arXiv subjects

Trang Le Ngan

Publications and source records attributed to Trang Le Ngan.

3 recordsLinked to original sources

arXiv2019

The Positive Maximum Principle on Lie Groups

We extend a classical theorem of Courrège to Lie groups in a global setting, thus characterising all linear operators on the space of smooth functions of compact support that satisfy the positive maximum principle. We show that these are Lévy type operators (with variable characteristics), and pseudo--differential operators when the group is compact. If the characteristics are constant, then the operator is the generator of the contraction semigroup associated to a convolution semigroup of sub--probability measures.

math.FA↗

arXiv2019

The Positive Maximum Principle on Symmetric Spaces

We investigate the Courrège theorem in the context of linear operators $A$ that satisfy the positive maximum principle on a space of continuous functions over a symmetric space. Applications are given to Feller--Markov processes. We also introduce Gangolli operators, which satisfy the positive maximum principle, and generalise the form associated with the generator of a Lévy process on a symmetric space. When the space is compact, we show that Gangolli operators are pseudo--differential operators having scalar symbols.

math.FA↗

arXiv2017

Transition Densities and Traces for Invariant Feller Processes on Compact Symmetric Spaces

We find necessary and sufficient conditions for a finite $K$-bi-invariant measure on a compact Gelfand pair $(G, K)$ to have a square-integrable density. For convolution semigroups, this is equivalent to having a continuous density in positive time. When $(G,K)$ is a compact Riemannian symmetric pair, we study the induced transition density for $G$-invariant Feller processes on the symmetric space $X = G/K$. These are obtained as projections of $K$-bi-invariant Lévy processes on $G$, whose laws form a convolution semigroup. We obtain a Fourier series expansion for the density, in terms of spherical functions, where the spectrum is described by Gangolli's Lévy-Khintchine formula. The density of returns to any given point on $X$ is given by the trace of the transition semigroup, and for subordinated Brownian motion, we can calculate the short time asymptotics of this quantity using recent work of Bañuelos and Baudoin. In the case of the sphere, there is an interesting connection with the Funk-Hecke theorem.

math.PR↗