[@MT99; @AT07], the $k$-nearest neighbors algorithm [@G78; @chaudhuri; @BD2015; @GKM16], support vector machines [@Steinwart], or random forests [@BiauScornet], just to name a few.

However there are situations where the inputs $X_i$ and $X$ are better modelled as functions; the set $\mathcal{H}$ is then infinite-dimensional. Practical examples can be found, [e.g.]{}, in stochastic population dynamics [@L03], in signal processing [@C02], or in finance [@LL96]. This binary supervised functional classification problem was tackled with nonparametric procedures such as kernel methods or the $k$-nearest neighbours algorithm. For example, [@CoH67] studied the nearest neighbour rule in any metric space, while [@KP95] analyzed the performances of the $k$-nearest neighbours algorithm in terms of a metric covering measure. Such metric entropy arguments were also used in [@CG06], or with kernel methods in [@ABC03].\
[**Our functional model.**  ]{} In the present work, we focus on one of the most elementary diffusion classification model: we suppose that the input $X=(X(t))_{t \in [0,1]}$ is a continuous trajectory, solution to the stochastic differential equation $$\label{eq:model}
\forall t \in [0,1], \qquad 
dX(t) = Y f(t) dt + (1-Y)g(t) dt +  dW(t) \,,$$ where $(W(t))_{0 {\leqslant}t {\leqslant}1}$ is a standard Brownian motion, and where $Y$ is a Bernoulli $\mathcal{B}(1/2)$ random variable independent 