Mykhailo Osypchuk | Searcharxiv

arXiv2024

Perturbation of an alpha-stable type stochastic process by a pseudo-gradient

We consider the Markov process defined by some pseudo-differential operator of the order $1<α<2$ as the process generator. Using a pseudo-gradient operator, that is, the operator defined by the symbol $iλ|λ|^{β-1}$ with some $0<β<α$, the perturbation of the Markov process by the pseudo-gradient with a multiplier integrable at some great enough power is constructed. Such perturbation defines a family of evolution operators, the properties of which are investigated.

math.PR↗

arXiv2023

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

In the paper, the transition probability density of 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 $|λ|^\varkappa$ and $λ|λ|^{\varkappa-1}$, where $\varkappa$ is some constant. The first operator with negative sign is fractional Laplacian and the second one multiplied by imaginary unit is fractional gradient.

math.PR↗