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Uniqueness properties of diffusion processes. (English) Zbl 1345.60093
Duong, Xuan (ed.) et al., AMSI international conference on harmonic analysis and applications. Proceedings of the conference, Macquarie University, Sydney, Australia, February 7–11, 2011. Canberra: Australian National University, Centre for Mathematics and its Applications. Proceedings of the Centre for Mathematics and its Applications, Australian National University 45, 124-135 (2013).
Summary: 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 $$\mathbb{R}^d$$. 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 $$H$$.
For the entire collection see [Zbl 1334.42001].
##### MSC:
 60J60 Diffusion processes 60H30 Applications of stochastic analysis (to PDEs, etc.)
##### Keywords:
diffusion processes; $$L_1$$-uniqueness; Markov uniqueness