$ is a symmetric stable process of index $\beta$ in ${\mathbb{R}}^{d}$. The density of $W(Y(t))$ is given by

[TABLE]

where $p_{t}(s)$ is the density of $Y(t).$ Then the function

[TABLE]

is a solution of

[TABLE]

where $-(-\Delta)^{\beta/2}$ is the fractional Laplacian with Fourier transform

[TABLE]

for functions $\psi$