Browsing by Author "Du Buisson, Johannes Petrus"

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    Dynamical large deviations of diffusions
    (Stellenbosch : Stellenbosch University, 2023-03) Du Buisson, Johannes Petrus; Touchette, Hugo; Muller-Nedebock, Kristian K.; Stellenbosch University. Faculty of Science. Dept. of Physics.
    ENGLISH ABSTRACT: We solve two problems related to the fluctuations o f t ime-integrated function- als of Markov diffusions, u sed i n p hysics t o m odel n onequilibrium s ystems. In the first w e d erive a nd i llustrate t he a ppropriate b oundary c onditions o n the spectral problem used to obtain the large deviations of current-type observables for reflected d iffusions. Fo r th e se cond pr oblem we st udy li near di ffusions and obtain exact results for the generating function associated with linear additive, quadratic additive and linear current-type observables by using the Feynman- Kac formula. We investigate the long-time behavior of the generating function for each of these observables to determine both the so-called rate function and the form of the effective p rocess r esponsible f or m anifesting t he fl uctuations of the associated observable. It is found that for each of these observables, the effective p rocess i s a gain a l inear d iffusion. We ap ply ou r ge neral re sults fo r a variety of linear diffusions i n R ², w ith p articular e mphasis o n i nvestigating the manner in which the density and current of the original process are modified in order to create fluctuations.
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    Large deviations of reflected diffusions
    (Stellenbosch : Stellenbosch University., 2020-04) Du Buisson, Johannes Petrus; Touchette, Hugo; Kastner, Michael; Stellenbosch University. Faculty of Science. Dept. of Physics.
    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.