Pseudo-differential operators and Markov processes
type: Item
instance of: scholarly article
title: Pseudo-differential operators and Markov processes
zbMATH Open document ID: 0539.60081
author: Takashi Komatsu
published in: Journal of the Mathematical Society of Japan
publication date: 1984-01-01T00:00:00Z
review text: A pseudo-differential operator L is said to be non-degenerate if there is a constant \(\alpha (0<\alpha \leq 2)\) and \(L=\psi^{(\alpha)}(x,\partial)+\phi^{(\alpha)}(x,\partial),\) where \(\psi^{(\alpha)}(x,\xi)\) is a homogeneous function in \(\xi\) with index \(\alpha\) such that Re \(\psi^{(\alpha)}(x,\xi)<0\) for \(| \xi | =1\), and \(\phi^{(\alpha)}(x,\xi)=o(| \xi |^{\alpha})\) for large \(| \xi |.\) A Markov process is associated with L in the sense that its resolvent \(\{R_{\lambda}\}\) satisfies the equality \(R_{\lambda}(\lambda -L)f=f\) for any test function f. It is proved that existence and uniqueness hold for Markov processes associated with operators in a class of non- degenerate pseudo-differential operators. The uniqueness of solutions of the martingale problem for L is discussed to prove the uniqueness of Markov processes.
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J75
zbMATH DE Number: 3858126
zbMATH Keywords: Lévy type generator
zbMATH Keywords: pseudo-differential operator
zbMATH Keywords: resolvent
zbMATH Keywords: martingale problem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03630387
OpenAlex ID: W2023219864
DOI: 10.2969/JMSJ/03630387
description: scientific article; zbMATH DE number 3858126