Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness

Abstract

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.

On the convergence of stochastic transport equations to a deterministic parabolic one

“…In general (UN) is satisfied under very mild assumptions on b, much weaker than those required for the associated deterministic transport equation to be well posed. This suggests the possibility to obtain uniqueness for the STLE under the same assumption (UN); in this direction, see the results given in [28], [5] and the references therein.…”

“…with W = 0), for which essentially sharp condition are given by [12], [1]. There is now an extensive literature on the topic of regularization by noise for transport equations, see the review [15] and the references in [5]. However, from the modelling point of view, spaceindependent noise is too simple, since formally the characteristics associated to (STLE) are given by…”

“…converges in probability (in a suitable topology) as N → ∞ to the unique deterministic solution u of the parabolic equation (5).…”

“…However, when considering the corresponding Itô formulation, the Itô-Stratonovich corrector gives rise to a Laplacian. It was intuited in [17] that equation (STLE) has some parabolic features at the mean level; this has become clear in [5].…”

“…iii) The statements holds for any sequence of energy solutions of (6), even when uniqueness is not known; existence of energy solutions can be shown under suitable assumptions on b. We only need well posedness for the limit problem (5) and that's why we require (UN) to hold. In general (UN) is satisfied under very mild assumptions on b, much weaker than those required for the associated deterministic transport equation to be well posed.…”