Noise Prevents Singularities in Linear Transport Equations

By E. Fedrizzi, Franco Flandoli. 2012

172 citations recorded by the source corpus.

A stochastic linear transport equation with multiplicative noise is considered and the question of no-blow-up is investigated. The drift is assumed only integrable to a certain power. Opposite to the deterministic case where smooth initial conditions may develop discontinuities, we prove that a certain Sobolev degree of regularity is maintained, which implies Holder continuity of solutions. The proof is based on a careful analysis of the associated stochastic flow of characteristics.

Subfields: Computational Mechanics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/602e3b4eb61a20b3efc9b68876d0de40dab94d3c

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Citations in the corpus

Regularity of Stochastic Kinetic Equations (2016, methodology): “To prove that some degree of Sobolev regularity of the initial condition is preserved on has to deal with weakly differentiable solutions, according to the definition introduced in [16] for solutions of the stochastic transport equation.”
Wellposedness for stochastic continuity equations with Ladyzhenskaya–Prodi–Serrin condition (2013, methodology): “Also concerning the persistence property, we recall from [10] that a certain Sobolev regularity is maintained under the Ladyzhenskaya-Prodi-Serrin condition, that is, u0 ∈ ⋂”
Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness (2014, methodology): “Similarly as in [43], the proof of regularity of solutions of the sTE from [36] is based on the construction of stochastic flows for the sDE and their regularity in terms of weak differentiability.”
Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws (2014, background): “Indeed, it has been shown by Flandoli, Gubinelli and Priola [27] that adding perturbations of the type (1.15) du+ b(x) · ∇u dt = ∂xu ◦ dβt, regularizes the dynamics in the sense that well-posedness can be obtained for (1.15) in cases for which (1.14) is ill-posed.”
Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients (2014, background): “It should be noticed that in all previous works (see [5, 15, 26]), the argument of mollifying coefficients is used to obtain the Sobolev differentiability of strong solutions.”
Stochastic differential equations with Sobolev diffusion and singular drift and applications (2016, background): “1) with singular drift is partly due to the discovery of Flandoli, Gubinelli and Priola [7] that noises can prevent the singularity for linear transport equations (see also [5]).”
Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation (2012, methodology): “2After completing the preparation of this article, personal communication with Flandoli indicated work in preparation with Fedrizzi [11] on similar issues regarding the regularity of stochastic flows for SDEs, using a different approach.”