The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e., to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in nonequilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.

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Particles and fields in fluid turbulence

This is a slightly expanded version of the text of a lecture I gave in a conference at Rutgers University in honor of David Ruelle and Yasha Sinai. In this lecture I reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle, and Bowen in the 1970s. SRB measures, as these objects are called, play an important role in the ergodic theory of dissipative dynamical systems with chaotic behavior. Roughly speaking,

What Are SRB Measures, and Which Dynamical Systems Have Them?

In this paper we extend the results of Meyn and Tweedie (1992b) from discrete-time parameter to continuous-parameter Markovian processes Φ evolving on a topological space. We consider a number of stability concepts for such processes in terms of the topology of the space, and prove connections between these and standard probabilistic recurrence concepts. We show that these structural results hold for a major class of processes (processes with continuous components) in a manner analogous to discrete-time results, and that complex operations research models such as storage models with state-dependent release rules, or diffusion models such as those with hypoelliptic generators, have this property. Also analogous to discrete time, ‘petite sets', which are known to provide test sets for stability, are here also shown to provide conditions for continuous components to exist. New ergodic theorems for processes with irreducible and countably reducible skeleton chains are derived, and we show that when these conditions do not hold, then the process may be decomposed into an uncountable orbit of skeleton chains.

http://www.meyn.ece.ufl.edu/archive/spm_files/Papers_pdf/meytweII.pdf

Stability of Markovian processes II: continuous-time processes and sampled chains

/pdf/variational-problems-on-flows-of-diffeomorphisms-for-image-223g1q133q.pdf

Variational problems on flows of diffeomorphisms for image matching

The phenomenon of super-resolution in time-reversal acoustics is analyzed theoretically and with numerical simulations. A signal that is recorded and then retransmitted by an array of transducers, propagates back though the medium, and refocuses approximately on the source that emitted it. In a homogeneous medium, the refocusing resolution of the time-reversed signal is limited by diffraction. When the medium has random inhomogeneities the resolution of the refocused signal can in some circumstances beat the diffraction limit. This is super-resolution. A theoretical treatment of this phenomenon is given, and numerical simulations which confirm the theory are presented.

/pdf/super-resolution-in-time-reversal-acoustics-4d3zeqr1bb.pdf

Super-resolution in time-reversal acoustics

1. Stochastic processes and random fields 2. Continuous semimartingales and stochastic integrals 3. Semimartingales with spatial parameter and stochastic integrals 4. Stochastic flows 5. Convergence of stochastic flows 6. Stochastic partial differential equations.

寛国田

Papers

Stochastic flows and stochastic differential equations

Lectures on stochastic flows and applications

Malliavin calculus of canonical stochastic differential equations with jumps

寛 国田

Papers

Stochastic flows and stochastic differential equations

Lectures on stochastic flows and applications

Malliavin calculus of canonical stochastic differential equations with jumps

寛国田