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Distortion of area and conditioned Brownian motion. (English) Zbl 0794.60081

In a simply connected planar domain \(D\) the expected lifetime of conditioned Brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain. We show that each hyperbolic geodesic \(\gamma\) induces a decomposition of \(D\) into disjoint subregions \(\Omega_ j\), \(\bigcup_ j \Omega_ j=D\), and that the subregions are obtained in a natural way by using Euclidean geometric quantities relating \(\gamma\) to \(D\). The lifetime associated with \(\gamma\) on each \(\Omega_ j\) is then shown to be bounded by the product of the diameter of the smallest ball containing \(\gamma \cap \Omega_ j\) and the diameter of the largest ball in \(\Omega_ j\). Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in \(\Omega_ j\), it leads to finite lifetime estimates in a variety of domains of infinite area.
Reviewer: P.S.Griffin

MSC:

60J65 Brownian motion
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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[1] Ahlfors, L. V., Conformal invariants (1973), New York: McGraw-Hill, New York · Zbl 0272.30012
[2] Baernstein, A., Integral means, univalent functions and circular symmetrization, Acta. Math., 133, 139-169 (1974) · Zbl 0315.30021
[3] Bañuelos, R.: Intrinsic ultracontractivity and eigenfunction estimates for Schrodinger operators. J. Funct. Anal.99 (1991) · Zbl 0766.47025
[4] Bañuelos, R.: Lifetime and heat kernel estimates in non-smooth domains. (to appear 1991) · Zbl 0797.60069
[5] Bañuelos, R., Carroll, R.: Conditioned brownian motion and hyperbolic geodesics in simply connected domains. (Preprint 1991) · Zbl 0805.60077
[6] Bañuelos, R., Davis, B.: A geometrical characterization of intrinsic ultracontractivity for planar domains with boundaries given by the graphs of functions. (Preprint 1992) · Zbl 0836.60082
[7] Bass, R.F., Burdzy, K.: Lifetimes of conditoned diffusions. (Preprint 1991)
[8] Brelot, M.; Choquet, G., Espaces et lignes de green, Ann. Inst. Fourier, 3, 199-263 (1951) · Zbl 0046.32701
[9] Churchhill, R. V., Fourier series and boundary value problems (1969), New York: McGraw-Hill, New York
[10] Cranston, M.; McConnell, T. R., The lifetime of conditioned brownian motion, Z. Wahrscheinlichkeitstheor. Verw. Geb., 65, 1-11 (1983) · Zbl 0506.60071
[11] Griffin, P.S., McConnell T.R., Verchota, G.C.: Conditioned brownian motion in simply connected planar domains. Ann. Inst. Henri Poincaré (to appear) · Zbl 0777.60073
[12] Hayman, W.K.: Subharmonic functions, vol. 2. Lond. Math. Soc. Monogr.20 (1989) · Zbl 0699.31001
[13] Helms, L., Introduction to potential theory (1969), New York: Interscience, New York · Zbl 0188.17203
[14] Jones, P. W., Extension theorems for bmo, Indiana Math. J., 29, 41-66 (1980) · Zbl 0432.42017
[15] Pommerenke, C., Univalent functions (1975), Göttingen: Vandenhoeck and Ruprecht, Göttingen · Zbl 0298.30014
[16] Stein, E. M., Singular integrals and differentiability properties of functions (1970), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 0207.13501
[17] Xu, J., The lifetime of conditioned brownian motion in planar domains of infinite area, Probab. Theory Relat. Fields, 87, 469-487 (1991) · Zbl 0718.60092
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