Stochastic differential equations with critical drifts

Nam, Ki‐Young

doi:10.1016/j.spa.2020.03.010

Cited by 10 publications

(7 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are some results on weak well-posedness for measure-valued drifts, see [6], and distributionvalued drifts, see [36,21,10], but it is unclear whether they apply to the limit case p = d: for example, the result in [6], when restricted to measures with density b with respect to the Lebesgue measure, requires p > d, see [6,Example 2.3]. The present paper is the first one to give information on sDEs in the limit case p = d (apart from [53,65]).…”

Section: Results For the Sdementioning

confidence: 95%

“…The papers [50,29] use Wiener chaos expansion techniques to obtain uniqueness for the sTE for drifts close to KR class, see [50], or even beyond, see [29], at the price of uniqueness in a smaller class (namely among solutions adapted to the Brownian filtration). A full solution of the uniqueness problem in the KR class was still open (apart from the recent preprints [53,65] mentioned above) and this is a by-product of this paper, which solves the problem in a stronger sense in two directions: i) path-by-path uniqueness instead of pathwise uniqueness;…”

Section: Uniqueness Results For the Spdesmentioning

confidence: 97%

“…Also here it requires a completely new approach. Under the condition d p + 2 q = 1, we do not know how to solve the sDE directly (see however the recent preprints [53,65] mentioned above); even in a weak sense, by Girsanov theorem, the strict inequality seems to be needed ( [43,57,40]). Similarly as in [32], the proof of regularity of solutions of the sTE from [26] is based on the construction of stochastic flows for the sDE and their regularity in terms of weak differentiability.…”

Section: Regularity Results For the Spdesmentioning

confidence: 99%

“…The deep connection of the LPS class, especially when d p + 2 q = 1, with the theory of 3D Navier-Stokes equations is one of our main motivations to analyze stochastic transport under such conditions. We finally note that, while preparing the second version of this work, two preprints [65] and [53] appeared, which go almost up to the critical case for weak and strong solution to the SDEs, the latter showing also Sobolev regularity of the stochastic flow. Respectively, the former preprint, while staying within the subcritical case in the interior [0, T [, allows a singularity at time T which matches, and actually goes slightly beyond, the critical case.…”

Section: Regularity Assumptions On Bmentioning

confidence: 99%

“…Respectively, the former preprint, while staying within the subcritical case in the interior [0, T [, allows a singularity at time T which matches, and actually goes slightly beyond, the critical case. In these preprints, the limiting case is d/p + 2/q = 1 is reached when replacing the L q integrability condition by a C q -continuity condition in [65] and by a L q,1 condition in [53] (for Lorenz space L q,1 L q ).…”

Section: Regularity Assumptions On Bmentioning

confidence: 99%

See 4 more Smart Citations

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

Beck

Flandoli

Gubinelli

et al. 2019

Electron. J. Probab.

137

View full text Add to dashboard Cite

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.

show abstract

Section: Results For the Sdementioning

confidence: 95%

Section: Uniqueness Results For the Spdesmentioning

confidence: 97%

Section: Regularity Results For the Spdesmentioning

confidence: 99%

Section: Regularity Assumptions On Bmentioning

confidence: 99%

Section: Regularity Assumptions On Bmentioning

confidence: 99%

See 3 more Smart Citations

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

Beck

Flandoli

Gubinelli

et al. 2019

Electron. J. Probab.

137

View full text Add to dashboard Cite

show abstract

Form-Boundedness and SDEs with Singular Drift

Kinzebulatov

2024

Springer INdAM Series

View full text Add to dashboard Cite

On diffusion processes with drift in $$L_{d}$$

Крылов

2020

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

This paper is a natural continuation of [11] and [12] where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in L d+1 (R d+1) and some properties of their Green's functions and probability of passing through narrow tubes are investigated. On the basis of this here we study some further properties of these processes such as Harnack inequality, Hölder continuity of potentials, Fanghua Lin estimates and so on.

show abstract

Stochastic differential equations with critical drifts

Cited by 10 publications

References 23 publications

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

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

Form-Boundedness and SDEs with Singular Drift

On diffusion processes with drift in $$L_{d}$$

Contact Info

Product

Resources

About