Fundamental solutions for Kolmogorov-Fokker-Planck operators with time-depending measurable coefficients

Abstract

We consider a Kolmogorov-Fokker-Planck operator of the kind: \[ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}\left( t\right) \partial_{x_{i}x_{j}} ^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u,\qquad (x,t)\in\mathbb{R}^{N+1} \] where $\left\{ a_{ij}\left(t\right) \right\} _{i,j=1}^{q}$ is a symmetric uniformly positive matrix on $\mathbb{R}^{q}$, $q\leq N$, of bounded measurable coefficients defined for $t\in\mathbb{R}$ and the matrix $B=\left\{ b_{ij}\right\} _{i,j=1}^{N}$ satisfies a structural assumption which makes the corresponding operator with constant $a_{ij}$ hypoelliptic. We construct an explicit fundamental solution $\Gamma$ for $\mathcal{L}$, study its properties, show a comparison result between $\Gamma$ and the fundamental solution of some model operators with constant $a_{ij}$, and show the unique solvability of the Cauchy problem for $\mathcal{L}$ under various assumptions on the initial datum.

Keywords

• degenerate kolmogorov-fokker-planck operators
• fundamental solution
• cauchy problem
• hypoellipticity