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PreprintCC BY-NC-ND 4.0arXiv

Generalized Images for Brownian First Passage to Monomial Boundaries

  • Gerardo Hernández-del-Valle
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Abstract

We study the first passage of Brownian motion to a monomial moving boundary. For every integer degree at least two, we construct a time-independent signed image measure whose heat potential is the killed Brownian transition density on a nonzero interval of the natural Brownian time scale. The image density is determined by a second-kind Volterra equation in the image coordinate, not by the classical Volterra equations in time for the hitting density. Its kernel is an explicit Wright function, and the first-passage density is recovered afterwards from the boundary flux. Exponential bounds establish convergence, while a beta-moment transform connects the Wright kernel with the finite-order differential hierarchy previously obtained from transported moving zeros. A cubic numerical experiment independently checks the resulting density against simulated Brownian paths.

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Cite

BibTeXRIS
BibTeX
@misc{hernandezdelvalle2026generalized,
  title = {{Generalized Images for Brownian First Passage to Monomial Boundaries}},
  author = {Gerardo Hernández-del-Valle},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30298},
  eprint = {2610.30298},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30298},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Generalized Images for Brownian First Passage to Monomial Boundaries
AU  - Gerardo Hernández-del-Valle
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.30298
UR  - https://arxiv.org/abs/2610.30298
AB  - We study the first passage of Brownian motion to a monomial moving boundary. For every integer degree at least two, we construct a time-independent signed image measure whose heat potential is the killed Brownian transition density on a nonzero interval of the natural Brownian time scale. The image density is determined by a second-kind Volterra equation in the image coordinate, not by the classical Volterra equations in time for the hitting density. Its kernel is an explicit Wright function, and the first-passage density is recovered afterwards from the boundary flux. Exponential bounds establish convergence, while a beta-moment transform connects the Wright kernel with the finite-order differential hierarchy previously obtained from transported moving zeros. A cubic numerical experiment independently checks the resulting density against simulated Brownian paths.
ER  -