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January  2019, 24(1): 109-125. doi: 10.3934/dcdsb.2019001

Convergence rates for semistochastic processes

 Department of Mathematics, University of Oklahoma, Norman, OK, 73019, USA

* Corresponding author. Present address: Quantitative Reasoning Program and Department of Mathematics, Bowdoin College, Brunswick, ME 04011, USA

J.B. and N.P.P. were partially supported by NSF grant DMS-0807658. A.G. was partially supported NSF grant DMS-1413428. N.P.P. was also generously supported by the Nancy Scofield Hester Presidential Professorship. We thank Martin Oberlack for useful suggestions

Received  April 2017 Revised  September 2018 Published  October 2018

We study processes that consist of deterministic evolution punctuated at random times by disturbances with random severity; we call such processes semistochastic. Under appropriate assumptions such a process admits a unique stationary distribution. We develop a technique for establishing bounds on the rate at which the distribution of the random process approaches the stationary distribution. An important example of such a process is the dynamics of the carbon content of a forest whose deterministic growth is interrupted by natural disasters (fires, droughts, insect outbreaks, etc.).

Citation: James Broda, Alexander Grigo, Nikola P. Petrov. Convergence rates for semistochastic processes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 109-125. doi: 10.3934/dcdsb.2019001
References:

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References:
Schematic for pre- and post- disturbance levels
On the construction of the minorizing measure in Theorem 2.1
Plots of $(1 - \epsilon_{\Delta t, \kappa})^{r / \Delta t }$ vs. $\Delta t$ for selected $\kappa$
Plots of $(1 - \epsilon_{\Delta t, \kappa})^{r / \Delta t }$ vs. $\kappa$ for selected $\Delta t$
Plot of $(1 - \epsilon_{\Delta t})^{1/\Delta t}$ vs. $\Delta t$
Plots of $(1 - \epsilon_{\Delta t})^{\lfloor t/\Delta t \rfloor}$ vs. $t$ for selected values of $\Delta t$
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