The Small-Noise Limit of the Most Likely Element is the Most Likely Element in the Small-Noise Limit
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910434162900992 |
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| author | Selk, Zachary Honnappa, Harsha |
| author_facet | Selk, Zachary Honnappa, Harsha |
| contents | In this paper, we study the Onsager-Machlup function and its relationship to the Freidlin-Wentzell function for measures equivalent to arbitrary infinite dimensional Gaussian measures. The Onsager-Machlup function can serve as a density on infinite dimensional spaces, where a uniform measure does not exist, and has been seen as the Lagrangian for the ``most likely element". The Freidlin-Wentzell rate function is the large deviations rate function for small-noise limits and has also been identified as a Lagrangian for the ``most likely element". This leads to a conundrum - what is the relationship between these two functions?
We show both pointwise and $Γ$-convergence (which is essentially the convergence of minimizers) of the Onsager-Machlup function under the small-noise limit to the Freidlin-Wentzell function - and give an expression for both. That is, we show that the small-noise limit of the most likely element is the most likely element in the small noise limit for infinite dimensional measures that are equivalent to a Gaussian. Examples of measures include the law of solutions to path-dependent stochastic differential equations and the law of an infinite system of random algebraic equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_04523 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Small-Noise Limit of the Most Likely Element is the Most Likely Element in the Small-Noise Limit Selk, Zachary Honnappa, Harsha Probability 60 In this paper, we study the Onsager-Machlup function and its relationship to the Freidlin-Wentzell function for measures equivalent to arbitrary infinite dimensional Gaussian measures. The Onsager-Machlup function can serve as a density on infinite dimensional spaces, where a uniform measure does not exist, and has been seen as the Lagrangian for the ``most likely element". The Freidlin-Wentzell rate function is the large deviations rate function for small-noise limits and has also been identified as a Lagrangian for the ``most likely element". This leads to a conundrum - what is the relationship between these two functions? We show both pointwise and $Γ$-convergence (which is essentially the convergence of minimizers) of the Onsager-Machlup function under the small-noise limit to the Freidlin-Wentzell function - and give an expression for both. That is, we show that the small-noise limit of the most likely element is the most likely element in the small noise limit for infinite dimensional measures that are equivalent to a Gaussian. Examples of measures include the law of solutions to path-dependent stochastic differential equations and the law of an infinite system of random algebraic equations. |
| title | The Small-Noise Limit of the Most Likely Element is the Most Likely Element in the Small-Noise Limit |
| topic | Probability 60 |
| url | https://arxiv.org/abs/2209.04523 |