In this paper linear stochastic transport and continuity equations with drift in critical L p spaces are considered. In this situation noise prevents shocks for the transport equation and singularities in the density for the continuity equation, starting from smooth initial conditions. Specifically, we first prove a result of Sobolev regularity of solutions, which is false for the corresponding deterministic equation. The technique needed to reach the critical case is new and based on parabolic equations satisfied by moments of first derivatives of the solution, opposite to previous works based on stochastic flows. The approach extends to higher order derivatives under more regularity of the drift term. By a duality approach, these regularity results are then applied to prove uniqueness of weak solutions to linear stochastic continuity and transport equations and certain well-posedness results for the associated stochastic differential equation (sDE) (roughly speaking, existence and uniqueness of flows and their C α regularity, strong uniqueness for the sDE when the initial datum has diffuse law). Finally, we show two types of examples: on the one hand, we present well-posed sDEs, when the corresponding ODEs are ill-posed, and on the other hand, we give a counterexample in the supercritical case.We first choose α = 2 such that the stochastic part λ α/2−1 ∇u λ (t, x) • W λ (t) is comparable to the derivative in time ∂ t u λ . Notice that this is the parabolic scaling, although sTE is not parabolic (but as we will see below, a basic idea of our approach is that certain expected values of the solution satisfy parabolic equations for which the above scaling is the relevant one). Next we require that, for small λ, the rescaled drift b λ becomes small (or at least controlled) in some suitable norm (in our case, L q (0, T ; L p (R d , R d ))). It is easy to see that b λ L q (0,T /λ 2 ;L p ) = λ 1−(2/q+d/p) b L q (0,T ;L p ) (here, the exponent d comes from rescaling in space and the exponent 2 from rescaling in time and the choice α = 2). In conclusion, we find that• if LPS holds with strict inequality, then b λ L q (0,T /λ 2 ;L p ) → 0 as λ → 0: the stochastic term dominates and we expect a regularizing effect (subcritical case);• if LPS holds with equality, then b λ L q (0,T /λ α ;L p ) = b L q (0,T ;L p ) remains constant: the deterministic drift and the stochastic forcing are comparable (critical case).This intuitively explains why the analysis of the critical case is more difficult. Notice that, if LPS does not hold, then we expect the drift to dominate, so that a general result for regularization by noise is probably false. In this sense, LPS condition should be regarded as an optimal condition for expecting regularity of solutions.

The strong existence and the pathwise uniqueness of solutions with L ∞ -vorticity of the 2D stochastic Euler equations are proved. The noise is multiplicative and it involves the first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. The stability under regularizations is also proved.

We take a pathwise approach to classical McKean-Vlasov stochastic differential equations with additive noise, as e.g. exposed in Sznitmann [38]. Our study was prompted by some concrete problems in battery modelling [23], and also by recent progrss on rough-pathwise McKean-Vlasov theory, notably Cass-Lyons [10], and then Bailleul, Catellier and Delarue [4]. Such a "pathwise McKean-Vlasov theory" can be traced back to Tanaka [40]. This paper can be seen as an attempt to advertize the ideas, power and simplicity of the pathwise appproach, not so easily extracted from [4,10,40], together with a number of novel applications. These include mean field convergence without a priori independence and exchangeability assumption; common noise, càdlàg noise, and reflecting boundaries. Last not least, we generalize Dawson-Gärtner large deviations and the central limit theorem to a non-Brownian noise setting.

A linear stochastic vector advection equation is considered; the equation may model a passive magnetic field in a random fluid. When the driving velocity field is rough but deterministic, in particular just Hölder continuous and bounded, one can construct examples of infinite stretching of the passive field, arising from smooth initial conditions. The purpose of the paper is to prove that infinite stretching is prevented if the driving velocity field contains in addition a white noise component.