Given a smooth geometrically connected curve C over a perfect field k and a smooth commutative group scheme G defined over the function field K of C, one can consider isomorphism classes of G-torsors locally trivial at completions of K coming from closed points of C. They form a generalized Tate-Shafarevich group which specializes to the classical one in the case when k is finite. Recently, these groups have been studied over other base fields k as well, for instance p-adic or number fields. Surprisingly, finiteness can be proven in some cases but there are also quite a few open questions which I plan to discuss  in my talk.

Keywords: SDE on the plane, Brownian sheet, path by path uniqueness, space time local time integral, Malliavin calculus

In this talk, we discuss the existence, uniqueness, and regularisation by noise for stochastic differential equations (SDEs) on the plane. These equations can also be interpreted as quasi-linear hyperbolic stochastic partial differential equations (HSPDEs). More specifically, we address path-by-path uniqueness for multidimensional SDEs on the plane, under the assumption that the drift coefficient satisfies a spatial linear growth condition and is componentwise non-decreasing. In the case where the drift is only measurable and uniformly bounded, we show that the corresponding additive HSPDE on the plane admits a unique strong solution that is Malliavin differentiable. Our approach combines tools from Malliavin calculus with variational techniques originally introduced by Davie (2007), which we non-trivially extend to the setting of SDEs on the plane.

This talk is based on a joint works with A. M. Bogso, M. Dieye and F. Proske.