An infinite-times renewal equation

In neuroscience, the time elapsed since the last discharge has been used to predict the probability of the next discharge. Such predictions can be improved taking into account the last two discharge times, and preferentially more. Such multi-times renewal processes arise in many other areas, and there is no universal limitation on the number of times to be used. This observation leads us to study the infinite-times renewal equation as a simple model to understand the meaning and properties of such partial differential equations depending on an infinite number of variables. This model also has a solid connection to the theory of Markov chain of infinite order. We rigorously construct solutions for the infinite-times renewal equation via the limit N→∞ of N-times equations, establishing notions of hierarchy and measure solutions. We then investigate the long-time convergence, overcoming the difficulty of vanishing rates as N→∞ and use two methods: a refined Doeblin analysis and a coupling approach with a specific distance yielding uniform-in-N convergences. This work provides a rigorous framework for the forward Kolmogorov equation of the infinite-times renewal process, demonstrates its approximation by finite-dimensional Markov processes, and aids in analyzing its ergodic or mixing properties, contributing insights relevant to infinite-dimensional dynamics.