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Math & Stats Department Colloquium

Friday, November 11th, 2022

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Zachary Selk (¾ÅÐãÖ±²¥)

Title: The Small Noise Limit of the Most Likely Element is the Most Likely Element in the Small Noise Limit

Abstract: In this talk, I discuss the Onsager-Machlup function and its relationship with the Freidlin-Wentzell rate function from large deviations. The Onsager-Machlup function can serve as a probability 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 functionals?

We show that the small noise limit of the Onsager-Machlup functional both pointwise and in the sense of minimizers converges to the Freidlin-Wentzell functional for measures equivalent to arbitrary Gaussian measures. 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 stochastic differential equations or the law of an infinite system of random algebraic equations.

Joint work with Harsha Honnappa