Bilateral estimates of some pseudo-derivatives of the transition probability density of an isotropic α-stable stochastic process

Keywords:
stable process, Green's function, fractional Laplacian, fractional gradientAbstract
In the paper, the transition probability density of an isotropic α-stable stochastic process in a finite dimensional Euclidean space is considered. The results of applying pseudo-differential operators with respect spatial variables to this function are estimated from the both side: above and below. Operators in the consideration are defined by the symbols |λ|ϰ and λ|λ|ϰ−1, where ϰ is some constant. The first operator with negative sign is fractional Laplacian and the second one multiplied by imaginary unit is fractional gradient.