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Derivative Formula and Gradient Estimates for Gruschin Type Semigroups

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  • Feng-Yu Wang

    (Beijing Normal University
    Swansea University)

Abstract

By solving a control problem and using Malliavin calculus, an explicit derivative formula is derived for the semigroup P t generated by the Gruschin type operator on ℝ m ×ℝ d : where σ∈C 1(ℝ m ;ℝ d ⊗ℝ d ) might be degenerate. In particular, if σ(x) is comparable with |x| l I d×d for some l≥1 in the sense of (1.5), then for any p>1 there exists a constant C p >0 such that which implies a new Harnack type inequality for the semigroup. A more general model is also investigated.

Suggested Citation

  • Feng-Yu Wang, 2014. "Derivative Formula and Gradient Estimates for Gruschin Type Semigroups," Journal of Theoretical Probability, Springer, vol. 27(1), pages 80-95, March.
    • Handle: RePEc:spr:jotpro:v:27:y:2014:i:1:d:10.1007_s10959-012-0427-2
    DOI: 10.1007/s10959-012-0427-2
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    References listed on IDEAS

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    1. Zhang, Xicheng, 2010. "Stochastic flows and Bismut formulas for stochastic Hamiltonian systems," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1929-1949, September.
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