# Large deviations of reflected diffusions

Du Buisson, Johannes Petrus (2020-04)

Thesis (MSc)--Stellenbosch University, 2020.

Thesis

ENGLISH ABSTRACT: We study the ﬂuctuations of time-integrated functionals of Markov diﬀusions evolving in a bounded domain. These ﬂuctuations can be described in large deviation theory by the so-called rate function, which encodes information about the probability distribution of such functionals in the long-time limit. In practice, the rate function is obtained by performing a spectral calculation. Furthermore, solving the spectral problem allows us to construct an eﬀective process which realizes a given ﬂuctuation away from the mean and explains how that ﬂuctuation is created dynamically in time. Most works in large deviation theory have considered Markov diﬀusions evolving in an unbounded domain (e.g. R or Rd). In this thesis we consider diﬀusions in bounded domains with perfect reﬂection at the boundaries. Considering the one-dimensional case, we derive the appropriate boundary conditions on the spectral problem and explore the implications for the eﬀective process. We apply this knowledge to obtain the rate function of the area of the reﬂected Ornstein-Uhlenbeck process and reﬂected Brownian motion with drift, and to obtain their eﬀective process. A variational representation of the rate function is used to construct accurate approximations of the eﬀective process for both of the systems considered.

AFRIKAANSE OPSOMMING: Ons bestudeer die ﬂuktuasies van tyd-geïntegreerde funksionale van Markov diffusies wat in ‘n begrensde domein ewolueer. Hierdie ﬂuktuasies kan in die teorie van groot afwykings deur die so-genoemde koers funksie, wat informasie rakende die waarskynlikheidsverspreiding van sulke funksionale in die lang-tyd limiet bevat, beskryf word. In die praktyk kan die koers funksie bepaal word deur ‘n spektrale berekening uit te voer. Verder, die oplossing van die spektrale probleem stel ons daartoe in staat om ‘n eﬀektiewe proses te konstrueer, wat ‘n gegewe ﬂuksuasie weg van die gemiddeld realiseer en wat verduidelik hoe hierdie ﬂuksuasie dinamies in tyd geskep word. Meeste navorsing in die teorie van groot afwykings handel met Markov diﬀusies wat in ‘n onbegrensde domein (bv. R of Rd) ewolueer. In hierdie tesis oorweeg ons diﬀusies in begrensde domeine met perfekte reﬂeksie by die g rense. Met betrekking tot die een-dimensionele geval, lei ons die gepaste grens toestande op die spektrale probleem af en ondersoek ons die implikasies vir die eﬀektiewe proses. Ons pas hierdie kennis toe om die koers funksie vir die area van die gereﬂekteerde Ornstein-Uhlenbeck proses en gereﬂekteerde Browniese beweging met drif te bereken, en om hul eﬀektiewe proses te bepaal. ‘n Variationele verteenwoordiging van die koers funksie word gebruik om akkurate benaderings van die eﬀektiewe proses vir beide van die stelses wat oorweeg is te konstrueer.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/108166

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