326 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
Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplicative noise
A linear stochastic transport equation with non-regular coefficients is
considered. Under the same assumption of the deterministic theory, all weak
-solutions are renormalized. But then, if the noise is nondegenerate,
uniqueness of weak -solutions does not require essential new
assumptions, opposite to the deterministic case where for instance the
divergence of the drift is asked to be bounded. The proof gives a new
explanation why bilinear multiplicative noise may have a regularizing effect
Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise
We prove that every Markov solution to the three dimensional Navier-Stokes
equation with periodic boundary conditions driven by additive Gaussian noise is
uniquely ergodic. The convergence to the (unique) invariant measure is
exponentially fast.
Moreover, we give a well-posedness criterion for the equations in terms of
invariant measures. We also analyse the energy balance and identify the term
which ensures equality in the balance.Comment: 32 page
Regularity of transition semigroups associated to a 3D stochastic Navier-Stokes equation
A 3D stochastic Navier-Stokes equation with a suitable non degenerate
additive noise is considered. The regularity in the initial conditions of every
Markov transition kernel associated to the equation is studied by a simple
direct approach. A by-product of the technique is the equivalence of all
transition probabilities associated to every Markov transition kernel.Comment: 17 page
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
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