272 research outputs found
Pathwise uniqueness and continuous dependence for SDEs with nonregular drift
A new proof of a pathwise uniqueness result of Krylov and R\"{o}ckner is
given. It concerns SDEs with drift having only certain integrability
properties. In spite of the poor regularity of the drift, pathwise continuous
dependence on initial conditions may be obtained, by means of this new proof.
The proof is formulated in such a way to show that the only major tool is a
good regularity theory for the heat equation forced by a function with the same
regularity of the drift
Markov selections for the 3D stochastic Navier-Stokes equations
We investigate the Markov property and the continuity with respect to the
initial conditions (strong Feller property) for the solutions to the
Navier-Stokes equations forced by an additive noise.
First, we prove, by means of an abstract selection principle, that there are
Markov solutions to the Navier-Stokes equations. Due to the lack of continuity
of solutions in the space of finite energy, the Markov property holds almost
everywhere in time. Then, depending on the regularity of the noise, we prove
that any Markov solution has the strong Feller property for regular initial
conditions.
We give also a few consequences of these facts, together with a new
sufficient condition for well-posedness.Comment: 59 pages; corrected several errors and typos, added reference
Strong uniqueness for stochastic evolution equations with unbounded measurable drift term
We consider stochastic evolution equations in Hilbert spaces with merely
measurable and locally bounded drift term and cylindrical Wiener noise. We
prove pathwise (hence strong) uniqueness in the class of global solutions. This
paper extends our previous paper (Da Prato, Flandoli, Priola and M. Rockner,
Annals of Prob., published online in 2012) which generalized Veretennikov's
fundamental result to infinite dimensions assuming boundedness of the drift
term. As in our previous paper pathwise uniqueness holds for a large class, but
not for every initial condition. We also include an application of our result
to prove existence of strong solutions when the drift is only measurable,
locally bounded and grows more than linearly.Comment: The paper will be published in Journal of Theoretical Probability.
arXiv admin note: text overlap with arXiv:1109.036
Multidimensional stochastic differential equations with distributional drift
This paper investigates a time-dependent multidimensional stochastic differential equation with drift being a distribution in a suitable class of Sobolev spaces with negative derivation order. This is done through a careful analysis of the corresponding Kolmogorov equation whose coefficient is a distribution
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