Research lectures, seminars and events
The events listed in this area are research seminars, workshops and lectures hosted by Durham University departments and research institutes. If you are not a member of the University, but wish to enquire about attending one of the events please contact the organiser or host department.
|May 2020||July 2020|
Events for 10 June 2020
The Fokker-Planck equation plays an important role when one considers problems that involve white noise. As such, it has a long and illustrious history with many applications in statistical physics, plasma physics, stochastic analysis and mathematical finances.
Recent studies have focused on the case where the diffusive part of the equation is degenerate, an issue that can impact the long time behaviour of the solution to this equation. This difficulty, however, can be corrected by the drift mechanism in the system, as long as it manages to â€œmixâ€ the diffusive and non-diffusive directions. When this happens, a simple and natural equilibrium emerges.
The strong connection between the Fokker-Planck equation and the world of statistical physics also fosters the tool to investigate the aforementioned convergence: The notion of an entropy for the system. One common methodology to investigate the long time behaviour under an entropy is the so-called entropy method, where one searches for a geometric functional inequality between the entropy and its formal production under the evolution flow. This methodology, however, is problematic when degeneracy appears.
In the case where the diffusion and drift parts of the equation are constant, Anton and Erb have introduced in 2014 a â€œmodifiedâ€ production functional (motivated by notions of hypocoercivity), which have managed to yield the sharp convergence rate for a large family of entropies, when the diffusion matrix is degenerate or not - as long as the drift matrix wasnâ€™t defective.
The problems the defectiveness of the drift bring become apparent when one examines the standard technique to obtain the desired inequality of the entropy method, the so-called Bakry-Ãˆmry method. In this technique one uses the entropy method again - but on the production functional.
In this talk we shall consider a different approach to the problem, which yields the sharp rate of convergence to equilibrium for a family of natural entropies to the system. We will circumvent the entropy method by carefully exploring the spectral properties of the Fokker-Planck operator in an appropriate Hilbert setting, resulting in the desired convergence for one particular entropy, which will then be cascaded to all other entropies by the use a newly found non-symmetric hypercontractivity result.
This talk is based on a joint work with Anton Arnold and Tobias WÃ¶hrer.
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Research Seminar: Understanding Headteacherâ€™s Instructional Leadership: A Theoretical and Empirical Analysis in China
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Jennifer Binczewski: â€˜Secret Rooms and Sacred Spaces: Widows and Clandestine Mass in Post-Reformation Englandâ€™
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