On some perturbations of a symmetric stable process and the corresponding Cauchy problems

M. M. Osypchuk
Theory of Stochastic Processes
Vol.21 (37), no.1, 2016, pp.64-72
A semigroup of linear operators on the space of all continuous bounded functions given on a $d$-dimensional Euclidean space $\mathbb{R}^d$ is constructed such that its generator can be written in the following form $ \mathbf{A}+(a(\cdot),\mathbf{B}), $ where $\mathbf{A}$ is the generator of a symmetric stable process in $\mathbb{R}^d$ with the exponent $\alpha\in(1,2]$, $\mathbf{B}$ is the operator that is determined by the equality $\mathbf{A}=c\ \mathbf{div}(\mathbf{B})$ ($c>0$ is a given parameter), and a given $\mathbb{R}^d$-valued function $a\in L_p(\mathbb{R}^d)$ for some $p>d+\alpha$ (the case of $p=+\infty$ is not exclusion). However, there is no Markov process in $\mathbb{R}^d$ corresponding to this semigroup because it does not preserve the property of a function to take on only non-negative values. We construct a solution of the Cauchy problem for the parabolic equation $\frac{\partial u}{\partial t}=(\mathbf{A}+(a(\cdot),\mathbf{B}))u$.