On the Radius of Self-Repellent Fractional Brownian Motion

Abstract

We study the radius of gyration $R_T$ of a self-repellent fractional Brownian motion $\left\{ B^H_t\right\} _{0\le t\le T}$ taking values in $\mathbb {R}^d$. Our sharpest result is for $d=1$, where we find that with high probability,

$$\begin{aligned} R_T \asymp T^\nu , \quad \text {with }\quad \nu =\frac{2}{3}\left( 1+H\right) . \end{aligned}$$

For $d>1$, we provide upper and lower bounds for the exponent $\nu$, but these bounds do not match.

Appendix A: Fractional Brownian Motions and Girsanov Theorem

In this section, we present some preliminaries about stochastic calculus for fBm’s. For a more detailed account of this topic, we refer the interested readers to [21].

A d-dimensional stochastic process $B^H = \left\{ \left( B^{H,1}_t,\dots , B^{H,d}_t\right) :t\in \mathbb {R}_+\right\}$ is a called a fraction Brownian motion (fBm) with the Hurst parameters $H \in (0,1)$ on a probability space $(\Omega , \mathcal {F}, \mathbb {P})$, if

• (i) $B^{H,i}$, $i=1,\dots ,d$, are independent;

• (ii) for each $1\le i\le d$, $\left\{ B^{H,i}_t :t \in \mathbb {R}_+\right\}$ is a centered Gaussian family with covariance

  $$\begin{aligned} \mathbb {E} \left[ B^{H,i}_t B^{H,i}_s\right] = \frac{1}{2}\left( t^{2H} + s^{2H} - |t-s|^{2H}\right) . \end{aligned}$$

Without loss of generality, we can assume that the filtration $\left\{ \mathcal {F}_t :t \in \mathbb {R}_+\right\}$ is the canonical filtration generated by $B^H$.

Next, we define the integration of deterministic functions against fBm’s. If $\phi$ is a smooth function on $\mathbb {R}_+$ with compact support, i.e., $\phi \in C_c^{\infty }(\mathbb {R}_+)$, then the integrals

$$\begin{aligned} B^{H,i}(\phi ) :=\int _0^{\infty } \phi (t) \textrm{d} B^{H,i}_t, \qquad i = 1,\dots , d, \end{aligned}$$

are centered Gaussian random variables with the following covariance structure:

for all $1\le i,j \le d$, and $\phi , \psi \in C_c^{\infty }(\mathbb {R}_+).$

By typical approximation arguments, one can extend the integration to the Hilbert space $\mathcal {H}$ of functions on $\mathbb {R}_+$, with inner product,

$$\begin{aligned} \left\langle \phi , \psi \right\rangle _{\mathcal {H}} :=\iint _{\mathbb {R}_+^2} \textrm{d} t \, \textrm{d} s\, \phi (s)\psi (t) |t-s|^{2H - 2}. \end{aligned}$$

In particular, for any $t\in \mathbb {R}_+$, the function $w(t,\cdot )$, given by

$$\begin{aligned} w (t,s) :=c_{1}\, s^{1/2 - H} (t - s)^{1/2 - H} \textbf{1}_{(0,t)} (s),\quad \text {for all } s\in \mathbb {R}, \end{aligned}$$

is an element of $\mathcal {H}$ (see [22, Proposition 2.1]), where, with $B(\cdot ,\cdot )$ denoting the Beta function,

This observation allows us to define the following Gaussian process $M = M^{\textbf{u}} = \{ M_t :t \in \mathbb {R}_+\}$ with parameter $\textbf{u} = (u_1,\dots , u_d)$ being a unit vector in $\mathbb {R}^d$ as follows:

$$\begin{aligned} M_t :=\sum _{i=1}^d u_i \int _0^t w(t,s) \textrm{d} B^{H,i}_s, \quad \text {for all } t\in \mathbb {R}_+. \end{aligned}$$

(3.1)

The following theorem is a straightforward extension of [22, Theorem 3.1] from $d=1$ to higher dimensional cases, thanks to the independence of the components of $B^H$:

Theorem A.1

Let $B^H$ be a d-dimensional fBm with $H \in (0,1)$, let $\textbf{u}$ be a unit vector in $\mathbb {R}^d$, and let $M = M^{\textbf{u}}$ be the Gaussian process given as in (3.1). Then M is a square-integrable martingale with quadratic variation

For any $\lambda > 0$ and $T > 0$, denote

$$\begin{aligned} \mathcal {Q}_T (M) :=\exp \left( M_T - \frac{1}{2} \langle M, M \rangle _T \right) , \end{aligned}$$

and let $\mathbb {P}_T^{\lambda } = \mathbb {P}_T^{\lambda , \textbf{u}}$ be a probability measure on $(\Omega , \mathcal {F}_T)$ that is equivalent to $\mathbb {P}_T$ with the Radon–Nikodym derivative

$$\begin{aligned} \frac{\textrm{d} \mathbb {P}_T^{\lambda }}{\textrm{d} \mathbb {P}_T} :=\mathcal {Q}_T(\lambda M). \end{aligned}$$

(3.2)

The next theorem, a Girsanov formula for fBm’s, is a straightforward extension of [22, Theorem 4.1].

Theorem A.2

Under probability $\mathbb {P}^{\lambda }_T$, the process $\left\{ B^H_t :0 \le t \le T\right\}$ is a d-dimensional fBm with a drift $\lambda \textbf{u}\in \mathbb {R}^d$, i.e., the distribution of the process $B^H$ up to time T under $\mathbb {P}^{\lambda }_T = \mathbb {P}^{\lambda ,\textbf{u}}_T$ is the same as $B^{H, \lambda , \textbf{u}, T} = \left\{ B^H_t + \lambda t\, \textbf{u} \,:0\le t\le T\right\}$ under $\mathbb {P}_T$.