On the Fundamental Solution of a Homogeneous Linearized Coagulation Equation

In this paper we study the fundamental solution of the equation obtained by the linearisation of the Smoluchowski coagulation equation with the multiplicative kernel $(x y)^{\lambda/2}$ with $\lambda\in (1, 2)$ around the steady state $f(x)=x^{-(3+\lambda)/2}$. An explicit representation formula as well as detailed estimates on its asymptotics are obtained. We also describe in a detailed form particle fluxes between different sizes for this linearised equation.