from $(W(t))_{0 {\leqslant}t {\leqslant}1}$. In particular, in the sample $\mathcal{S}_n$, trajectories $X_i$ labeled with $Y_i=1$ correspond to observations of the signal $f$, while trajectories $X_i$ labeled with $Y_i=0$ correspond to $g$.\
The white noise model has played a key role in statistical theoretical developments; see, [e.g.]{}, the seminal contributions of [@IH81] in nonparametric estimation and of [@L90] in adaptive nonparametric estimation. In our supervised classification setting, the goal is not to estimate $f$ and $g$ but to predict the value of $Y$ given an observed continuous trajectory $(X(t))_{0 {\leqslant}t {\leqslant}1}$. Of course, we assume that both functions $f$ and $g$ are unknown so that the joint distribution $\mathbb{P}_{X,Y}$ of the pair $((X(t))_{t\in[0,1]},Y)$ is unknown. Without any assumption on $f$ and $g$, there is no hope to solve this problem in general. However, learning the functions $f$ and $g$ (and thus $\mathbb{P}_{X,Y}$) from the sample $\mathcal{S}_n$ becomes statistically feasible when $f$ and $g$ are smooth enough.

The functional model considered in this paper is very close to the one studied by [@Cadre]. Actually our setting is less general since [@Cadre] considered more general diffusions driven by state-dependent drift terms $t \longmapsto f(t,X(t))$ and $t \longmapsto g(t,X(t))$. We focus on a simpler 