On the identifiability of interaction functions in systems of interacting particles

Understand

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.