Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2021
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2105.02495 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915175887536128 |
|---|---|
| author | Boubel, Charles Juillet, Nicolas |
| author_facet | Boubel, Charles Juillet, Nicolas |
| contents | Let $μ$ = ($μ$t)t$\in$R be a 1-parameter family of probability measures on R. In [11] we introduced its ``Markov-quantile''process: a process X= (Xt)t$\in$R that resembles as much as possible the quantile process attached to $μ$, among the Markov processesattached to $μ$, i.e. whose family of marginal laws is $μ$.In this article we look at the case where $μ$ is absolutely continuous in the Wasserstein space P2(R). Then X is solution of adynamical transport problem with marginals ($μ$t)t. It provides a Markov minimal Lagrangian probabilistic representative of $μ$, whichis moreover unique among the processes obtained as certain types of limits: limits for the finite dimensional topology of quantileprocesses where the past is made independent of the future conditionally on the present at finitely many times, or limits of processeslinearly interpolating $μ$.This raises new questions about ways to obtain Markov Lagrangian representatives, and to seek uniqueness properties in thisframework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_02495 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On absolutely continuous curves in the Wasserstein space over R and their representation by an optimal Markov process Boubel, Charles Juillet, Nicolas Analysis of PDEs Probability Let $μ$ = ($μ$t)t$\in$R be a 1-parameter family of probability measures on R. In [11] we introduced its ``Markov-quantile''process: a process X= (Xt)t$\in$R that resembles as much as possible the quantile process attached to $μ$, among the Markov processesattached to $μ$, i.e. whose family of marginal laws is $μ$.In this article we look at the case where $μ$ is absolutely continuous in the Wasserstein space P2(R). Then X is solution of adynamical transport problem with marginals ($μ$t)t. It provides a Markov minimal Lagrangian probabilistic representative of $μ$, whichis moreover unique among the processes obtained as certain types of limits: limits for the finite dimensional topology of quantileprocesses where the past is made independent of the future conditionally on the present at finitely many times, or limits of processeslinearly interpolating $μ$.This raises new questions about ways to obtain Markov Lagrangian representatives, and to seek uniqueness properties in thisframework. |
| title | On absolutely continuous curves in the Wasserstein space over R and their representation by an optimal Markov process |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2105.02495 |