Convergence rates for semistochastic processes

James Broda; Alexander Grigo; Nikola P. Petrov

doi:10.3934/dcdsb.2019001

Article Contents

2019, Volume 24, Issue 1: 109-125. Doi: 10.3934/dcdsb.2019001

This issue Previous Article Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping Next Article A dimension splitting and characteristic projection method for three-dimensional incompressible flow

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

Received: March 31, 2017

Revised: August 31, 2018

Early access: October 2018

Published: January 2019

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Abstract

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.).

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Mathematics Subject Classification: Primary: 34F05, 60J25, 92D25; Secondary: 60Gxx, 92Bxx.

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Figure 1. Schematic for pre- and post- disturbance levels

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Figure 2. On the construction of the minorizing measure in Theorem 2.1

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Figure 5. Plots of $(1 - \epsilon_{\Delta t, \kappa})^{r / \Delta t }$ vs. $\Delta t$ for selected $\kappa$

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Figure 6. Plots of $(1 - \epsilon_{\Delta t, \kappa})^{r / \Delta t }$ vs. $\kappa$ for selected $\Delta t$

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Figure 3. Plot of $(1 - \epsilon_{\Delta t})^{1/\Delta t}$ vs. $\Delta t$

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Figure 4. Plots of $(1 - \epsilon_{\Delta t})^{\lfloor t/\Delta t \rfloor}$ vs. $t$ for selected values of $\Delta t$

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