We consider statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses, under the assumption that a suitable single test (and corresponding $p$-value) is known for each individual hypothesis. We extend to this setting the notion of false discovery rate (FDR) as a measure of type I error. This extends the standard statistical setting for multiple hypotheses testing, which is restricted to a finite set. This work is motivated by numerous modern applications where the observed signal is modeled by a stochastic process over a continuum.
Our main result studies specific procedures called "step-up" based on the observation of the $p$-value process. Control of the FDR at a nominal level is ensured either under arbitrary dependence of $p$-values, or under the assumption that the finite dimensional distributions of the $p$-value process have positive correlations of a specific type (weak PRDS). The interest of this approach is demonstrated in several non-parametric examples.