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 $C^{\alpha }$ 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.

/pdf/stochastic-odes-and-stochastic-linear-pdes-with-critical-2th9gggh58.pdf

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

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 $C^\alpha$ 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.

Un resultat de A.M. Davie (Int. Math. Res. Not. 24 (2007) rnm124) etablit qu’une equation stochastique multi-dimensionnelle $dX_{t}=b(t,X_{t})\,dt+dW_{t}$, $X_{0}=x$, dirigee par un processus de Wiener $W=(W_{t})$ avec un coefficient $b$ qui est seulement borne et mesurable admet une unique solution pour presque tout choix de la trajectoire du processus $W$ la dirigeant. Nous considerons un probleme similaire lorsque $W$ est remplace par un processus de Levy $L=(L_{t})$ et $b$ est $\beta$-Holder continu en espace. Nous supposons que $L_{1}$ a un moment fini d’ordre $\theta$ pour un certain $\theta>0$. En utilisant un nouveau resultat de regularite cadlag, nous prouvons que l’existence et unicite forte pour l’EDS, associees a une $L^{p}$-Lipschitz continuite de la solution forte par rapport a $x$, impliquent une unicite de type Davie pour presque tout choix de la trajectoire de Levy. Nous appliquons ce resultat a une classe d’EDS dirigees par un processus de Levy $\alpha$-stable non degenere pour $\alpha\in(0,2)$ et $\beta>1-\alpha/2$.

Davie’s type uniqueness for a class of SDEs with jumps

The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $\alpha$-Holder drift in the recent literature the rate $\alpha/2$ was proved in many related situations. By exploiting the regularising effect of the noise more efficiently, we show that the rate is in fact arbitrarily close to $1/2$ for all $\alpha>0$. The result extends to Dini continuous coefficients, while in $d=1$ also to all bounded measurable coefficients.

On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift

We analyse the effect of a generic continuous additive perturbation to the well-posedness of ordinary differential equations. Genericity here is understood in the sense of prevalence. This allows us to discuss these problems in a setting where we do not have to commit ourselves to any restrictive assumption on the statistical properties of the perturbation. The main result is that a generic continuous perturbation renders the Cauchy problem well-posed for arbitrarily irregular vector fields. Therefore we establish regularisation by noise "without probability".

Noiseless regularisation by noise

Estimates of the Kantorovich norm on manifolds

Lower bounds for the Kantorovich distance

We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space.

Sobolev functions on infinite-dimensional domains

New inequalities relating Sobolev and Kantorovich norms for functions on Riemannian manifolds satisfying certain curvature conditions are obtained.

On inequalities relating the Sobolev and Kantorovich norms

On extensions of Sobolev functions on the wiener space

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