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 $\alpha\in(0;1)$ 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 $\mathcal{P}_\alpha(F)$ 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 $\alpha$-wandering model of the Lévy particle in the Euclidean space $\mathbb{R}^3$ 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 $\mathbb{R}^3$.