Abstract
We give an exposition of a result of Borell (Commun Math Phys 86:143–147, 1982) that the probability function that Brownian motion hits the inner boundary before time t and before hitting the outer boundary is a space-time quasiconcave function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borell, C.: Brownian motion in a convex ring and quasi-concavity. Commun. Math. Phys. 86(1), 143–147 (1982)
Chau, A., Weinkove, B.: Counterexamples to the quasiconcavity for heat equation, preprint arXiv:1802.04770
Diaz, J.I., Kawohl, B.: On convexity and starshapedness of level sets for some non-linear elliptic and parabolic problems on convex rings. J. Math. Anal. Appl. 177, 263–286 (1993)
Gabriel, R.: A result concerning convex level surface of 3-dimensional harmonic functions. J. Lond. Math. Soc. 32, 286–294 (1957)
Lewis, J.L.: Capacitary function in convex rings. Arch. Ration. Mech. Anal. 66, 201–224 (1977)
Acknowledgements
This article is written under the guidance of my adviser Professor Elton P Hsu. I would like to offer my sincerest thanks for his help and support. The author gratefully acknowledges financial support from China Scholarship Council. The author is supported by Chinese Government Scholarship (Grant No. 201606340046).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Suo, J. A Quasiconcavity Property for the Heat Equation in a Convex Ring. Commun. Math. Stat. 8, 453–462 (2020). https://doi.org/10.1007/s40304-020-00207-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40304-020-00207-6