Remark on the analyticity of the fractional Fokker-Planck equation

In this note, we study the Cauchy problem of the fractional Fokker-Planck equation. We prove that the solution to the Cauchy problem enjoys the analytic smoothing effect with an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> {L}² initial datum for positive time, and the evolution of the analytic radius is similar to the heat equation.