Loss of regularity for Kolmogorov equations

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Title: Loss of regularity for Kolmogorov equations
Author(s): Hairer, M
Hutzenthaler, M
Jentzen, A
Item Type: Journal Article
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.
Publication Date: 2-Feb-2015
Date of Acceptance: 1-Feb-2015
URI: http://hdl.handle.net/10044/1/55728
DOI: https://dx.doi.org/10.1214/13-AOP838
ISSN: 0091-1798
Publisher: Institute of Mathematical Statistics
Start Page: 468
End Page: 527
Journal / Book Title: Annals of Probability
Volume: 43
Issue: 2
Copyright Statement: © Institute of Mathematical Statistics, 2015
Keywords: Science & Technology
Physical Sciences
Statistics & Probability
Mathematics
Kolmogorov equation
loss of regularity
roughening effect
smoothing effect
hypoellipticity
Hormander condition
viscosity solution
degenerate noise
nonglobally Lipschitz continuous
STOCHASTIC DIFFERENTIAL-EQUATIONS
LIPSCHITZ CONTINUOUS COEFFICIENTS
HAMILTON-JACOBI EQUATIONS
STRONG-CONVERGENCE RATES
STEP-SIZE CONTROL
DIFFUSION-COEFFICIENTS
UNIFORM APPROXIMATION
VISCOSITY SOLUTIONS
BACKWARD EULER
SYSTEMS
math.AP
math.AP
math.PR
Science & Technology
Physical Sciences
Statistics & Probability
Mathematics
Kolmogorov equation
loss of regularity
roughening effect
smoothing effect
hypoellipticity
Hormander condition
viscosity solution
degenerate noise
nonglobally Lipschitz continuous
STOCHASTIC DIFFERENTIAL-EQUATIONS
LIPSCHITZ CONTINUOUS COEFFICIENTS
HAMILTON-JACOBI EQUATIONS
STRONG-CONVERGENCE RATES
STEP-SIZE CONTROL
DIFFUSION-COEFFICIENTS
UNIFORM APPROXIMATION
VISCOSITY SOLUTIONS
BACKWARD EULER
SYSTEMS
0104 Statistics
Statistics & Probability
Publication Status: Published
Appears in Collections:Pure Mathematics
Mathematics
Faculty of Natural Sciences



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