Cole–Hopf-Feynman–Kac formula and quasi-invariance for Navier–Stokes equations

Abstract

We make a refined comparison between the Navier–Stokes equations and their dynamically-scaled Leray equations solely on the basis of their scaling property. Previously it was observed using the vector potentials that they differ only by one drift term (Ohkitani 2017 J. Phys. A: Math. Theor. 50 045501). The Duhamel principle recasts the equations in path integral forms, which differ by two Maruyama–Girsanov densities. In this brief paper we simplify the concept of quasi-invariance (or, near-invariance) by combining the result with a Cole–Hopf transform and the Feynman–Kac formula. That way, as a multiplicative characterisation we can place those equations just one Maruyama–Girsanov density apart. Furthermore, as an additive characterisation we express the difference in terms of the Malliavin H-derivative.