Rough linear PDE’s with discontinuous coefficients – existence of solutions via regularization by fractional Brownian motion

Nilssen, Torstein

doi:10.1214/20-ejp437

Cited by 4 publications

(4 citation statements)

References 24 publications

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“…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. Finally, [52] and [55] treat the case of bounded measurable drift and, in [55], fractional Brownian motion (the classical work under this condition on pathwise uniqueness for the sDE is [64]), again starting from a weak differentiability result for stochastic flows, proved however with methods different from [26].…”

Section: Regularity Results For the Spdesmentioning

confidence: 99%

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

Beck

Flandoli

Gubinelli

et al. 2019

Electron. J. Probab.

137

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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.

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Section: Regularity Results For the Spdesmentioning

confidence: 99%

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

Beck

Flandoli

Gubinelli

et al. 2019

Electron. J. Probab.

137

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“…x , without assuming any regularity on the distribution b. To the best of our knowledge, this case has never been considered in literature so far; although perturbed linear PDEs have been previously treated in [11,43], it is always assumed therein at least b 2 L 1 t;x (which can be treated analogously to Section 5.1). However, our approach in the "Young regime", namely for time regularity > 1=2, is undoubtedly similar (and even simpler) to that in the "rough regime" 2 .1=3; 1=2 treated in [5].…”

Section: The Case Of Distributional Bmentioning

confidence: 99%

Noiseless regularisation by noise

Galeati¹,

Gubinelli²

2021

Rev. Mat. Iberoam.

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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".

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“…To name a few, this s filtering theory [34], McKean-Vlasov equations [41], or pathwise stochastic control problems (see for instance [11,Example 2] and references therein). In the more general context of a degenerate left hand side, this type of noise appears in stochastic transport equations (with X 0 = 0), where a regularization by noise phenomenon is observed [12,22,52,54], or in stochastic conservation laws, see [33] for an overview. We also mention the works [9,15] where the authors solve an equation similar to (1.1), with the difference that they consider a vector field X i t (x) which is rough with respect to the space-like variable.…”

Section: Introductionmentioning

confidence: 99%

An Itô Formula for rough partial differential equations and some applications

Hocquet

Nilssen

2020

Potential Anal

Self Cite

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We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form ∂ t u−A t u−f = (Ẋ t (x)•∇+Ẏ t (x))u on [0, T ]×R d. To do so, we introduce a concept of "differential rough driver", which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces W k,p. We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e. when Y = 0), we use this framework to prove an Itô Formula (in the sense of a chain rule) for Nemytskii operations of the form u → F (u), where F is C 2 and vanishes at the origin. Our method is based on energy estimates, and a generalization of the Moser Iteration argument to prove boundedness of a dense class of solutions of parabolic problems as above. In particular, we avoid the use of flow transformations and work directly at the level of the original equation. We also show the corresponding chain rule for F (u) = |u| p with p ≥ 2, but also when Y = 0 and p ≥ 4. As an application of these results, we prove existence and uniqueness of a suitable class of L p-solutions of parabolic equations with multiplicative noise. Another related development is the homogeneous Dirichlet boundary problem on a smooth domain, for which a weak maximum principle is shown under appropriate assumptions on the coefficients.

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Rough linear PDE’s with discontinuous coefficients – existence of solutions via regularization by fractional Brownian motion

Cited by 4 publications

References 24 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

Noiseless regularisation by noise

An Itô Formula for rough partial differential equations and some applications

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