**Zu Chongzhi Center Mathematics Research Seminar**

**Date and Time (China standard time): **Friday, April 7, 4:30-5:30 pm

**Zoom ID**: 994 9819 4337

**Passcode: **dkumath

**Title: **An infinite-times renewal equation: finite-dimensional approximation and long-time convergence of the infinite-dimensional PDE

**Speaker**: Chenjiayue Qi

**Bio**: Chenjiayue Qi is a fourth-year undergraduate at Peking University, and is going to the University of Paris-Saclay to pursue a graduate degree this September. Her research supervisors are Prof. Zhennan Zhou (BICMR, Peking University) and Prof. Benoît Perthame (LJLL, Sorbonne University). Her current research interest includes renewal equations with motivations from neuroscience (such as the elapsed time model and the nonlinear noisy integrate-and-fire model), and the derivation of Boltzmann equation from Hamiltonian dynamics.

**Abstract**: 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 by taking into account the last two discharge times, and possibly more. Such multi-time 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.

We define two notions of solutions, prove existence and uniqueness of solutions, possibly measures. We also prove the long-time convergence, with an exponential rate, to the steady state in different, strong or weak, topologies depending on assumptions on the coefficients.