Mean exit times and hitting probabilities of Brownian motion in geodesic balls and tubular neighborhoods

1986 ◽  
pp. 216-223 ◽  
Author(s):  
Mark A. Pinsky
Keyword(s):  
Brownian Motion ◽  
Exit Times ◽  
Hitting Probabilities ◽  
Tubular Neighborhoods

scholarly journals The surface limit of Brownian motion in tubular neighborhoods of an embedded Riemannian manifold

2004 ◽  
Vol 206 (2) ◽  
pp. 391-413 ◽  
Author(s):  
Nadezda A. Sidorova ◽  
Oleg G. Smolyanov ◽  
Heinrich v. Weizsäcker ◽  
Olaf Wittich
Keyword(s):  
Brownian Motion ◽  
Riemannian Manifold ◽  
Tubular Neighborhoods

scholarly journals Brownian motion with variable drift: 0-1 laws, hitting probabilities and Hausdorff dimension

2012 ◽  
Vol 153 (2) ◽  
pp. 215-234 ◽  
Author(s):  
YUVAL PERES ◽  
PERLA SOUSI
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Lévy Process ◽  
Dirichlet Space ◽  
Levy Process ◽  
Standard Brownian Motion ◽  
Double Points ◽  
Hitting Probabilities

AbstractBy the Cameron–Martin theorem, if a function f is in the Dirichlet space D, then B + f has the same a.s. properties as standard Brownian motion, B. In this paper we examine properties of B + f when fD. We start by establishing a general 0-1 law, which in particular implies that for any fixed f, the Hausdorff dimension of the image and the graph of B + f are constants a.s. (This 0-1 law applies to any Lévy process.) Then we show that if the function f is Hölder(1/2), then B + f is intersection equivalent to B. Moreover, B + f has double points a.s. in dimensions d ≤ 3, while in d ≥ 4 it does not. We also give examples of functions which are Hölder with exponent less than 1/2, that yield double points in dimensions greater than 4. Finally, we show that for d ≥ 2, the Hausdorff dimension of the image of B + f is a.s. at least the maximum of 2 and the dimension of the image of f.

Super-Brownian motion: Path properties and hitting probabilities

1989 ◽  
Vol 83 (1-2) ◽  
pp. 135-205 ◽  
Author(s):  
D. A. Dawson ◽  
I. Iscoe ◽  
E. A. Perkins
Keyword(s):  
Brownian Motion ◽  
Motion Path ◽  
Hitting Probabilities ◽  
Super Brownian Motion ◽  
Path Properties

Sur certaines fonctionnelles exponentielles du mouvement brownien réel

1992 ◽  
Vol 29 (01) ◽  
pp. 202-208 ◽  
Author(s):  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Bivariate Distribution ◽  
Exit Times ◽  
Bessel Processes ◽  
Gamma Variable ◽  
Negative Drift ◽  
Mouvement Brownien

Dufresne [1] recently showed that the integral of the exponential of Brownian motion with negative drift is distributed as the reciprocal of a gamma variable. In this paper, it is shown that this result is another formulation of the distribution of last exit times for transient Bessel processes. A bivariate distribution of such integrals of exponentials is also obtained explicitly.

Sur certaines fonctionnelles exponentielles du mouvement brownien réel

1992 ◽  
Vol 29 (1) ◽  
pp. 202-208 ◽  
Author(s):  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Bivariate Distribution ◽  
Exit Times ◽  
Bessel Processes ◽  
Gamma Variable ◽  
Negative Drift ◽  
Mouvement Brownien

Dufresne [1] recently showed that the integral of the exponential of Brownian motion with negative drift is distributed as the reciprocal of a gamma variable. In this paper, it is shown that this result is another formulation of the distribution of last exit times for transient Bessel processes. A bivariate distribution of such integrals of exponentials is also obtained explicitly.

scholarly journals Bayesian sequential testing with expectation constraints

2020 ◽  
Vol 26 ◽  
pp. 51
Author(s):  
Stefan Ankirchner ◽  
Maike Klein
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Posterior Probability ◽  
Drift Rate ◽  
Stopping Rules ◽  
Sequential Testing ◽  
Exit Times ◽  
Intermediate Point ◽  
Optimal Stopping Rule ◽  
Optimal Stopping Rules

We study a stopping problem arising from a sequential testing of two simple hypotheses H0 and H1 on the drift rate of a Brownian motion. We impose an expectation constraint on the stopping rules allowed and show that an optimal stopping rule satisfying the constraint can be found among the rules of the following type: stop if the posterior probability for H1 attains a given lower or upper barrier; or stop if the posterior probability comes back to a fixed intermediate point after a sufficiently large excursion. Stopping at the intermediate point means that the testing is abandoned without accepting H0 or H1. In contrast to the unconstrained case, optimal stopping rules, in general, cannot be found among interval exit times. Thus, optimal stopping rules in the constrained case qualitatively differ from optimal rules in the unconstrained case.

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