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Brownian motion with drift conditioned to have restricted L2-norm

  • Frank Aurzada
  • Yuvraj Dutta
  • Max Wiegand
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Abstract

We study the long-time behaviour of Brownian motion with drift under a rare-event conditioning of the quadratic additive functional ZT=∫0T Ws2ds, where (Wt)t≥ 0 is a Brownian motion with drift μ. More precisely, we consider the conditional law of W given the event ZT≤ θT, as T→∞, allowing the drift μ to depend on T. We derive sharp, uniform small-deviation asymptotics for ZT, including the exact prefactor, and use them to show that the conditioned process converges weakly to an Ornstein--Uhlenbeck process. Remarkably, the limiting dynamics are independent of the original drift.

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Cite

BibTeXRIS
BibTeX
@misc{aurzada2026brownian,
  title = {{Brownian motion with drift conditioned to have restricted $L^2$-norm}},
  author = {Frank Aurzada and Yuvraj Dutta and Max Wiegand},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12354},
  eprint = {2610.12354},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12354},
}
RIS
TY  - UNPB
TI  - Brownian motion with drift conditioned to have restricted L^2-norm
AU  - Frank Aurzada
AU  - Yuvraj Dutta
AU  - Max Wiegand
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12354
UR  - https://arxiv.org/abs/2610.12354
AB  - We study the long-time behaviour of Brownian motion with drift under a rare-event conditioning of the quadratic additive functional Z_T=∫_0^T W_s^2ds, where (W_t)_t≥ 0 is a Brownian motion with drift μ. More precisely, we consider the conditional law of W given the event Z_T≤ θT, as T→∞, allowing the drift μ to depend on T. We derive sharp, uniform small-deviation asymptotics for Z_T, including the exact prefactor, and use them to show that the conditioned process converges weakly to an Ornstein--Uhlenbeck process. Remarkably, the limiting dynamics are independent of the original drift.
ER  -