A practical criterion for positivity of transition densities
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© 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).
Published Version (Please cite this version)10.1088/0951-7715/28/8/2823
Publication InfoHerzog, DP; & Mattingly, JC (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.
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James B. Duke Distinguished Professor
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