  In this paper we show the existence of stochastic Lagrangian particle
trajectory for Leray's solution of 3D Navier-Stokes equations. More precisely,
for any Leray's solution ${\mathbf u}$ of 3D-NSE and each
$(s,x)\in\mathbb{R}_+\times\mathbb{R}^3$, we show the existence of weak
solutions to the following SDE, which has a density $\rho_{s,x}(t,y)$ belonging
to $\mathbb{H}^{1,p}_q$ provided $p,q\in[1,2)$ with
$\frac{3}{p}+\frac{2}{q}>4$: $$ \mathrm{d} X_{s,t}={\mathbf u}
(s,X_{s,t})\mathrm{d} t+\sqrt{2\nu}\mathrm{d} W_t,\ \ X_{s,s}=x,\ \ t\geq s, $$
where $W$ is a three dimensional standard Brownian motion, $\nu>0$ is the
viscosity constant. Moreover, we also show that for Lebesgue almost all
$(s,x)$, the solution $X^n_{s,\cdot}(x)$ of the above SDE associated with the
mollifying velocity field ${\mathbf u}_n$ weakly converges to $X_{s,\cdot}(x)$
so that $X$ is a Markov process in almost sure sense.
