The Brownian transport map
Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage o...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Springer Science and Business Media LLC
2024
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| Online Access: | https://hdl.handle.net/1721.1/154997 |
| _version_ | 1811079904020660224 |
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| author | Mikulincer, Dan Shenfeld, Yair |
| author_facet | Mikulincer, Dan Shenfeld, Yair |
| author_sort | Mikulincer, Dan |
| collection | MIT |
| description | Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage of the infinite-dimensional nature of the Gaussian measure and construct a new transport map, based on the Föllmer process, which pushes forward the Wiener measure onto probability measures on Euclidean spaces. Utilizing the tools of the Malliavin and stochastic calculus in Wiener space, we show that this Brownian transport map is a contraction in various settings where the analogous questions for optimal transport maps are open. The contraction properties of the Brownian transport map enable us to prove functional inequalities in Euclidean spaces, which are either completely new or improve on current results. Further and related applications of our contraction results are the existence of Stein kernels with desirable properties (which lead to new central limit theorems), as well as new insights into the Kannan–Lovász–Simonovits conjecture. We go beyond the Euclidean setting and address the problem of contractions on the Wiener space itself. We show that optimal transport maps and causal optimal transport maps (which are related to Brownian transport maps) between the Wiener measure and other target measures on Wiener space exhibit very different behaviors. |
| first_indexed | 2024-09-23T11:22:23Z |
| format | Article |
| id | mit-1721.1/154997 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T11:22:23Z |
| publishDate | 2024 |
| publisher | Springer Science and Business Media LLC |
| record_format | dspace |
| spelling | mit-1721.1/1549972024-09-19T05:54:44Z The Brownian transport map Mikulincer, Dan Shenfeld, Yair Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage of the infinite-dimensional nature of the Gaussian measure and construct a new transport map, based on the Föllmer process, which pushes forward the Wiener measure onto probability measures on Euclidean spaces. Utilizing the tools of the Malliavin and stochastic calculus in Wiener space, we show that this Brownian transport map is a contraction in various settings where the analogous questions for optimal transport maps are open. The contraction properties of the Brownian transport map enable us to prove functional inequalities in Euclidean spaces, which are either completely new or improve on current results. Further and related applications of our contraction results are the existence of Stein kernels with desirable properties (which lead to new central limit theorems), as well as new insights into the Kannan–Lovász–Simonovits conjecture. We go beyond the Euclidean setting and address the problem of contractions on the Wiener space itself. We show that optimal transport maps and causal optimal transport maps (which are related to Brownian transport maps) between the Wiener measure and other target measures on Wiener space exhibit very different behaviors. 2024-05-20T14:24:51Z 2024-05-20T14:24:51Z 2024-05-16 2024-05-19T03:12:55Z Article http://purl.org/eprint/type/JournalArticle 0178-8051 1432-2064 https://hdl.handle.net/1721.1/154997 Mikulincer, D., Shenfeld, Y. The Brownian transport map. Probab. Theory Relat. Fields (2024). PUBLISHER_CC en 10.1007/s00440-024-01286-0 Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer Science and Business Media LLC Springer Berlin Heidelberg |
| spellingShingle | Mikulincer, Dan Shenfeld, Yair The Brownian transport map |
| title | The Brownian transport map |
| title_full | The Brownian transport map |
| title_fullStr | The Brownian transport map |
| title_full_unstemmed | The Brownian transport map |
| title_short | The Brownian transport map |
| title_sort | brownian transport map |
| url | https://hdl.handle.net/1721.1/154997 |
| work_keys_str_mv | AT mikulincerdan thebrowniantransportmap AT shenfeldyair thebrowniantransportmap AT mikulincerdan browniantransportmap AT shenfeldyair browniantransportmap |