An H-theorem for a conditional McKean-Vlasov process related to interacting diffusions on regular trees
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910768311566336 |
|---|---|
| author | Hu, Kevin Ramanan, Kavita |
| author_facet | Hu, Kevin Ramanan, Kavita |
| contents | We study the long-time behavior of the $κ$-Markov local-field equation ($κ$-MLFE), which is a conditional McKean-Vlasov equation associated with interacting diffusions on the $κ$-regular tree. Under suitable assumptions on the coefficients, we prove well-posedness of the $κ$-MLFE. We also establish an H-theorem by identifying an energy functional, referred to as the sparse free energy, whose derivative along the measure flow of the $κ$-MLFE is given by a nonnegative functional that can be viewed as a modified Fisher information. Moreover, we show that the zeros of the latter functional coincide with the set of stationary distributions of the $κ$-MLFE and are also marginals of splitting Gibbs measures on the $κ$-regular tree. Furthermore, we show that for a natural class of initial conditions, the corresponding measure flow converges to one of the stationary distributions, thus demonstrating that the sparse free energy acts as a global Lyapunov function. Under mild additional conditions, in the case $κ= 2$ we prove that the sparse free energy arises naturally as the renormalized limit of certain relative entropies. We exploit this characterization to prove a modified logarithmic Sobolev inequality and establish an exponential rate of convergence of the $2$-MLFE measure flow to its unique stationary distribution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07710 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An H-theorem for a conditional McKean-Vlasov process related to interacting diffusions on regular trees Hu, Kevin Ramanan, Kavita Probability Analysis of PDEs 60K35, 60J60 (Primary) 60J70, 82C22, 35Q84, 82C31 (Secondary) We study the long-time behavior of the $κ$-Markov local-field equation ($κ$-MLFE), which is a conditional McKean-Vlasov equation associated with interacting diffusions on the $κ$-regular tree. Under suitable assumptions on the coefficients, we prove well-posedness of the $κ$-MLFE. We also establish an H-theorem by identifying an energy functional, referred to as the sparse free energy, whose derivative along the measure flow of the $κ$-MLFE is given by a nonnegative functional that can be viewed as a modified Fisher information. Moreover, we show that the zeros of the latter functional coincide with the set of stationary distributions of the $κ$-MLFE and are also marginals of splitting Gibbs measures on the $κ$-regular tree. Furthermore, we show that for a natural class of initial conditions, the corresponding measure flow converges to one of the stationary distributions, thus demonstrating that the sparse free energy acts as a global Lyapunov function. Under mild additional conditions, in the case $κ= 2$ we prove that the sparse free energy arises naturally as the renormalized limit of certain relative entropies. We exploit this characterization to prove a modified logarithmic Sobolev inequality and establish an exponential rate of convergence of the $2$-MLFE measure flow to its unique stationary distribution. |
| title | An H-theorem for a conditional McKean-Vlasov process related to interacting diffusions on regular trees |
| topic | Probability Analysis of PDEs 60K35, 60J60 (Primary) 60J70, 82C22, 35Q84, 82C31 (Secondary) |
| url | https://arxiv.org/abs/2412.07710 |