Probabilistic Approach to Regularity

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2038)

Abstract

We here gather in a single note several original probabilistic works 4 devoted to the analysis of the C1,1 regularity of the solution to the possibly 5 degenerate complex Monge–Ampère equation. The whole analysis relies on a 6 probabilistic writing of the solution as the value function of a stochastic 7 optimal control problem. Such a representation has been introduced by 8 Gaveau [Gav77] in the late 1970s and used in an exhaustive way by Krylov in 9 a series of papers published in the late 1980s up to the final paper [Kry89] in 10 which the C1,1-estimate is eventually established. All the arguments we here 11 use follow from these seminal works.

Keywords

Brownian Motion Probabilistic Approach Representation Formula Bellman Equation Martingale Property 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Laboratoire J.-A. DieudonnéUniversité de Nice Sophia-AntipolisNice Cedex 02France

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