We derive concentration inequalities for maxima of empirical processes associated with Poisson point processes. The proofs are based on a careful application Ledoux's entropy method. demonstrate the utility obtained by to adaptive intensity estimation.

Let X be a discrete random variable with support S and f : S → S′ be a bijection. Then it is wellknown that the entropy of X is the same as the entropy of f(X). This entropy preservation property has been well-utilized to establish non-trivial properties of discrete stochastic processes, e.g. queuing process [1]. Entropy as well as entropy preservation is well-defined only in the context of pur...

The SBV regularity of weak entropy solutions to the Burgers-Poisson equation for initial data in L 1 ( R ) is considered. We show that derivative a solution consists only absolutely continuous part and jump part.

Kinetic theory describes a dilute monatomic gas using distribution function $f(q,p,t)$, the expected phase-space density of particles. The evolves according to collisionless Boltzmann equation in high Knudsen number limit. Fluid dynamics is an alternative description hydrodynamic variables that are functions position and time only. These evolve compressible Euler equations inviscid Both systems...

Let (G,μ) be a discrete group equipped with a generating probability measure, and let Γ be a finite index subgroup of G. A μ-random walk on G, starting from the identity, returns to Γ with probability one. Let θ be the hitting measure, or the distribution of the position in which the random walk first hits Γ. We prove that the Furstenberg entropy of a (G,μ)-stationary space, with respect to the...

We consider the possibility that massive primordial black holes are the dominant form of dark matter. Black hole formation generates entropy fluctuations that adds a Poisson noise to the matter power spectrum. We use Ly-α forest observations to constrain this Poisson term in matter power spectrum, then we constrain the mass of black holes to be less than ∼ few × 10 M⊙. We also find that structu...

We announce a deterministic analog of Bartlett’s displacement theorem. The result is that a Poisson property is stable with respect to deterministic Hamiltonian displacements. While the random point configurations move according to an n-body evolution, the mean measure P satisfies a nonlinear Vlasov type equation Ṗ + y · ∇xP − ∇y · E(P ) = 0. Combined with Bartlett’s theorem, the result general...

The asymptotic behavior for the Vlasov–Poisson–Fokker–Planck system in bounded domains is analyzed in this paper. Boundary conditions defined by a scattering kernel are considered. It is proven that the distribution of particles tends for large time to a Maxwellian determined by the solution of the Poisson–Boltzmann equation with Dirichlet boundary condition. In the proof of the main result, th...

The spectral efficiency (channel capacity) of the optical direct-detection channel is studied. The modeling of the optical direct-detection channel as a discrete-time Poisson channel is reviewed. Closed-form integral representations for the entropy of random variables with Poisson and negative binomial distributions are derived. The spectral efficiency achievable with an arbitrary input gamma d...