It is well known that the Laplacian is the infinitesimal generator of a Brownian Motion, that is,
$$
\lim_{t \to 0} \frac{E[f(x+B_t)-f(x)]}{t}= \Delta f(x).
$$
Is it true that for the Fractional Brownian Motion $B_H$ with hurst parameter $H \in (0,1)$, that is, the continuous-time Stochastic Process with stationary gaussian increments, mean $0$ and covariance
$$
\mathbb{E}[(B_H(t)-B_H(s))^2]= |t|^{2H} + |s|^{2H} – |t-s|^2H
$$
has the infinitesimal generator to be a fractional Laplacian $-(-\Delta)^{\alpha}$for some $\alpha \in (0,\infty)$? If so, what is the relation between $H$ and $\alpha$?
Best Answer
As saz points out, the fractional Laplacian is not the generator of fractional Brownian motion. In fact, since fractional Brownian motion is not a Markov process when $H \neq 1/2$, it does not even make sense to talk about a generator for it (e.g., the limit of the above difference quotient will simply be zero or $\infty$ for $H \neq 1/2$).
Instead, the fractional Laplacian for $\alpha \in (0,1]$ is the generator for the (radially symmetric) $2\alpha$-stable Levy process, i.e., the Levy process associated to the infinitely divisible probability distribution whose characteristic function is given by $\phi_{2\alpha}(t) = e^{-|t|^{2\alpha}}$. When $\alpha \neq 1$, one should notice that $-(-\Delta)^{\alpha}$ is a non-local differential operator, and this is perfectly embodied by the fact that the $2\alpha$-stable Levy process will have jumps (i.e., it is not a continuous path like Brownian motion when $\alpha \neq 1$).
With that being said, the fractional Laplacian actually does have a different kind of relationship to frational Brownian motion. Specifically, since $(B_H(t))_{t \in [0,1]}$ is a Gaussian variable in $C[0,1]$, it has a Cameron-Martin space which encodes its covariance structure. In the case of fractional Brownian motion with Hurst parameter $H$, the Cameron martin space is the Sobolev space $$W_0^{H+1/2,2}([0,1]) := \{ f \in C[0,1]: f(0)=0, \;\;\langle f, (-\Delta)^{H+1/2}f \rangle_{L^2[0,1]}<+\infty\}.$$ Since Sobolev spaces extend to all real parameters (corresponding to the exponent on the fractional Laplacian), this means that fractional Brownian motion can be naturally generalized to all $H \in \Bbb R$ and can also be generalized to $d$-dimensional indexing space, i.e., $\{B_H(t)\}_{t \in [0,1]^d}$ is the Gaussian random variable on $\mathcal D'([0,1]^d)$ whose Cameron-Martin space is $W^{H+d/2,2}_0([0,1]^d)$.
In the case $H=-d/2$, this is just white noise on $[0,1]^d$. In the case that $H=0$ and $d=2$, this object (called the Gaussian Free Field) is conformally invariant and a Fields medal has even been awarded for a detailed study of its geometry. These fractional fields may be coupled for $H<H'$ by the identity $$B_H = (-\Delta)^{(H’-H)/2}B_{H’}.$$In particular fractional BM may be obtained from ordinary BM just by applying a fractional Laplacian to the latter. Take a look at this nice survey paper for more information (and more general types of boundary conditions) on these fractional Gaussian fields.