Sharp Non-uniqueness in Law for Stochastic Differential Equations on the Whole Space

In this paper, we investigate the stochastic differential equation on $\mathbb{R}^d,d\geq2$: \begin{align*} \dif X_t&=v(t,X_t)\dif t+\sqrt{2} \dif W_t. \end{align*} For any finite collection of initial probability measures $\{\mu^i_0\}_{1\leq i\leq M}$ on $\mathbb{R}^d$ and $\frac{d}{p}+\frac{1}{r}>1$, we construct a divergence-free drift field $v\in L_t^rL^p\cap C_tL^{d-}$ such that the associated SDE admits at least two distinct weak solutions originating from each initial measure $\mu^i_0$. This result is sharp in view of the well-known uniqueness of strong solutions for drifts in $C_tL^{d+}$, as established in \cite{KR05}. As a corollary, there exists a measurable set $A\subset\mathbb{R}^d$ with positive Lebesgue measure such that for any $x\in A$, the SDE with drift $v$ admits at least two weak solutions when with start in $x\in A$. The proof proceeds by constructing two distinct probability solutions to the associated Fokker-Planck equation via a convex integration method adapted to all of $\mathbb{R}^d$ (instead of merely the torus), together with refined heat kernel estimate.