Default functions and Liouville type theorems based on symmetric diffusions

Atsushi ATSUJI

doi:10.2969/jmsj/82398239

Abstract

Default functions appear when one discusses conditions which ensure that a local martingale is a true martingale. We show vanishing of default functions of Dirichlet processes enables us to obtain Liouville type theorems for subharmonic functions and holomorphic maps.

Funding Statement

Partially supported by the Grant-in-Aid for Scientific Research 17K18741, Japan Society for the Promotion of Science.

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Atsushi ATSUJI. "Default functions and Liouville type theorems based on symmetric diffusions." J. Math. Soc. Japan 73 (2) 525 - 551, April, 2021. https://doi.org/10.2969/jmsj/82398239

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Received: 7 April 2019; Revised: 14 December 2019; Published: April, 2021

First available in Project Euclid: 1 December 2020

Digital Object Identifier: 10.2969/jmsj/82398239

Subjects:

Primary: 31C05

Secondary: 58J65

Keywords: $L^1$-Liouville theorem , Brownian motion on manifolds , Liouville theorem for holomorphic maps , strictly local submartingale , subharmonic function

Abstract

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