Baire category results for quasi–copulas

The aim of this manuscript is to determine the relative size of several functions (copulas, quasi– copulas) that are commonly used in stochastic modeling. It is shown that the class of all quasi–copulas that are (locally) associated to a doubly stochastic signed measure is a set of first category in the class of all quasi– copulas. Moreover, it is proved that copulas are nowhere dense in the class of quasi-copulas. The results are obtained via a checkerboard approximation of quasi–copulas.

ON THE HAHN–JORDAN DECOMPOSITION FOR SIGNED MEASURE VALUED STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Let N-point particles be distributed over ℝd, d ∈ ℕ. The position of the ith particle at time t will be denoted r(t, qi) where qi is the position at t = 0. mi ∈ ℝ\{0} is the "weight" of the ith particle. Let δr be the point measure concentrated in r and [Formula: see text] the initial mass distribution of the N-point particles. The empirical mass distribution (also called the "empirical process") at time t is then given by [Formula: see text] i.e. by the N-particle flow. In Kotelenez (2008) the weights are positive and the motion of the positions of the point particles is described by a stochastic ordinary differential equation (SODE). Further, the resulting empirical process is the solution of a stochastic partial differential equation (SPDE) which, by a continuum limit, can be extended to an SPDE in smooth positive measures. Some generalizations to the case of signed measures with applications in 2D fluid mechanics have been made. Kotelenez (2010) shows that the signed measure valued solutions of the SPDEs preserve the Hahn–Jordan decomposition of the initial distributions under the assumption that the coefficients of the SODEs are Lipschitz and that the flows of SODEs are smooth. Employing a certain metric on the signed measures, we prove the preservation of the Hahn–Jordan decomposition assuming only Lipschitz conditions on the coefficients of the SODEs.

ENTROPY AND RENORMALIZED SOLUTIONS FOR THE p(x)-LAPLACIAN EQUATION WITH MEASURE DATA

In this paper we prove the existence and uniqueness of both entropy solutions and renormalized solutions for the p(x)-Laplacian equation with variable exponents and a signed measure in L1(Ω)+W−1,p′(⋅)(Ω). Moreover, we obtain the equivalence of entropy solutions and renormalized solutions.

ON SOME SIMILARITIES AND DIFFERENCES BETWEEN FRACTIONAL PROBABILITY DENSITY SIGNED MEASURE OF PROBABILITY AND QUANTUM PROBABILITY

A probability density of fractional (or fractal) order is defined by the probability increment pr{x < X ≤ x+dx} = pα(x)(dx)α, 0 < α < 1, and appears to be quite suitable to deal with random variables defined in a fractal space. Combining this definition with the fractional Taylor's series [Formula: see text] denotes the Mittag–Leffler function) provided by the modified Riemann–Liouville derivative, one can expand a probability calculus parallel to the standard one. This approach could be considered as a framework for the derivation of some space fractional partial differential diffusion equations in coarse-grained spaces. It is shown firstly that there is some relation between fractional probability and signed measure of probability, and secondly that when α = 1/2, there is some identity between this fractal probability and quantum probability. Shortly, a wavefunction could be thought of as a fractal probability density of order 1/2. One exhibits further relations with possibility theory and relative information. Lastly, one arrives at a new informational entropy based on the inverse of the Mittag–Leffler function.