Having read this [link](https://math.stackexchange.com/questions/49763/relationship-between-brownian-motion-and-1-2-delta) on math stackexchange, I would like to submit to your wisdom the following questions.


Is it possible, mutatis mutandis, to repeat the same reasoning for a fractional Brownian motion?


More specifically: a real valued Gaussian process $B^H:=\{B^H(t)\}\_{t\geq 0}$
 in a probability space
 $(\Omega,\mathscr{F},\mathbb{P})$ is a fractional Brownian
 motion (fBm) with Hurst parameter $H \in(0,1)$ if for all $s,t\in \mathbb{R}\_+$


1. $B^H(0)=0$,
2. $\mathbb{E}B^H(t)=0$,
3. $\operatorname{Cov}[B^H(t),B^H(s)]=\frac{1}{2}
 \left(t^{2 H}+s^{2 H}-|t-s|^{2 H}\right)$.


In addition, the It\^o formula for fBm is written as:
$$
f(B^H(t))= \displaystyle\int\_0^t f'\left(B^H(s)\right)\, d B^H(s) + H \displaystyle\int\_0^t f''\left(B^H(s)\right) s^{2H-1}\, ds.
$$ 
Taking the expectation in both sides of the above equality, we obtain:
$$
\mathbb{E}\left[f(B^H(t))\right]= H \displaystyle\int\_0^t \mathbb{E}\left[f''\left(B^H(s)\right)\right] s^{2H-1}\, ds.
$$ 
 I might continue as;
 changing the expectation by the conditional expectation $\mathbb{E}\_x$ with respect to the event $\{X\_0=x\}$ where $X(t)=B^H(t)+ x $, it follows:
$$
\mathbb{E}\_x\left[f(X^H(t))\right]= H \displaystyle\int\_0^t \mathbb{E}\_x\left[f''\left(X^H(s)\right)\right] s^{2H-1}\, ds.
$$ 
And if we put
$$
m(x,t; H)= \mathbb{E}\_x\left[f\left(X^H(t)\right)\right].
$$
We get:
$$
\displaystyle\frac{\partial}{\partial t} m(x,t; H)= H \, t^{2H-1}\displaystyle\frac{\partial^2}{\partial x^2 }m(x,t; H)
$$
Knowledge that fBm is not a semimartingale nor a Markov process except for cases $H=\frac{1}{2}$.
 I have some doubts about the last deduction.1.



Fractional Brownian motion and Laplacian
Your construction has been carried out for much more general situations. See


Baudoin, F., Coutin, L. Operators associated with a stochastic differential equation driven
by fractional Brownian motions, Stoch. Proc. Appl. 117, 5, 550–574, 2007. ArXiv: math/0509511.


