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Convergence of Martingales with Jumps on Submanifolds of Euclidean Spaces and its Applications to Harmonic Maps

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  • Fumiya Okazaki

    (Tohoku University)

Abstract

Martingales with jumps on Riemannian manifolds and harmonic maps with respect to Markov processes are discussed in this paper. Discontinuous martingales on manifolds were introduced in Picard (Séminaire de Probabilités de Strasbourg 25:196–219, 1991). We obtain results about the convergence of martingales with finite quadratic variations on Riemannian submanifolds of higher-dimensional Euclidean space as $$t\rightarrow \infty $$ t → ∞ and as $$t\rightarrow 0$$ t → 0 . Furthermore, we apply the result about martingales with jumps on submanifolds to harmonic maps with respect to Markov processes such as fractional harmonic maps.

Suggested Citation

  • Fumiya Okazaki, 2024. "Convergence of Martingales with Jumps on Submanifolds of Euclidean Spaces and its Applications to Harmonic Maps," Journal of Theoretical Probability, Springer, vol. 37(2), pages 1168-1198, June.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:2:d:10.1007_s10959-023-01273-6
    DOI: 10.1007/s10959-023-01273-6
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