542 research outputs found
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
The Vector Valued Quartile Operator
Certain vector-valued inequalities are shown to hold for a Walsh analog of
the bilinear Hilbert transform. These extensions are phrased in terms of a
recent notion of quartile type of a UMD (Unconditional Martingale Differences)
Banach space X. Every known UMD Banach space has finite quartile type, and it
was recently shown that the Walsh analog of Carleson's Theorem holds under a
closely related assumption of finite tile type. For a Walsh model of the
bilinear Hilbert transform however, the quartile type should be sufficiently
close to that of a Hilbert space for our results to hold. A full set of
inequalities is quantified in terms of quartile type.Comment: 32 pages, 5 figures, incorporates referee's report, to appear in
Collect. Mat
Averaging for SDE-BSDE with null recurrent fast component Application to homogenization in a non periodic media
We establish an averaging principle for a family of
solutions
of a system of
SDE-BSDEwith a null recurrent fast component . Incontrast
to the classical periodic case, we can not rely on aninvariant probability and
the slow forward component cannot be approximated by a
diffusion process.On the other hand, we assume that the coefficients admit a
limit in a\`{C}esaro sense. In such a case, the limit coefficients may
havediscontinuity. We show that we can approximate the
triplet bya
system of SDE-BSDE where is aMarkov diffusion
which is the unique (in law) weak solution of theaveraged forward component and
is the unique solution to the averaged backward component. This is done
with a backward component whosegenerator depends on the variable .
Asapplication, we establish an homogenization result for semilinearPDEs when
the coefficients can be neither periodic nor ergodic. Weshow that the averaged
BDSE is related to the averaged PDE via aprobabilistic representation of the
(unique) Sobolev --solution of the
limitPDEs. Our approach combines PDE methods and probabilistic argumentswhich
are based on stability property and weak convergence of BSDEsin the S-topology
On the rate of convergence of some stochastic processes
We present a general technique for obtaining bounds on the deviation of the optimal value of some stochastic combinatorial problems from their mean. As a particular application, we prove an exponential rate of convergence for the length of a shortest path through n random points in the unit square. This strengthens a previous result of Steele (Steele, J.M. 1981. Complete convergence of short paths and Karp's algorithm for the TPS. Math.Oper.Res.)
- …