PERTURBATION OF AN α-STABLE STOCHASTIC PROCESS AND THE FUNDAMENTAL SOLUTION OF A CERTAIN PARABOLIC PSEUDODIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS

Abstract

Using the method of perturbation of a rotationally invariant alpha-stable stochastic process, an explicit representation of the fundamental solution of the pseudodifferential equation \(\frac{\partial}{\partial t}u(t,x)=-c(-\Delta)^{\alpha/2}u(t,\cdot)(x)+ (a,\nabla_\beta u(t,\cdot)(x)), t>0, x\in\mathbb{R}^d\), where \(\Delta\) is the Laplace operator, \(\nabla_\beta\) is the "pseudogradient" operator of order \(0<\beta<\alpha\) (given by the symbol \(i|\lambda|^{\beta-1}\lambda\)). The case of fixed \(0<\alpha<2,\) \(c>0\) and constant \(a\in\mathbb{R}^d\) is considered.