  Non-uniqueness of three-dimensional Euler equations and Navier-Stokes
equations forced by random noise, path-wise and more recently even in law, have
been proven by various authors. We prove non-uniqueness in law of the
three-dimensional Navier-Stokes equations forced by random noise and diffused
via a fractional Laplacian that has power between zero and one half. The
solution we construct has H$\ddot{\mathrm{o}}$lder regularity with a small
exponent rather than Sobolev regularity with a small exponent. For the power
sufficiently small, the non-uniqueness in law holds at the level of Leray-Hopf
regularity. In particular, in order to handle transport error, we consider
phase functions convected by not only a mollified velocity field but a sum of
that with a mollified Ornstein-Uhlenbeck process if noise is additive and a
product of that with a mollified exponential Brownian motion if noise is
multiplicative.
