Stochastic transport equation with L\'evy noise

：Jianliang Zhai (University of Science Technology of China)
： 2025-07-04 09:00
：Conference Room C610 at Administration Building at Haiyun Campus

Speaker：Jianliang Zhai (University of Science and Technology of China)

Time：2025-7-4 9:00

Location：Conference Room C610 at Administration Building at Haiyun Campus

Abstract:

We study the linear transport equation with a globally H\"older continuous and bounded vector field driven by a non-degenerate L\'evy noise of $\alpha$-stable type: \begin{align}\label{strong3}& \frac{\partial{u(t,x)}}{\partial t}+(b(x)\cdot {\nabla})u(t,x)\,+\sum_{i=1}^de_i\cdot  {{\nabla}u(t-,x)}\diamond \frac{d L_t^i}{dt}=0,~t>0;\\& u(0,x)=u_0(x),\ \ x\in\mathbb{R}^d,\end{align} where the initial data $u_0$ is a bounded Borel function and the stochastic integration is understood in the Marcus form. Assuming also an integrability condition on the divergence of $b$ we mainly consider $\alpha \in [1,2)$. In particular we prove well-posedness theorems for $\mathrm{L}^{\infty}$-weak solutions. We can even prove a uniqueness result for $\mathrm{L}^{\infty}$-weak solutions in the case of $\alpha \in (0,1)$ assuming in addition that $L_t=(L_t^1,\cdots,L_t^d),~t\geq0$ is rotationally invariant. This extends the results proved in [F. Flandoli, M. Gubinelli and E. Priola 2010] in the Brownian case and shows regularization by noise phenomena with L\'evy noises. Indeed uniqueness is restored by the presence of $L$ since without the noise term the transport equation \eqref{strong3} has in general many solutions.