Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-02-05T09:07:23.464Z Has data issue: false hasContentIssue false
  • English
  • Français

Killed Brownian motion and the Brunn–Minkowski inequalities

Published online by Cambridge University Press:  22 February 2012

HUILING LE
HUILING LE*
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD. e-mail: huiling.le@nottingham.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct triplets of killed Brownian motions to obtain the Brunn–Minkowski inequalities concerning the solutions of the equation (1/2)Δψ − h ψ = g on three interrelated compact sets in Euclidean space. These, in particular, include inequalities relating to the solutions of the Schrödinger equation and the Poisson equation on the three compact convex sets and an inequality relating to harmonic functions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 1 , July 2012 , pp. 111 - 121
Copyright
Copyright © Cambridge Philosophical Society 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Borell, C.Capacitary inequalitties of the Brunn–Minkowski type. Math. Ann. 263 (1983), 179–184.CrossRefGoogle Scholar
[2]Borell, C.Greenian potentials and concavity. Math. Ann. 272 (1985), 155–160.CrossRefGoogle Scholar
[3]Borell, C.Diffusion equations and geometric inequalities. Potential Analysis 12 (2000), 49–71.CrossRefGoogle Scholar
[4]Bramscamp, H. J. and Lieb, E.On extension of the Brunn–Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to a diffusion equation. J. Funct. Anal. 22 (1976), 366–389.CrossRefGoogle Scholar
[5]Colesanti, A.Brunn–Minkowski inequalities for variational problems and related problems. Adv. Math. 194 (2005), 105–140.CrossRefGoogle Scholar
[6]Fleming, F. S. and Soner, H. M.Controlled Markov Processes and Viscosity Solutions (Springer, 1993).Google Scholar
[7]Gardner, R. J.The Brunn–Minkowski inequality. Bull. Amer. Math. Soc. 39 (2002), 355–405.CrossRefGoogle Scholar
[8]Hu, H. and Zhou, S.Brunn–Minkowski inequality for variational functional involving the p-Laplacian operator. Acta Math. Sci. 29(B) (2009), 1143–1154.CrossRefGoogle Scholar
[9]Le, H.Killed Brownian motion and inequalities among solutions of the Schrödinger equation. Stochastic Process. Appl. 119 (2009), 1257–1269.CrossRefGoogle Scholar
[10]Le, H. and Barden, D.Semimartingales and geometric inequalities on manifolds. J. London Math. Soc. 75 (2007), 522–544.CrossRefGoogle Scholar
[11]Le, H. and Barden, D.Semimartingales and geometric inequalities on locally symmetric manifolds. Probab. Theory Related Fields 142 (2008), 285–311.CrossRefGoogle Scholar
[12]Rogers, L. C. G. and Williams, D.Diffusions, Markov Processes and Martingales Vol. 2 (Wiley, 1987).Google Scholar