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In this work we show that the s-harmonic extension in \(\mathbb R^{n+1}_+\) of a function \(u \in \mathbb R^n\) is the solution to an appropriate stochastic Dirichlet problem associated to a Bessel process. We compute the solution to the stochastic Dirichlet problem and we show that it is equal to the convolution between the boundary datum u and the kernel associated to the s-harmonic extension of u. This result gives a stochastic interpretation of the Caffarelli-Silvestre extension theorem for the fractional Laplacian.

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A Note on a Stochastic Approach to Caffarelli–Silvestre Theorem

  • 会议论文

  • 发表日期：

    2026年

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  • Michelangelo Cavina

摘要

In this work we show that the s-harmonic extension in \(\mathbb R^{n+1}_+\) of a function \(u \in \mathbb R^n\) is the solution to an appropriate stochastic Dirichlet problem associated to a Bessel process. We compute the solution to the stochastic Dirichlet problem and we show that it is equal to the convolution between the boundary datum u and the kernel associated to the s-harmonic extension of u. This result gives a stochastic interpretation of the Caffarelli-Silvestre extension theorem for the fractional Laplacian.