Arkiv för Matematik

Volume 57 (2019)

Number 2

Pluripotential theory and convex bodies: large deviation principle

Pages: 247 – 283

DOI: https://dx.doi.org/10.4310/ARKIV.2019.v57.n2.a2

Authors

Turgay Bayraktar (Sabanci University, Istanbul, Turkey)

Thomas Bloom (Department of Mathematics, University of Toronto, Ontario, Canada)

Norman Levenberg (Indiana University, Bloomington, In., U.S.A.)

Chinh H. Lu (Université Paris-Sud, Orsay, France)

Abstract

We continue the study in [2] in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $(\mathbb{R}^{+})^d$. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of $P$-pluripotential-theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge–Ampère equation in an appropriate finite energy class. This is achieved using a variational approach.

Keywords

convex body, $P$-extremal function, large deviation principle

2010 Mathematics Subject Classification

31C15, 32U15, 32U20

Received 18 August 2018

Received revised 10 February 2019

Accepted 26 February 2019

Published 7 October 2019