On the variances of occupation times of conditioned Brownian motion

Zhang, Biao

doi:10.1090/S0002-9947-96-01486-9

On the variances of occupation times of conditioned Brownian motion
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Trans. Amer. Math. Soc. 348 (1996), 173-185 Request permission

Abstract:

We extend some bounds on the variance of the lifetime of two–dimensional Brownian motion, conditioned to exit a planar domain at a given point, to certain domains in higher dimensions. We also give a short “analytic” proof of some existing results.

References

Rodrigo Bañuelos and Burgess Davis, A geometrical characterization of intrinsic ultracontractivity for planar domains with boundaries given by the graphs of functions, Indiana Univ. Math. J. 41 (1992), no. 4, 885–913. MR 1206335, DOI 10.1512/iumj.1992.41.41049
M. Cranston, Conditional Brownian motion, Whitney squares and the conditional gauge theorem, Seminar on Stochastic Processes, 1988 (Gainesville, FL, 1988) Progr. Probab., vol. 17, Birkhäuser Boston, Boston, MA, 1989, pp. 109–119. MR 990477
Burgess Davis, Conditioned Brownian motion in planar domains, Duke Math. J. 57 (1988), no. 2, 397–421. MR 962513, DOI 10.1215/S0012-7094-88-05718-3
Burgess Davis, Intrinsic ultracontractivity and the Dirichlet Laplacian, J. Funct. Anal. 100 (1991), no. 1, 162–180. MR 1124297, DOI 10.1016/0022-1236(91)90106-F
Burgess Davis and Biao Zhang, Moments of the lifetime of conditioned Brownian motion in cones, Proc. Amer. Math. Soc. 121 (1994), no. 3, 925–929. MR 1195717, DOI 10.1090/S0002-9939-1994-1195717-5
J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
Richard A. Hunt and Richard L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc. 147 (1970), 507–527. MR 274787, DOI 10.1090/S0002-9947-1970-0274787-0
S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089, DOI 10.1007/BF03046994
David S. Jerison and Carlos E. Kenig, Boundary value problems on Lipschitz domains, Studies in partial differential equations, MAA Stud. Math., vol. 23, Math. Assoc. America, Washington, DC, 1982, pp. 1–68. MR 716504
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095