Abstract
Our concern in this paper is the energy form induced by an eigenfunction of a self-adjoint extension of the restriction of the Laplace operator to \(C_c^\infty ({\mathbb{R}^3}\backslash \{ 0\} )\)). We will prove that this energy form is a regular Dirichlet form with core \(C_c^\infty ({\mathbb{R}^3})\). The associated diffusion X behaves like a 3-dimensional Brownian motion with a mild radial drift when far from 0, subject to an ever-stronger push toward 0 near that point. In particular, {0} is not a polar set with respect to X. The diffusion X is rotation invariant, and admits a skew-product representation before hitting {0}: its radial part is a diffusion on (0, ∞) and its angular part is a time-changed Brownian motion on the sphere S2. The radial part of X is a “reflected” extension of the radial part of X0 (the part process of X before hitting {0}). Moreover, X is the unique reflecting extension of X0, but X is not a semi-martingale.
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Acknowledgements
The second author was supported by National Natural Science Foundation of China (Grant Nos. 11688101 and 11801546) and Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (Grant No. 2008DP173182).
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In Memory of Professor Kai Lai Chung on the 100th Anniversary of His Birth
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Fitzsimmons, P.J., Li, L. On the Dirichlet form of three-dimensional Brownian motion conditioned to hit the origin. Sci. China Math. 62, 1477–1492 (2019). https://doi.org/10.1007/s11425-017-9400-8
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DOI: https://doi.org/10.1007/s11425-017-9400-8