703 research outputs found
Weak vorticity formulation of 2D Euler equations with white noise initial condition
The 2D Euler equations with random initial condition distributed as a certain
Gaussian measure are considered. The theory developed by S. Albeverio and A.-B.
Cruzeiro is revisited, following the approach of weak vorticity formulation. A
solution is constructed as a limit of random point vortices. This allows to
prove that it is also limit of L^\infty-vorticity solutions. The result is
generalized to initial measures that have a continuous bounded density with
respect to the original Gaussian measure.Comment: 45 p
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
An infinite-dimensional approach to path-dependent Kolmogorov equations
In this paper, a Banach space framework is introduced in order to deal with
finite-dimensional path-dependent stochastic differential equations. A version
of Kolmogorov backward equation is formulated and solved both in the space of
paths and in the space of continuous paths using the associated
stochastic differential equation, thus establishing a relation between
path-dependent SDEs and PDEs in analogy with the classical case. Finally, it is
shown how to establish a connection between such Kolmogorov equation and the
analogue finite-dimensional equation that can be formulated in terms of the
path-dependent derivatives recently introduced by Dupire, Cont and Fourni\'{e}.Comment: Published at http://dx.doi.org/10.1214/15-AOP1031 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Noise Prevents Singularities in Linear Transport Equations
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 H\"older continuity
of solutions. The proof is based on a careful analysis of the associated
stochastic flow of characteristics
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
Propagation of chaos for interacting particles subject to environmental noise
A system of interacting particles described by stochastic differential
equations is considered. As oppopsed to the usual model, where the noise
perturbations acting on different particles are independent, here the particles
are subject to the same space-dependent noise, similar to the (noninteracting)
particles of the theory of diffusion of passive scalars. We prove a result of
propagation of chaos and show that the limit PDE is stochastic and of inviscid
type, as opposed to the case when independent noises drive the different
particles.Comment: Published at http://dx.doi.org/10.1214/15-AAP1120 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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