Statistical aspects of the fractional stochastic calculus

We apply the techniques of stochastic integration with respect to fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by a fractional Brownian motion with any level of H\"{o}lder-regularity (any Hurst parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We also prove that a version of the MLE using only discrete observations is still a strongly consistent estimator.Comment: Published at http://dx.doi.org/10.1214/009053606000001541 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonlinear Radiation Pressure and Stochasticity in Ultraintense Laser Fields

The radiation force on a single electron in an ultraintense plane wave ($a = eE/mc\omega \sim 1$) is calculated and shown to be proportional to $a^4$ in the high-$a$ limit for arbitrary waveform and polarization. The cyclotron motion of an electron in a constant magnetic field and an ultraintense plane wave is numerically found to be quasiperiodic even in the high-$a$ limit if the magnetic field is not too strong, as suggested by previous analytical work. A strong magnetic field causes highly chaotic electron motion and the boundary of the highly chaotic region of parameter space is determined numerically. Applications to experiments and astrophysics are briefly discussed.Comment: 5 pages, 4 figures; uses RevTex, epsf macros. Corrected, expanded versio

Equation of motion for relativistic compact binaries with the strong field point particle limit: Third post-Newtonian order

An equation of motion for relativistic compact binaries is derived through the third post-Newtonian (3 PN) approximation of general relativity. The strong field point particle limit and multipole expansion of the stars are used to solve iteratively the harmonically relaxed Einstein equations. We take into account the Lorentz contraction on the multipole moments defined in our previous works. We then derive a 3 PN acceleration of the binary orbital motion of the two spherical compact stars based on a surface integral approach which is a direct consequence of local energy momentum conservation. Our resulting equation of motion admits a conserved energy (neglecting the 2.5 PN radiation reaction effect), is Lorentz invariant and is unambiguous: there exist no undetermined parameter reported in the previous works. We shall show that our 3 PN equation of motion agrees physically with the Blanchet and Faye 3 PN equation of motion if $\lambda = - 1987/3080$, where $\lambda$ is the parameter which is undetermined within their framework. This value of $\lambda$ is consistent with the result of Damour, Jaranowski, and Sch\"afer who first completed a 3 PN iteration of the ADM Hamiltonian in the ADMTT gauge using the dimensional regularization.Comment: 52 pages, no figure, Appendices B and D added. Phys. Rev. D in pres

Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces

This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton-Cabrera-Frank (BCF) type model following the work [Xiang, SIAM J. Appl. Math. 2002]. We prove that as the lattice parameter goes to zero, for a finite time interval, a modified discrete model converges to the strong solution of the limiting PDE with first order convergence rate.Comment: 52 page

Noise-induced macroscopic bifurcations in globally-coupled chaotic units

Large populations of globally-coupled identical maps subjected to independent additive noise are shown to undergo qualitative changes as the features of the stochastic process are varied. We show that for strong coupling, the collective dynamics can be described in terms of a few effective macroscopic degrees of freedom, whose deterministic equations of motion are systematically derived through an order parameter expansion.Comment: Phys. Rev. Lett., accepte