A practical criterion for positivity of transition densities
Abstract
© 2015 IOP Publishing Ltd & London Mathematical Society.We establish a simple criterion
for locating points where the transition density of a degenerate diffusion is strictly
positive. Throughout, we assume that the diffusion satisfies a stochastic differential
equation (SDE) on Rd with additive noise and polynomial drift. In this setting, we
will see that it is often the case that local information of the flow, e.g. the Lie
algebra generated by the vector fields defining the SDE at a point x ∈ Rd, determines
where the transition density is strictly positive. This is surprising in that positivity
is a more global property of the diffusion. This work primarily builds on and combines
the ideas of Arous and Lé andre (1991 Décroissance exponentielle du noyau de la chaleur
sur la diagonale. II Probab. Theory Relat. Fields 90 377-402) and Jurdjevic and Kupka
(1985 Polynomial control systems Math. Ann. 272 361-8).
Type
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https://hdl.handle.net/10161/9510Published Version (Please cite this version)
10.1088/0951-7715/28/8/2823Publication Info
Herzog, David P; & Mattingly, Jonathan C (2015). A practical criterion for positivity of transition densities. Nonlinearity, 28(8). pp. 2823-2845. 10.1088/0951-7715/28/8/2823. Retrieved from https://hdl.handle.net/10161/9510.This is constructed from limited available data and may be imprecise. To cite this
article, please review & use the official citation provided by the journal.
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Jonathan Christopher Mattingly
Kimberly J. Jenkins Distinguished University Professor of New Technologies
Jonathan Christopher Mattingly grew up in Charlotte, NC where he attended Irwin Ave
elementary and Charlotte Country Day. He graduated from the NC School of Science
and Mathematics and received a BS is Applied Mathematics with a concentration in physics
from Yale University. After two years abroad with a year spent at ENS Lyon studying
nonlinear and statistical physics on a Rotary Fellowship, he returned to the US to
attend Princeton University where he obtained a PhD in Applied and

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