Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha$-stable noise

TL;DR

This paper develops a new parametrix method to construct and estimate the transition probability density of solutions to certain SDEs driven by alpha-stable noise, including cases with non-zero drift and alpha less than or equal to one.

Contribution

It introduces a novel parametrix approach capable of handling SDEs with alpha-stable noise and non-zero drift for the full range of alpha in (0,2], including the challenging case alpha ≤ 1.

Findings

Constructed the transition density for the SDE solution.

Derived two-sided estimates for the transition density.

Extended parametrix method to handle non-zero drift and alpha ≤ 1.

Abstract

Let $L:= -a(x) (-\Delta)^{\alpha/2}+ (b(x), \nabla)$, where $\alpha\in (0,2)$, and $a:\rd\to (0,\infty)$, $b: \rd\to \rd$. Under certain regularity assumptions on the coefficients $a$ and $b$, we associate with the $C_\infty(\rd)$-closure of $(L, C_\infty^2(\rd))$ a Feller Markov process $X$, which possesses a transition probability density $p_t(x,y)$. To construct this transition probability density and to obtain the two-sided estimates on it, we develop a new version of the parametrix method, which allows us to handle the case $0<\alpha\leq 1$ and $b\neq 0$, i.e. when the gradient part of the generator is not dominated by the jump part..