scholarly journals On the continuous dependence of the stationary distribution of a piecewise deterministic Markov process on its jump intensity

2020 ◽  
Author(s):  
Dawid Czapla ◽  
Sander C. Hille ◽  
Katarzyna Horbacz ◽  
Hanna Wojewódka-Ściążko
Keyword(s):  
Markov Process ◽  
Stationary Distribution ◽  
Continuous Dependence ◽  
Piecewise Deterministic Markov Process ◽  
Jump Intensity

scholarly journals On level crossings for a general class of piecewise-deterministic Markov processes

2008 ◽  
Vol 40 (03) ◽  
pp. 815-834 ◽  
Author(s):  
K. Borovkov ◽  
G. Last
Keyword(s):  
Markov Process ◽  
Point Process ◽  
Conditional Distribution ◽  
Compound Poisson Process ◽  
Rate Function ◽  
Intensity Function ◽  
Level Crossing ◽  
Piecewise Deterministic Markov Process ◽  
Piecewise Deterministic Markov Processes ◽  
Jump Intensity

We consider a piecewise-deterministic Markov process (Xt) governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. The paper deals with the point processof upcrossings of some levelbby (Xt). We prove a version of Rice's formula relating the stationary density of (Xt) to level crossing intensities and show that, for a wide class of processes (Xt), asb→ ∞, the scaled point processwhere ν+(b) denotes the intensity of upcrossings ofb, converges weakly to a geometrically compound Poisson process.

scholarly journals Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process

2020 ◽  
Vol 17 (2) ◽  
pp. 1059-1073
Author(s):  
Dawid Czapla ◽  
◽  
Sander C. Hille ◽  
Katarzyna Horbacz ◽  
Hanna Wojewódka-Ściążko ◽  
...  
Keyword(s):  
Markov Process ◽  
Invariant Measure ◽  
Continuous Dependence ◽  
Piecewise Deterministic Markov Process ◽  
Jump Rate

scholarly journals On level crossings for a general class of piecewise-deterministic Markov processes

2008 ◽  
Vol 40 (3) ◽  
pp. 815-834 ◽  
Author(s):  
K. Borovkov ◽  
G. Last
Keyword(s):  
Markov Process ◽  
Point Process ◽  
Conditional Distribution ◽  
Compound Poisson Process ◽  
Rate Function ◽  
Intensity Function ◽  
Level Crossing ◽  
Piecewise Deterministic Markov Process ◽  
Piecewise Deterministic Markov Processes ◽  
Jump Intensity

We consider a piecewise-deterministic Markov process (Xt) governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. The paper deals with the point process of upcrossings of some level b by (Xt). We prove a version of Rice's formula relating the stationary density of (Xt) to level crossing intensities and show that, for a wide class of processes (Xt), as b → ∞, the scaled point process where ν+(b) denotes the intensity of upcrossings of b, converges weakly to a geometrically compound Poisson process.

scholarly journals ON/OFF STORAGE SYSTEMS WITH STATE-DEPENDENT INPUT, OUTPUT, AND SWITCHING RATES

2005 ◽  
Vol 19 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Onno Boxma ◽  
Haya Kaspi ◽  
Offer Kella ◽  
David Perry
Keyword(s):  
Markov Process ◽  
Stationary Distribution ◽  
Existence And Uniqueness ◽  
Storage Systems ◽  
Input Output ◽  
Piecewise Deterministic Markov Process ◽  
Storage Model ◽  
Dependent Rate ◽  
State Dependent ◽  
Special Case

We consider a storage model that can be on or off. When on, the content increases at some state-dependent rate and the system can switch to the off state at a state-dependent rate as well. When off, the content decreases at some state-dependent rate (unless it is at zero) and the system can switch to the on position at a state-dependent rate. This process is a special case of a piecewise deterministic Markov process. We identify the stationary distribution and conditions for its existence and uniqueness.

The reverse Galton-Watson process

1975 ◽  
Vol 12 (03) ◽  
pp. 574-580 ◽  
Author(s):  
Warren W. Esty
Keyword(s):  
Markov Process ◽  
Stationary Distribution ◽  
Probability Function ◽  
Transition Probabilities ◽  
Initial State ◽  
Reverse Process ◽  
Watson Process ◽  
Step Transition

Consider the following path, Zn (w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn , which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN –1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ 2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

scholarly journals Entropic Equilibria Selection of Stationary Extrema in Finite Populations

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 631
Author(s):  
Marc Harper ◽  
Dashiell Fryer
Keyword(s):  
Markov Process ◽  
Population Size ◽  
Mutation Rate ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Local Maxima ◽  
Moran Process ◽  
The Stability ◽  
Definition Of ◽  
Selection Of

We propose the entropy of random Markov trajectories originating and terminating at the same state as a measure of the stability of a state of a Markov process. These entropies can be computed in terms of the entropy rates and stationary distributions of Markov processes. We apply this definition of stability to local maxima and minima of the stationary distribution of the Moran process with mutation and show that variations in population size, mutation rate, and strength of selection all affect the stability of the stationary extrema.

The reverse Galton-Watson process

1975 ◽  
Vol 12 (3) ◽  
pp. 574-580 ◽  
Author(s):  
Warren W. Esty
Keyword(s):  
Markov Process ◽  
Stationary Distribution ◽  
Probability Function ◽  
Transition Probabilities ◽  
Initial State ◽  
Reverse Process ◽  
Watson Process ◽  
Step Transition

Consider the following path, Zn(w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn, which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN–1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

Markov-renewal fluid queues

2004 ◽  
Vol 41 (3) ◽  
pp. 746-757 ◽  
Author(s):  
Guy Latouche ◽  
Tetsuya Takine
Keyword(s):  
Markov Process ◽  
Exponential Distribution ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Input Rate ◽  
Fluid Queue ◽  
Birth And Death Processes ◽  
Fluid Queues ◽  
Semi Markov Process ◽  
Semi Markov Processes

We consider a fluid queue controlled by a semi-Markov process and we apply the Markov-renewal approach developed earlier in the context of quasi-birth-and-death processes and of Markovian fluid queues. We analyze two subfamilies of semi-Markov processes. In the first family, we assume that the intervals during which the input rate is negative have an exponential distribution. In the second family, we take the complementary case and assume that the intervals during which the input rate is positive have an exponential distribution. We thoroughly characterize the structure of the stationary distribution in both cases.

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