Loss of regularity for Kolmogorov equations

Abstract

The celebrated Hörmander condition is a sufficient (and nearly necessary) condition for a second-order linear Kolmogorov partial differential equation (PDE) with smooth coefficients to be hypoelliptic. As a consequence, the solutions of Kolmogorov PDEs are smooth at all positive times if the coefficients of the PDE are smooth and satisfy Hörmander’s condition even if the initial function is only continuous but not differentiable. First-order linear Kolmogorov PDEs with smooth coefficients do not have this smoothing effect but at least preserve regularity in the sense that solutions are smooth if their initial functions are smooth. In this article, we consider the intermediate regime of nonhypoelliptic second-order Kolmogorov PDEs with smooth coefficients. The main observation of this article is that there exist counterexamples to regularity preservation in that case. More precisely, we give an example of a second-order linear Kolmogorov PDE with globally bounded and smooth coefficients and a smooth initial function with compact support such that the unique globally bounded viscosity solution of the PDE is not even locally Hölder continuous. From the perspective of probability theory, the existence of this example PDE has the consequence that there exists a stochastic differential equation (SDE) with globally bounded and smooth coefficients and a smooth function with compact support which is mapped by the corresponding transition semigroup to a function which is not locally Hölder continuous. In other words, degenerate noise can have a roughening effect. A further implication of this loss of regularity phenomenon is that numerical approximations may converge without any arbitrarily small polynomial rate of convergence to the true solution of the SDE. More precisely, we prove for an example SDE with globally bounded and smooth coefficients that the standard Euler approximations converge to the exact solution of the SDE in the strong and numerically weak sense, but at a rate that is slower then any power law.