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 u\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{u }$$\end{document} of 3D-NSE and each (s,x)∈R+×R3\documentclass[12pt]{minimal}
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\begin{document}$$(s,x)\in \mathbb {R}_+\times \mathbb {R}^3$$\end{document}, we show the existence of weak solutions to the following SDE, which have densities ρs,x(t,y)\documentclass[12pt]{minimal}
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\begin{document}$$\rho _{s,x}(t,y)$$\end{document} belonging to Hq1,p\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {H}^{1,p}_q$$\end{document} with p,q∈[1,2)\documentclass[12pt]{minimal}
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\begin{document}$$p,q\in [1,2)$$\end{document} and 3p+2q>4\documentclass[12pt]{minimal}
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\begin{document}$$\frac{3}{p}+\frac{2}{q}>4$$\end{document}: dXs,t=u(s,Xs,t)dt+2νdWt,Xs,s=x,t⩾s,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \text {d} X_{s,t}=\mathbf{u } (s,X_{s,t})\text {d} t+\sqrt{2\nu }\text {d} W_t,\ \ X_{s,s}=x,\ \ t\geqslant s, \end{aligned}$$\end{document}where W is a three dimensional standard Brownian motion, ν>0\documentclass[12pt]{minimal}
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\begin{document}$$\nu >0$$\end{document} is the viscosity constant. Moreover, we also show that for Lebesgue almost all (s, x), the solution Xs,·n(x)\documentclass[12pt]{minimal}
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\begin{document}$$X^n_{s,\cdot }(x)$$\end{document} of the above SDE associated with the mollifying velocity field un\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{u }_n$$\end{document} weakly converges to Xs,·(x)\documentclass[12pt]{minimal}
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\begin{document}$$X_{s,\cdot }(x)$$\end{document} so that X is a Markov process in almost sure sense.
