| Full text | |
| Author(s): |
Total Authors: 3
|
| Affiliation: | [1] Univ Sao Paulo, Sao Paulo, SP - Brazil
[2] Univ Paris 1 Pantheon Sorbonne, Paris - France
[3] Univ Fed Rio de Janeiro, Rio De Janeiro - Brazil
Total Affiliations: 3
|
| Document type: | Journal article |
| Source: | ESAIM-PROBABILITY AND STATISTICS; v. 23, p. 770-796, DEC 20 2019. |
| Web of Science Citations: | 0 |
| Abstract | |
Non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels are considered. It is shown that their stability properties can be studied in terms of an associated class of piecewise deterministic Markov processes, called Markovian cascades of successive memory terms. Explicit conditions implying the positive Harris recurrence of these processes are presented. The proof is based on integration by parts with respect to the jump times. A crucial property is the non-degeneracy of the transition semigroup which is obtained thanks to the invertibility of an associated Vandermonde matrix. For Lipschitz continuous rate functions we also show that these Markovian cascades converge to equilibrium exponentially fast with respect to the Wasserstein distance. Finally, an extension of the classical thinning algorithm is proposed to simulate such Markovian cascades. (AU) | |
| FAPESP's process: | 16/17791-9 - Interacting processes with variable range memory in neurobiological models |
| Grantee: | Aline Duarte de Oliveira |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
| FAPESP's process: | 13/07699-0 - Research, Innovation and Dissemination Center for Neuromathematics - NeuroMat |
| Grantee: | Oswaldo Baffa Filho |
| Support Opportunities: | Research Grants - Research, Innovation and Dissemination Centers - RIDC |
| FAPESP's process: | 16/17789-4 - Stability, statistical selection of models and hydrodynamic boundary for interacting chains with variable reach memory |
| Grantee: | Guilherme Ost de Aguiar |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |