Sharp Non-uniqueness in Law for Stochastic Differential Equations on the Whole Space
A talk in the Bielefeld Stochastic Afternoon series by
Huaxiang Lyu
| Abstract: | In this paper, we investigate the stochastic differential equation on
$\mathbb{R}^d,d\geq2$:
\begin{align*}
d X_t&=v(t,X_t)d t+\sqrt{2}d 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. Within the CRC this talk is associated to the project(s): B1 |