# Noise Prevents Singularities in Linear Transport Equations

@article{Fedrizzi2013NoisePS,
title={Noise Prevents Singularities in Linear Transport Equations},
author={Ennio Fedrizzi and Franco Flandoli},
journal={Journal of Functional Analysis},
year={2013},
volume={264},
pages={1329-1354}
}
• Published 24 May 2012
• Mathematics
• Journal of Functional Analysis
122 Citations
Remarks on the stochastic transport equation with Holder drift
• Mathematics
• 2012
We consider a stochastic linear transport equation with a globally Holder continuous and bounded vector field. Opposite to what happens in the deterministic case where shocks may appear, we show that
Well-Posedness of the Stochastic Transport Equation with Unbounded Drift
• Mathematics
• 2015
The Cauchy problem for a multidimensional linear transport equation with unbounded drift is investigated. Provided the drift is Holder continuous , existence, uniqueness and strong stability of
Impacts of noise on a class of partial differential equations
• Mathematics
Journal of Differential Equations
• 2015
Stochastic transport equation in bounded domains
• Mathematics
• 2014
This paper is concerned with the initial-boundary value problem \; for stochastic transport equations in bounded domains. For a given stochastic perturbation of the drift vector field, we prove
Regularization by Noise in One-Dimensional Continuity Equation
A linear stochastic continuity equation with non-regular coefficients is considered. We prove existence and uniqueness of strong solution, in the probabilistic sense, to the Cauchy problem when the
On a class of stochastic transport equations for L2loc vector fields
• Mathematics
• 2014
We study in this article the existence and uniqueness of solutions to a class of stochastic transport equations with irregular coefficients. Asking only boundedness of the divergence of the
Regularity of Stochastic Kinetic Equations
• Mathematics
• 2016
We consider regularity properties of stochastic kinetic equations with multiplicative noise and drift term which belongs to a space of mixed regularity ($L^p$-regularity in the velocity-variable and
Rough linear PDE’s with discontinuous coefficients – existence of solutions via regularization by fractional Brownian motion
We consider two related linear PDE's perturbed by a fractional Brownian motion. We allow the drift to be discontinuous, in which case the corresponding deterministic equation is ill-posed. However,
Global existence and non-existence of stochastic parabolic equations
• Mathematics
• 2019
This paper is concerned with the blowup phenomenon of stochastic parabolic equations both on bounded domain and in the whole space. We introduce a new method to study the blowup phenomenon on bounded
Initial-boundary value problem for stochastic transport equations
• Mathematics
• 2020
This paper concerns the Dirichlet initial-boundary value problem for stochastic transport equations with non-regular coefficients. First, the existence and uniqueness of the strong stochastic traces

## References

SHOWING 1-10 OF 27 REFERENCES
Well-posedness of the transport equation by stochastic perturbation
• Mathematics
• 2008
We consider the linear transport equation with a globally Hölder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the
Strong solutions of stochastic equations with singular time dependent drift
• Mathematics
• 2005
Abstract.We prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time. We only assume local
Stochastic Homeomorphism Flows of SDEs with Singular Drifts and Sobolev Diffusion Coefficients
In this paper we prove the stochastic homeomorphism flow property and the strong Feller property for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion
STOCHASTIC FLOWS OF DIFFEOMORPHISMS FOR ONE-DIMENSIONAL SDE WITH DISCONTINUOUS DRIFT
The existence of a stochastic flow of class $C^{1,\alpha}$, for $\alpha < 1/2$, for a 1-dimensional SDE will be proved under mild conditions on the regularity of the drift. The diffusion coefficient
Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise
• Mathematics
• 2005
We study the influence of a multiplicative Gaussian noise, white in time and correlated in space, on the blow-up phenomenon in the supercritical nonlinear Schrodinger equation. We prove that any
Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift
• Mathematics
• 2013
We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing
FINITE-TIME BLOW-UP IN THE ADDITIVE SUPERCRITICAL STOCHASTIC NONLINEAR SCHRÖDINGER EQUATION : THE REAL NOISE CASE
We review some results concerning the apparition of finite time singularities in nonlinear Schrödinger equations with a Gaussian additive noise which is white in time and correlated in space. We then
On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation
• Mathematics
• 2002
Abstract We investigate the influence of a random perturbation of white noise type on the finite time blow up of solutions of a focusing supercritical nonlinear Schrödinger equation. We prove that,
Ordinary differential equations, transport theory and Sobolev spaces
• Mathematics
• 1989
SummaryWe obtain some new existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces. These results are deduced from corresponding results on
Pathwise uniqueness and continuous dependence for SDEs with non-regular drift
• Mathematics
• 2011
A new proof of a pathwise uniqueness result of Krylov and Röckner is given. It concerns SDEs with drift having only certain integrability properties. In spite of the poor regularity of the drift,