Browsing by Author "Herzog, David P"

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    A practical criterion for positivity of transition densities
    (Nonlinearity, 2015-07-10) Herzog, David P; Mattingly, Jonathan C
    © 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).
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    Noise-induced stabilization of planar flows ii
    (Electronic Journal of Probability, 2015-10-25) Herzog, David P; Mattingly, Jonathan C
    © 2015 University of Washington. All rights reserved.We continue the work started in Part I [6], showing how the addition of noise can stabilize an otherwise unstable system. The analysis makes use of nearly optimal Lyapunov functions. In this continuation, we remove the main limiting assumption of Part I by an inductive procedure as well as establish a lower bound which shows that our construction is radially sharp. We also prove a version of Peskir’s [7] generalized Tanaka formula adapted to patching together Lyapunov functions. This greatly simplifies the analysis used in previous works.
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    Noise-Induced Stabilization of Planar Flows II
    (ArXiv e-prints, 2014-04) Herzog, David P; Mattingly, Jonathan Christopher