Diffusive behaviour of some linear kinetic equations

We consider linear kinetic equations of the form $\partial_t f + \frac{1}{\epsilon} v \nabla_x f = \frac{1}{\epsilon^2} L(f)$, for an unknown $f$ which depends on time $t$, position $x$ and velocity $v$, and where $L$ is a linear operator which acts only in the velocity variable, and which typically has a probability equilibrium in $v$. Important examples include the Fokker-Planck operator, nonlocal diffusion operators, linear BGK-type operators, or linear Boltzmann operators. This PDE typically represents a mesoscopic physical model, where we keep track of the probability distribution of the position and velocity of particles. It is well known that when $\epsilon$ tends to $0$, this type of equation has a macroscopic or diffusive limit for the density $\rho(t,x) := \int f(t,x,v) dv$, which is either the standard heat equation, or the fractional heat equation. As a new result, we show that for a fixed epsilon, the behaviour of this equation for large times also follows the standard or fractional heat equation, and that the long-time and small-epsilon limits are actually interchangeable in many cases. This is a work in collaboration with Stéphane Mischler (U. Paris-Dauphine) and Niccolò Tassi (U. Granada).

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