Volume 13, Issue 4

We consider some tests of Gaussianity for random fields on a spherical surface, based on regression methods in harmonic space. The asymptotic properties of the suggested procedures are discussed in detail.

In this paper we give a probabilistic representation for the solution of the heat equation of convolution type with a generalized function ƒ as initial condition. The method uses a combination between convolution calculus and the generalized stochastic calculus, namely Itô's formula for generalized functions. Finally, generalization to the stochastic heat equation with a gradient term and generalized coefficients is presented.

Consider a random vector x n = ( x 1 , ..., x n ) in the space R n where { x k , k ≤ n } are independent identically distributed random variables in R with a common distribution F ( dx ) in R . Denote by F n the distribution of x n . Let { x n (1), ... , x n ( m )} be independent identically distributed vectors in R n with the common distribution F n . We investigate asymptotic behavior of empirical distribution in R n which is determined by the relation as n → ∞, m → ∞. The main tool of investigation is the theory of large deviation ( see: Richard S. Ellis, Entropy, Large Deviations and Statistical Mechanics ). We will consider continuous distributions F on a bounded set.

For an additive process ( X t ) t≥0 with values in the dual of a nuclear Fréchet space and for each finite time T > 0, the existence of an equivalent additive process which takes values in a Hilbert space is shown. The additive process takes values in a cone if and only if it has a special Lévy-Khintchine representation and in this case for each T > 0 there exists a pathwise version in some Hilbert space .

We extend to white noise integrals the scalar type integrator inequalities introduced by Accardi, Fagnola and Quaegebeur [1] as a generalization of the Hudson–Parthasarathy basic estimates on stochastic integrals. We use these estimates as "regularity results", showing that some Hida distributions are in fact elements of the Fock space. We also use them to prove an analogue regularity result for solutions of white noise equations with bounded coefficients.

In this paper we study a one-dimensional random motion by having a general Erlang distribution for the sojourn times and we obtain higher order hyperbolic equations for this case. We apply the methodology of random evolutions to find the partial differential equations governing the particle motion and we obtain a factorization of these equations. As a particular case we find the linear biwave equation for the symmetric motion case and 2-Erlang distributions for the sojourn times of a semi-Markov evolution.