2000 character limit reached
Brownian motion conditioned to have restricted $L_2$-norm (2312.12982v2)
Published 20 Dec 2023 in math.PR
Abstract: We condition a Brownian motion on having an atypically small $L_2$-norm on a long time interval. The obtained limiting process is a non-stationary Ornstein-Uhlenbeck process.
- P. Billingsley. Convergence of probability measures. Wiley, New York, 1968.
- A. N. Borodin and P. Salminen. Handbook of Brownian motion—facts and formulae. Probability and its Applications. Birkhäuser Verlag, Basel, 1996.
- J. L. Doob. Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France, 85:431–458, 1957.
- Exact L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT small balls of Gaussian processes. J. Theoret. Probab., 17(2):503–520, 2004.
- I. Karatzas. On a stochastic representation for the principal eigenvalue of a second-order differential equation. Stochastics, 3, 1980.
- F. B. Knight. Brownian local times and taboo processes. Trans. Amer. Math. Soc., 143:173–185, 1969.
- M. Kolb and M. Savov. Transience and recurrence of a Brownian path with limited local time. Ann. Probab., 44(6):4083–4132, 2016.
- M. A. Lifshits. On the lower tail probabilities of some random series. Ann. Probab., 25(1):424–442, 1997.
- W. V. Li and W. Linde. Small ball problems for non-centered Gaussian measures. Probab. Math. Statist., 14(2):231–251, 1993.
- M. Lifshits and E. Setterqvist. Energy of taut strings accompanying Wiener process. Stochastic Process. Appl., 125(2):401–427, 2015.
- A. Nazarov and Y. Petrova. L2subscript𝐿2L_{2}italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-small ball asymptotics for Gaussian random functions: a survey. Probab. Surv., 20:608–663, 2023.
- J. Rosen and B. Simon. Fluctuations in p(ϕ)1𝑝subscriptitalic-ϕ1p(\phi)_{1}italic_p ( italic_ϕ ) start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT processes. Ann.Probab., 4(2):155–174, 1976.
- Limiting laws associated with Brownian motion perturbed by normalized exponential weights. I. Studia Sci. Math. Hungar., 43(2):171–246, 2006.
- W. Whitt. Weak convergence of probability measures on the function space C[0,∞)𝐶0C[0,\,\infty)italic_C [ 0 , ∞ ). Ann. Math. Statist., 41:939–944, 1970.
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