219 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
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
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
Euler-Lagrangian approach to 3D stochastic Euler equations
3D stochastic Euler equations with a special form of multiplicative noise are
considered. A Constantin-Iyer type representation in Euler-Lagrangian form is
given, based on stochastic characteristics. Local existence and uniqueness of
solutions in suitable Hoelder spaces is proved from the Euler-Lagrangian
formulation
Noise based on vortex structures in 2D and 3D
A new noise, based on vortex structures in 2D (point vortices) and 3D (vortex filaments), is introduced. It is defined as the scaling limit of a jump process, which explores vortex structures, and it can be defined in any domain, also with boundary. The link with fractional Gaussian fields and Kraichnan noise is discussed. The vortex noise is finally shown to be suitable for the investigation of the eddy dissipation produced by small scale turbulence
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