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Motivated by the stochastic dynamics of some ecohydrological and neuronal processes, we discuss probabilistic dynamics of systems driven by state-dependent jumps resulting from a non-homogeneous Poisson process. Starting from a one-dimensional system described by a Langevin-type of equation, we construct the master or Chapman Kolmogorov equation and present the general solution in steady state conditions for the case of exponentially distributed jumps. Specific cases of soil moisture and neuronal dynamics are illustrated.
We then move to the analysis of the inter-times between jumps, beginning from the case of state-dependent renewal processes. For these systems the pdf of the jump inter-times is linked to the so-called survivor function and the steady state pdf of the state variable. The case of wild fires in arid ecosystems is briefly discussed as an example. More complicated cases of renewal processes with Gaussian noise and non renewal processes are also discussed, with particular attention to a simple model of neuronal activity. Finally, the phenomenon of noise induced transitions is analyzed, giving rise to temporal persistence and preferential states in spatially extended systems. We also discuss some recent generalizations to n-dimensional systems with specific examples for linear oscillators and chaotic flows. Porporato A., D'Odorico P., Phase transitions driven by state-dependent Poisson noise, Phys. Rev. Lett. 92(11), 110601, 2004. Daly, E., and A. Porporato, State-dependent fire models and related renewal processes, Phys. Rev. E, 74, 041112, 2006. Daly, E., and A. Porporato, Inter-time jump statistics of state-dependent Poisson processes, Phys. Rev. E, 75, 011119, 2007. |
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