Probabilistic Representation of a Solution of the Cauchy Problem for Evolution Equations with Riemann--Liouville Operators

Abstract

This paper studies properties of probabilistic approximation of a solution of the Cauchy problem for evolution equations with fractional differential operators of order more than two. To this end we construct analogous one-sided $\alpha$-stable distributions for noninteger $\alpha>2$. Although densities of these distributions are signed functions, using generalized functions methods, it is possible to give them an exact probability sense.