Stochastic regularization for transport equations (Q2045408)

In this article, the authors deal with the stochastic transport equation \[ \partial_t u(t,x) + b(t,x) \cdot \nabla u(t,x) + \sum_{i=1}^d \partial_{x_i} u(t,x) \circ \dot{B}_i(t) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^d, \] \[ u(t,x)|_{t=0} = u_0(x), \,\, x \in \mathbb{R}^d, \] where \(B\) is a \(d\)-dimensional standard Brownian motion. Their first main result (Theorem 1.1) is an existence and uniqueness result for this equation with drift coefficients \(b \in L^q(0,T;\mathcal{C}_b^{\alpha}(\mathbb{R}^d))\), where \(\alpha > 2/q\). Then the authors deal with the question whether the noise has a regularizing effect for the stochastic transport equation. This is indeed the case: As Theorem 1.2 shows, for a suitably chosen drift \(b\) one has nonexistence of solutions for the corresponding deterministic transport equation. Theorem 1.3 presents a nonexistence result for the stochastic transport equation in case \(\alpha + 1 < 2/q\) and \(d \geq 2\). The authors also provide new results on existence and uniqueness of stochastic flows of diffeomorphisms on stochastic differential equations, which are used for the proofs of the above mentioned main results.