| Abstract: |
| According to the theory of Diperna/Lions, the continuity equation associated to a Sobolev (or BV) vector field has a unique weak solution in $L^p$. Under the addition of white in time stochastic perturbations to the characteristics of the continuity equation, it is known that uniqueness can be obtained under a relaxation of the regularity conditions. In this talk, we will consider the general stochastic continuity equation associated to an It\^{o} diffusion with irregular drift and diffusion coefficients and discuss conditions under which the equation has a unique solution. Using the renormalization approach of DiPerna/Lions we will present another proof of uniqueness of solutions to the stochastic transport with additive noise and a drift in $L^q_t L^p_x$, satisfying the subcritical Ladyzhenskaya/Prodi/Serrin criterion $2/q + d/p < 1$. |
|