  We address a fundamental issue in the nonparametric inference for systems of
interacting particles: the identifiability of the interaction functions. We
prove that the interaction functions are identifiable for a class of
first-order stochastic systems, including linear systems with general initial
laws and nonlinear systems with stationary distributions. We show that a
coercivity condition is sufficient for identifiability and becomes necessary
when the number of particles approaches infinity. The coercivity is equivalent
to the strict positivity of related integral operators, which we prove by
showing that their integral kernels are strictly positive definite by using
M\"untz type theorems.
