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Loss of regularity for Kolmogorov equations

by M. Hairer and M. Hutzenthaler and A. Jentzen

(Report number 2013-37)

Abstract
The celebrated Hormander condition is a sucient (and nearly necessary) condition for a second-order linear Kolmogorov partial dierential equation (PDE) with smooth coecients to be hypoelliptic. As a consequence, the solutions of Kolmogorov PDEs are smooth at all positive times if the coecients of the PDE are smooth and satisfy Hormander's condition even if the initial function is only continuous but not dierentiable. First-order linear Kolmogorov PDEs with smooth coecients do not have this smoothing eect 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 non-hypoelliptic second-order Kolmogorov PDEs with smooth coecients. 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 coecients and a smooth initial function with compact support such that the unique globally bounded viscosity solution of the PDE is not even locally Holder continuous. From the perspective of probability theory the existence of this example PDE has the consequence that there exists a stochastic dierential equation (SDE) with globally bounded and smooth coecients and a smooth function with compact support which is mapped by the corresponding transition semigroup to a non-locally Holder continuous function. In other words, degenerate noise can have a roughening eect. A further implication of this loss of regularity phenomenon is that numerical approximations may convergence 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 coecients 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.

Keywords: Kolmogorov equation, loss of regularity, roughening eect, smoothing eect, hypoellipticity, Hormander condition, viscosity solution, degenerate noise, non-globally Lipschitz continuous

@Techreport{HHJ13_537,
  author = {M. Hairer and M. Hutzenthaler and A. Jentzen},
  title = {Loss of regularity for Kolmogorov equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-37},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-37.pdf },
  year = {2013}
}

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