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State-dependent fire models and related renewal processes.

Edoardo Daly¹, Amilcare Porporato

¹Department of Civil and Environmental Engineering & Nicholas School of the Environment and Earth Sciences, Duke University, Durham, North Carolina 27708-0287, USA. edaly@pratt.duke.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|December 13, 2006

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We present a new class of stochastic processes with state-dependent random "fires" that reset system variables. This research models complex dynamics in physical systems, offering insights into renewal point processes.

Area of Science:

Stochastic Processes
Mathematical Physics
Dynamical Systems

Background:

Many physical systems exhibit dynamics influenced by sudden, random events or

Purpose of the Study:

To introduce and analyze a general class of stochastic processes driven by state-dependent random fires.
To investigate the relationship between fire occurrence frequency and system state.
To derive exact probability distributions for system variables and inter-fire times.

Main Methods:

Analysis of various functions relating fire occurrence frequency to the state variable x(t).
Derivation of the probabilistic dynamics of x(t) and inter-fire statistics.
Examination of systems with deterministic drift, additive, and multiplicative Gaussian white noise.

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Main Results:

Established that state-dependent fire dynamics lead to renewal point processes.
Obtained exact probability distributions for the state variable x(t) and inter-fire times.
Characterized the behavior of these processes under different forcing conditions, including Gaussian white noise.

Conclusions:

The introduced class of stochastic processes provides a versatile framework for modeling systems with reset events.
The derived probability distributions offer valuable tools for analyzing and predicting the behavior of such systems.
The study highlights the importance of state-dependent fire activity in shaping the statistical properties of stochastic processes.