Models written in terms of stochastic delay differential equations (SDDE's) have recently appeared in a number of fields, such as physiology, optics, and climatology. Unfortunately, the development of a Fokker-Planck approach for these equations is being hampered by their non-Markovian nature. In this thesis, an exact Fokker-Planck equation (FPE) is formulated for univariate SDDE's involving Gaussian white noise. Although this FPE is not self-sufficient, it is found to be helpful in at least two different contexts: with a short delay approximation and under an appropriate separation of time scales. In the short delay approximation, a Taylor expansion is applied to an SDDE with nondelayed diffusion and yields a nondelayed stochastic differential equation. The aforementioned FPE then allows the derivation of an alternate and complementary approximation of the original SDDE. This method is illustrated with linear and logistic SDDE's. Under the separation of time scales assumption, the FPE of a bistable system is reduced to a form that is uniquely determined by the steady-state probability density when the diffusion term of the SDDE is nondelayed. In the context of an overdamped particle with delayed coupling to a symmetrical and stochastically driven potential, the resulting FPE is used with standard techniques to express the transition rate between wells in terms of the noise amplitude and of the steady-state probability density. The same is also accomplished for the mean first passage time from one point to another. This whole approach is then applied to the case of a quartic potential, for which all realisations eventually stabilise on an oscillatory trajectory with an ever increasing amplitude. Although this latter phenomenon prevents the existence of a steady-state limit, a pseudo-steady-state probability density can be defined and used instead of the non-existent steady-state one when the transition rate to these unbounded oscillatory trajectories is sufficiently small. The transition to this peculiar attractor is investigated in more detail for a family of single-well potentials, and interestingly, the transition rate follows Arrhenius' law when the noise amplitude is small. Overall, it is found that the Fokker-Planck approach can play a significant role in the analysis of SDDE's.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/9127 |
| Date | January 2001 |
| Creators | Guillouzic, Steve. |
| Contributors | L'Heureux, Ivan, |
| Publisher | University of Ottawa (Canada) |
| Source Sets | Université d’Ottawa |
| Detected Language | English |
| Type | Thesis |
| Format | 100 p. |
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