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A Relationship Between Fixed Time Wiener Measures and Wiener Measures with Fixed Endpoints

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Abstract

Wiener measures are measures on curves that are derived from two-dimensional Brownian motion. We prove a relationship between two types of Wiener measures: measures on paths with fixed starting point (say the origin \(0\)) and fixed time duration (say \(1\)); and measures on paths with fixed endpoints (say \(0\) and \(i\)). The relationship is that if we take a curve from the first type, weight it by a suitable power of the distance to the endpoint of the curve and then apply the conformal map that takes the endpoint to \(i\), then we get the curve from the second type.

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Acknowledgments

The author wishes to thank Tom Kennedy for suggesting this problem to him and for many helpful discussions along the way. Thanks are also owed to the anonymous referees for many valuable comments and suggestions. Finally, the research was supported in part by NSF grant DMS-0758649.

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  1. Department of Mathematics, University of Arizona, Tucson, AZ , 85721-0089, USA

    Jianping Jiang

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  1. Jianping Jiang
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Correspondence to Jianping Jiang.

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Jiang, J. A Relationship Between Fixed Time Wiener Measures and Wiener Measures with Fixed Endpoints. J Stat Phys 156, 177–188 (2014). https://doi.org/10.1007/s10955-014-0998-7

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  • DOI: https://doi.org/10.1007/s10955-014-0998-7

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