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A relationship between three-dimensional Bessel processes in terms of Pitman's transformation

  • Yuu Hariya
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Abstract

For every x≥ 0, let Rx={ Rxt} t≥ 0 be a three-dimensional Bessel process started at x. Given x>0, with the aid of Pitman's theorem, a path transformation Tx under which the process Tx(Ry) is distributed as Rx for any y>0 is constructed. In fact, we prove that, for every x,y>0, the two three-dimensional processes ( Rx,Ty(Rx),𝒫(−Rx)+x) && and && ( Tx(Ry),Ry,𝒫(−Ry)+y) are identical in law, where 𝒫 is Pitman's transformation. In the course of our exploration, it will be revealed that, for any x>0, the process 𝒫(−Rx)+x has the same law as the standard three-dimensional Bessel process R0, which is then applied to derive a disintegration formula for the law of Rx in terms of Pitman's transformation. An application of the formula to three-dimensional Bessel bridges is also presented.

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Cite

BibTeXRIS
BibTeX
@misc{hariya2026relationship,
  title = {{A relationship between three-dimensional Bessel processes in terms of Pitman's transformation}},
  author = {Yuu Hariya},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25205},
  eprint = {2610.25205},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25205},
}
RIS
TY  - UNPB
TI  - A relationship between three-dimensional Bessel processes in terms of Pitman's transformation
AU  - Yuu Hariya
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25205
UR  - https://arxiv.org/abs/2610.25205
AB  - For every x≥ 0, let R^x={ R^x_t} _t≥ 0 be a three-dimensional Bessel process started at x. Given x>0, with the aid of Pitman's theorem, a path transformation T_x under which the process T_x(R^y) is distributed as R^x for any y>0 is constructed. In fact, we prove that, for every x,y>0, the two three-dimensional processes ( R^x,T_y(R^x),𝒫(−R^x)+x) && and && ( T_x(R^y),R^y,𝒫(−R^y)+y) are identical in law, where 𝒫 is Pitman's transformation. In the course of our exploration, it will be revealed that, for any x>0, the process 𝒫(−R^x)+x has the same law as the standard three-dimensional Bessel process R^0, which is then applied to derive a disintegration formula for the law of R^x in terms of Pitman's transformation. An application of the formula to three-dimensional Bessel bridges is also presented.
ER  -