Sample-path large deviation principle for a 2-D stochastic interacting vortex dynamics with singular kernel

We consider a stochastic interacting vortex system of N particles, approximating the vorticity formulation of the 2-D Navier–Stokes equation on a torus. The singular interaction kernel is given by the Biot–Savart law. Assuming only that the initial state has finite energy, we derive a sample-path large deviation principle for the empirical measure as the number of vortices tends to infinity. This contrasts with previous studies that require the initial state to have finite entropy. The rate function is characterized by an explicit formula that takes finite values only on sample paths with finite energy and finite integrals of L2 norms over time. The proof employs a symmetrization technique for the representation of the singular kernel, originating from Delort (J. Amer. Math. Soc. 4 (1991) 553–586), and a carefully modified energy dissipation structure to establish a sharp prior estimate for an auxiliary functional as a modification of the rate function. The key step is to prove that the singular term after symmetrization can be bounded by the integral of L2 norms along sample paths.