scholarly journals Corrigendum to: Asymptotic shape and the speed of propagation of continuous-time continuous-space birth processes

2020 ◽  
Vol 52 (4) ◽  
pp. 1325-1327
Author(s):  
V. Bezborodov
Keyword(s):  
Continuous Time ◽  
Continuous Space ◽  
Asymptotic Shape ◽  
Speed Of Propagation ◽  
Birth Processes

AbstractWe would like to correct the statement of Lemma 4.1 in [BDK+18].

scholarly journals Asymptotic shape and the speed of propagation of continuous-time continuous-space birth processes

2018 ◽  
Vol 50 (01) ◽  
pp. 74-101 ◽  
Author(s):  
Viktor Bezborodov ◽  
Luca Di Persio ◽  
Tyll Krueger ◽  
Mykola Lebid ◽  
Tomasz Ożański
Keyword(s):  
Continuous Time ◽  
Birth Rate ◽  
Growth Models ◽  
Classical Lattice ◽  
Continuous Space ◽  
Asymptotic Shape ◽  
Subadditive Ergodic Theorem ◽  
Speed Of Propagation ◽  
Birth Processes ◽  
Precise Expression

AbstractWe formulate and prove a shape theorem for a continuous-time continuous-space stochastic growth model under certain general conditions. Similar to the classical lattice growth models, the proof makes use of the subadditive ergodic theorem. A precise expression for the speed of propagation is given in the case of a truncated free-branching birth rate.

Exponential ergodicity for single-birth processes

2004 ◽  
Vol 41 (4) ◽  
pp. 1022-1032 ◽  
Author(s):  
Yong-Hua Mao ◽  
Yu-Hui Zhang
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
New Method ◽  
Sufficient Condition ◽  
Exponential Ergodicity ◽  
Comparison Methods ◽  
Birth Processes ◽  
Single Birth ◽  
Continuous Time Branching Processes

An explicit, computable, and sufficient condition for exponential ergodicity of single-birth processes is presented. The corresponding criterion for birth–death processes is proved using a new method. As an application, some sufficient conditions are obtained for exponential ergodicity of an extended class of continuous-time branching processes and of multidimensional Q-processes, by comparison methods.

On the identification of continuous-time Markov chains with a given invariant measure

1994 ◽  
Vol 31 (04) ◽  
pp. 897-910
Author(s):  
P. K. Pollett
Keyword(s):  
Invariant Measure ◽  
Continuous Time ◽  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Birth Process ◽  
Continuous Time Markov Chains ◽  
Branching Process With Immigration ◽  
Birth Processes ◽  
Necessary And Sufficient

In [14] a necessary and sufficient condition was obtained for there to exist uniquely a Q-process with a specified invariant measure, under the assumption that Q is a stable, conservative, single-exit matrix. The purpose of this note is to demonstrate that, for an arbitrary stable and conservative q-matrix, the same condition suffices for the existence of a suitable Q-process, but that this process might not be unique. A range of examples is considered, including pure-birth processes, a birth process with catastrophes, birth-death processes and the Markov branching process with immigration.

Strong ergodicity for single-birth processes

2001 ◽  
Vol 38 (01) ◽  
pp. 270-277 ◽  
Author(s):  
Yu-Hui Zhang
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Strong Ergodicity ◽  
Finite Dimensional ◽  
Comparison Methods ◽  
Schlögl Model ◽  
Birth Processes ◽  
Single Birth ◽  
Continuous Time Branching Processes

An explicit and computable criterion for strong ergodicity of single-birth processes is presented. As an application, some sufficient conditions are obtained for strong ergodicity of an extended class of continuous-time branching processes and multi-dimensional Q-processes by comparison methods respectively. Consequently strong ergodicity of the Q-process corresponding to the finite-dimensional Schlögl model is proven.

Exponential ergodicity for single-birth processes

2004 ◽  
Vol 41 (04) ◽  
pp. 1022-1032 ◽  
Author(s):  
Yong-Hua Mao ◽  
Yu-Hui Zhang
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
New Method ◽  
Sufficient Condition ◽  
Exponential Ergodicity ◽  
Comparison Methods ◽  
Birth Processes ◽  
Single Birth ◽  
Continuous Time Branching Processes

An explicit, computable, and sufficient condition for exponential ergodicity of single-birth processes is presented. The corresponding criterion for birth–death processes is proved using a new method. As an application, some sufficient conditions are obtained for exponential ergodicity of an extended class of continuous-time branching processes and of multidimensional Q-processes, by comparison methods.

scholarly journals Boundary effect in competition processes

2019 ◽  
Vol 56 (3) ◽  
pp. 750-768
Author(s):  
Vadim Shcherbakov ◽  
Stanislav Volkov
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Boundary Effect ◽  
Competitive Interaction ◽  
Continuous Time Markov Chain ◽  
Urn Models ◽  
Birth Processes ◽  
Special Case

AbstractThis paper is devoted to studying the long-term behaviour of a continuous-time Markov chain that can be interpreted as a pair of linear birth processes which evolve with a competitive interaction; as a special case, they include the famous Lotka–Volterra interaction. Another example of our process is related to urn models with ball removal. We show that, with probability one, the process eventually escapes to infinity by sticking to the boundary in a rather unusual way.

scholarly journals General methods and properties to evaluate continuum limits of the 1D discrete time quantum walk

2020 ◽  
Vol 19 (10) ◽  
Author(s):  
Michael Manighalam ◽  
Mark Kon
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Systems ◽  
Quantum Walks ◽  
Time Limit ◽  
Continuous Limit ◽  
Continuous Space ◽  
Time Quantum ◽  
Space Limit

Abstract Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and nonrelativistic regimes (Molfetta GD, Arrighi P. A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in Molfetta and Arrighi (A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions (“coins”) admit nontrivial continuum limits. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. Finally, we demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk when the coin is allowed to transition through the continuous limit process.

scholarly journals Geometric fluid approximation for general continuous-time Markov chains

2019 ◽  
Vol 475 (2229) ◽  
pp. 20190100 ◽  
Author(s):  
Michalis Michaelides ◽  
Jane Hillston ◽  
Guido Sanguinetti
Keyword(s):  
Population Structure ◽  
Continuous Time ◽  
Transition Matrix ◽  
Continuous Process ◽  
Gaussian Process Regression ◽  
Great Success ◽  
Fluid Approximation ◽  
Continuous Space ◽  
Continuous State ◽  
Macro Scale

Fluid approximations have seen great success in approximating the macro-scale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuous-time Markov chain (CTMC) having a particular population structure which suggests a natural continuous state-space endowed with a dynamics for the approximating process. We construct here a general method based on spectral analysis of the transition matrix of the CTMC, without the need for a population structure. Specifically, we use the popular manifold learning method of diffusion maps to analyse the transition matrix as the operator of a hidden continuous process. An embedding of states in a continuous space is recovered, and the space is endowed with a drift vector field inferred via Gaussian process regression. In this manner, we construct an ordinary differential equation whose solution approximates the evolution of the CTMC mean, mapped onto the continuous space (known as the fluid limit).

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