803 research outputs found
Uniform in time estimates for the weak error of the Euler method for SDEs and a Pathwise Approach to Derivative Estimates for Diffusion Semigroups
We present a criterion for uniform in time convergence of the weak error of
the Euler scheme for Stochastic Differential equations (SDEs). The criterion
requires i) exponential decay in time of the space-derivatives of the semigroup
associated with the SDE and ii) bounds on (some) moments of the Euler
approximation. We show by means of examples (and counterexamples) how both i)
and ii) are needed to obtain the desired result. If the weak error converges to
zero uniformly in time, then convergence of ergodic averages follows as well.
We also show that Lyapunov-type conditions are neither sufficient nor necessary
in order for the weak error of the Euler approximation to converge uniformly in
time and clarify relations between the validity of Lyapunov conditions, i) and
ii).
Conditions for ii) to hold are studied in the literature. Here we produce
sufficient conditions for i) to hold. The study of derivative estimates has
attracted a lot of attention, however not many results are known in order to
guarantee exponentially fast decay of the derivatives. Exponential decay of
derivatives typically follows from coercive-type conditions involving the
vector fields appearing in the equation and their commutators; here we focus on
the case in which such coercive-type conditions are non-uniform in space. To
the best of our knowledge, this situation is unexplored in the literature, at
least on a systematic level. To obtain results under such space-inhomogeneous
conditions we initiate a pathwise approach to the study of derivative estimates
for diffusion semigroups and combine this pathwise method with the use of Large
Deviation Principles.Comment: 47 pages and 9 figure
Solution properties of a 3D stochastic Euler fluid equation
We prove local well-posedness in regular spaces and a Beale-Kato-Majda
blow-up criterion for a recently derived stochastic model of the 3D Euler fluid
equation for incompressible flow. This model describes incompressible fluid
motions whose Lagrangian particle paths follow a stochastic process with
cylindrical noise and also satisfy Newton's 2nd Law in every Lagrangian domain.Comment: Final version! Comments still welcome! Send email
Fluctuation conductivity in layered d-wave superconductors near critical disorder
We consider the fluctuation conductivity in the critical region of a disorder
induced quantum phase transition in layered d-wave superconductors. We
specifically address the fluctuation contribution to the system's conductivity
in the limit of large (quasi-two-dimensional system) and small
(quasi-three-dimensional system) separation between adjacent layers of the
system. Both in-plane and c-axis conductivities were discussed near the point
of insulator-superconductor phase transition. The value of the dynamical
critical exponent, , permits a perturbative treatment of this quantum
phase transition under the renormalization group approach. We discuss our
results for the system conductivities in the critical region as function of
temperature and disorder.Comment: Final version to be published in Eur. Phys. J.
Bose-Einstein condensation of magnons
We use the Renormalization Group method to study the Bose-Einstein
condensation of the interacting dilute magnons which appears in three
dimensional spin systems in magnetic field. The obtained temperature dependence
of the critical field is different from the recent
self-consistent Hartree-Fock-Popov calculations (cond-mat/0405422) in which a
dependence was reported . The origin of this difference is discussed
in the framework of quantum critical phenomena.Comment: 11 pages, revtex
Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes
We study the long time behaviour of a large class of diffusion processes on
, generated by second order differential operators of (possibly)
degenerate type. The operators that we consider {\em need not} satisfy the
H\"ormander condition. Instead, they satisfy the so-called UFG condition,
introduced by Herman, Lobry and Sussman in the context of geometric control
theory and later by Kusuoka and Stroock, this time with probabilistic
motivations. In this paper we study UFG diffusions and demonstrate the
importance of such a class of processes in several respects: roughly speaking
i) we show that UFG processes constitute a family of SDEs which exhibit
multiple invariant measures and for which one is able to describe a systematic
procedure to determine the basin of attraction of each invariant measure
(equilibrium state). ii) We use an explicit change of coordinates to prove that
every UFG diffusion can be, at least locally, represented as a system
consisting of an SDE coupled with an ODE, where the ODE evolves independently
of the SDE part of the dynamics. iii) As a result, UFG diffusions are
inherently "less smooth" than hypoelliptic SDEs; more precisely, we prove that
UFG processes do not admit a density with respect to Lebesgue measure on the
entire space, but only on suitable time-evolving submanifolds, which we
describe. iv) We show that our results and techniques, which we devised for UFG
processes, can be applied to the study of the long-time behaviour of
non-autonomous hypoelliptic SDEs and therefore produce several results on this
latter class of processes as well. v) Because processes that satisfy the
(uniform) parabolic H\"ormander condition are UFG processes, our paper contains
a wealth of results about the long time behaviour of (uniformly) hypoelliptic
processes which are non-ergodic, in the sense that they exhibit multiple
invariant measures.Comment: 66 page
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