## Local Polya fluctuations of Riesz gravitational fields and the Cauchy problem

##### Abstract

We consider a pseudodifferential equation of parabolic type with a fractional power of the
Laplace operator of order a ∈ (0; 1) acting with respect to the spatial variable. This equation naturally
generalizes the well-known fractal diffusion equation. It describes the local interaction of
moving objects in the Riesz gravitational field. A simple example of such system of objects is stellar
galaxies, in which interaction occurs according to Newton’s gravitational law. The Cauchy problem
for this equation is solved in the class of continuous bounded initial functions. The fundamental
solution of this problem is the Polya distribution of probabilities Pa(F) of the force F of local interaction
between these objects. With the help of obtained solution estimates the correct solvability
of the Cauchy problem on the local field fluctuation coefficient under certain conditions is determined.
In this case, the form of its classical solution is found and the properties of its smoothness
and behavior at the infinity are studied. Also, it is studied the possibility of local strengthening of
convergence in the initial condition. The obtained results are illustrated on the a-wandering model
of the L´evy particle in the Euclidean space R3 in the case when the particle starts its motion from
the origin. The probability of this particle returning to its starting position is investigated. In particular,
it established that this probability is a descending to zero function, and the particle “leaves”
the space R3.