On bivariate distributions of the local time of Itô-McKean diffusions (Q6178557)

The problem of finding the joint distribution of a couple of processes \((X_t,L_t)\), where \(X\) is an Itô-McKean diffusion process and \(L\) its local time at 0, motivated the research of the authors reported in this article. The authors produce a new explicit representation of the bivariate distribution of \((X_t,L_t)\), where \(X\) is an Itô-McKean diffusion process and \(L\) its local time at 0. The key idea is the use of excursion theory in two respects: first, it allows one to obtain the transition density of the couple \((X.L)\), secondly, they get a simple connection formula for the distribution of excursions from a hyperplane of the couple \((X,L)\). Formulas were obtained for several functionals of \(L\), expressed in terms of the transition density \(p\) with respect to the velocity (speed) measure, recalling that the Green kernel for the above class of diffusions is \[G_\lambda(x, z)=\int_0^{\infty} e^{-\lambda t} p_t(x, z) d t, \quad \lambda>0, x, z \in E,\] where \(E\subset\mathbb{R}\) contains \(0\) and this is a boundary and instantaneous reflecting point. The paper is organized into three sections and various subsections. After a well-documented historical review on the subject, covered in the introductory section, the presentation of the theoretical core is given in Section two, finishing the paper in the third one, with the rigorous proofs of all new results. This is an interesting paper, the main features of which are summarized here. First, there is a new explicit description of the distribution of \(L_t\) in terms of a convolution exponent. This is done in subsection 2.1. The probability distribution \(Q(t,z)\) of \(L_t\) for an Itô-McKean diffusion starting from 0, with transition density \(p\) as before satisfies the equation \[p * \frac{\partial Q}{\partial z}+Q-1=0.\] The authors characterize in Theorem 2.1 the class of functions that solves the above equation, which provides an important tool to determine \(Q\). This is the first step to provide a convolution representation of the density of \(L_t\) for fixed \(t>0\). This is an integral representation in terms of a Bessel function and the transition probability. To do that, they define a convolution exponent as follows. Let \(\mathcal{A}\) be the set of all complex-valued locally integrable functions on \([0, \infty)\). For any \(f \in \mathcal{A}\), and define \[ \mathcal{E}^*(f)=\delta_0+\sum_{i \geq 1} \frac{f^{* i}}{i !}, \] Then an ingenious use of Bessel functions together with the introduction of a function \(\varphi (t,s)=1*\mathcal{E}^*(-tp)(s)\) allows the authors to prove Theorem 2.4 giving the distribution of \(L_t\). The paper further explores the excursion theory of Markov processes to determine the distribution of \((X_t, L_tt)\) and the excursions from a hyperplane for a bivariate diffusion. So, a description of the transition density of the pair \((X, L)\) using excursion theory is given in subsection 2.2, through Theorems 2.7 and 2.8, where convolution and class \(\mathcal{A}\) again play a major role. Going on, the paper shows a simple connection formula for the excursion distribution of a bivariate Itô-McKean diffusion from a hyperplane in subsection 2.3 (see Theorem 2.9 and Corollary 2.10). Examples include a local time distribution and a formula for the distribution of \((X_t, L_\infty)\) for transient diffusion (Theorem 2.15). Other examples of applications of local-time bivariate distributions are presented, including a probabilistic representation of the solution to the generalized Stroock-Williams equation. The article also mentions previous studies on the probabilistic structure of local time \(L\) for a 0-reflected Itô diffusion. Processes that meet the assumptions of the study include squared Bessel processes with index \(\mu\in (-1,0)\), radial Ornstein-Uhlenbeck processes, or Pearson diffusion.