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Chernoff approximations of Feller semigroups in Riemannian manifolds

Author

Listed:
  • Sonia Mazzucchi
  • Valter Moretti
  • Ivan Remizov
  • Oleg Smolyanov

Abstract

Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non‐compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non‐compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks.

Suggested Citation

  • Sonia Mazzucchi & Valter Moretti & Ivan Remizov & Oleg Smolyanov, 2023. "Chernoff approximations of Feller semigroups in Riemannian manifolds," Mathematische Nachrichten, Wiley Blackwell, vol. 296(3), pages 1244-1284, March.
    • Handle: RePEc:bla:mathna:v:296:y:2023:i:3:p:1244-1284
    DOI: 10.1002/mana.202100291
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    References listed on IDEAS

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    1. Marcelo Disconzi & Yuanzhen Shao & Gieri Simonett, 2016. "Some remarks on uniformly regular Riemannian manifolds," Mathematische Nachrichten, Wiley Blackwell, vol. 289(2-3), pages 232-242, February.
    2. Remizov, Ivan D., 2018. "Approximations to the solution of Cauchy problem for a linear evolution equation via the space shift operator (second-order equation example)," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 243-246.
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