A Partial Fourier Transform Method for a Class of Hypoelliptic Kolmogorov Equations

Abstract

We consider hypoelliptic Kolmogorov equations in n+1 spatial dimensions with $n\geq 1$, where the differential operator in the first $n$ spatial variables featuring in the equation is second-order elliptic, and with respect to the (n+1)st spatial variable the equation contains a pure transport term only and is therefore first-order hyperbolic. If the two differential operators, in the first $n$ and in the (n+1)st co-ordinate directions, do not commute, we benefit from hypoelliptic regularization in time, and the solution for $t>0$ is smooth even for a Dirac initial datum prescribed at t=0. We study specifically the case where the coefficients depend only on the first $n$ variables. In that case, a Fourier transform in the last variable and standard central finite difference approximation in the other variables can be applied for the numerical solution. We prove second-order convergence in the spatial grid size for the model hypoelliptic equation $\frac{\partial u}{\partial t} + x \frac{\partial u}{\partial y} = \frac{\partial^2 u}{\partial x^2}$ subject to the initial condition $u(x,y,0) = \delta (x) \otimes \delta (y)$ with $(x,y) \in \mathbb{R} \times\mathbb{R}$ and $t>0$, proposed by Kolmogorov, and we also consider an extension of this problem with n=2. We demonstrate exponential convergence of an approximation of the inverse Fourier transform based on the trapezium rule.