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Brownian subordinators and fractional Cauchy problems
A Brownian time process is a Markov process subordinated to the absolute
value of an independent one-dimensional Brownian motion. Its transition
densities solve an initial value problem involving the square of the generator
of the original Markov process. An apparently unrelated class of processes,
emerging as the scaling limits of continuous time random walks, involve
subordination to the inverse or hitting time process of a classical stable
subordinator. The resulting densities solve fractional Cauchy problems, an
extension that involves fractional derivatives in time. In this paper, we will
show a close and unexpected connection between these two classes of processes,
and consequently, an equivalence between these two families of partial
differential equations.Comment: 18 pages, minor spelling correction
Hydrodynamic Nambu Brackets derived by Geometric Constraints
A geometric approach to derive the Nambu brackets for ideal two-dimensional
(2D) hydrodynamics is suggested. The derivation is based on two-forms with
vanishing integrals in a periodic domain, and with resulting dynamics
constrained by an orthogonality condition. As a result, 2D hydrodynamics with
vorticity as dynamic variable emerges as a generic model, with conservation
laws which can be interpreted as enstrophy and energy functionals. Generalized
forms like surface quasi-geostrophy and fractional Poisson equations for the
stream-function are also included as results from the derivation. The formalism
is extended to a hydrodynamic system coupled to a second degree of freedom,
with the Rayleigh-B\'{e}nard convection as an example. This system is
reformulated in terms of constitutive conservation laws with two additive
brackets which represent individual processes: a first representing inviscid 2D
hydrodynamics, and a second representing the coupling between hydrodynamics and
thermodynamics. The results can be used for the formulation of conservative
numerical algorithms that can be employed, for example, for the study of fronts
and singularities.Comment: 12 page
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