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VOL. 45 | 2013 Uniqueness properties of diffusion processes
Derek W. Robinson

Editor(s) Xuan Duong, Jeff Hogan, Chris Meaney, Adam Sikora


We review recent results on the uniqueness of solutions of the diffusion equation \[ \partial \psi_{t} / \partial t + H \psi_{t} = 0 \] where $H$ is a strictly elliptic, symmetric, second-order operator on an open subset $\Omega$ of $\mathbf{R}^d$. In particular we discuss $L_1$-uniqueness, the existence of a unique continuous solution on $L_1(\Omega)$, and Markov uniqueness, the existence of a unique submarkovian solution on the spaces $L_p(\Omega)$. We give various criteria for uniqueness in terms of capacity estimates and the Riemannian geometry associated with $H$.


Published: 1 January 2013
First available in Project Euclid: 3 December 2014

zbMATH: 1345.60093
MathSciNet: MR3424871

Rights: Copyright © 2013, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.


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