The Markov property of local times of Brownian motion indexed by the Brownian tree (Q6118757)

The current work focuses on the Markov property for the process of local times in the model of a Brownian motion indexed by the Browninan tree \(\mathcal{T}\), tree that can be seen as a random continuous tree coded by a Brownian excursion under the Itô measure. For a given Brownian tree \(\mathcal{T}\), the author considers Brownian motion indexed by \(\mathcal{T}\), process denoted by \((V_a)_{a\in\mathcal{T}}\), and its total occupation measure is denoted by \(\mathcal{Y}\). The pair \((\mathcal{T},(V_a)_{a\in\mathcal{T}})\) has been intensively investigated in a variety of works and plays an important role in probability theory. The continuous density \(\ell^x,x\in\mathbb{R}\) of the measure \(\mathcal{Y}\) is called the \textit{local time of \((V_a)_{a\in\mathcal{T}}\) at level \(x\)}. The function \(x\mapsto \ell^x\) is continuously differentiable, and its derivative is denoted by \(\dot{\ell^x}\). The main result of this paper states that the process \(((\ell^x,\dot{\ell^x}), x\geq 0)\) is a time-homogeneous Markov process under \(\mathbb{N}_0\), where \(\mathbb{N}_0\) is the (\(\sigma\)-finite) measure under which \(\mathcal{T}\) and \((V_a)_{a\in\mathcal{T}}\) are defined. A similar result is also proven for super-Brownian motion.