Spatial Birth-death Processes Research Articles

Functional central limit theorems for local statistics of spatial birth–death processes in the thermodynamic regime

Functional central limit theorems for local statistics of spatial birth–death processes in the thermodynamic regime

Reaction diffusion systems and extensions of quantum stochastic processes

Reaction diffusion systems describe the behaviour of dynamic, interacting, particulate systems. Quantum stochastic processes generalise Brownian motion and Poisson processes, having operator valued Itô calculus machinery. Here it is shown that the three standard noises of quantum stochastic processes can be extended to model reaction diffusion systems, the methods being exemplified with spatial birth–death processes. The usual approach for these systems are master equations, or Doi-Peliti path integration techniques. The machinery described here provide efficient analyses for many systems of interest, and offer an alternative set of tools to investigate such problems.

Percolation and connection times in multi-scale dynamic networks

We study the effects of mobility on two crucial characteristics in multi-scale dynamic networks: percolation and connection times. Our analysis provides insights into the question, to what extent long-time averages are well-approximated by the expected values of the corresponding quantities, i.e., the percolation and connection probabilities. In particular, we show that in multi-scale models, strong random effects may persist in the limit. Depending on the precise model choice, these may take the form of a spatial birth–death process or a Brownian motion. Despite the variety of structures that appear in the limit, we show that they can be tackled in a common framework with the potential to be applicable more generally in order to identify limits in dynamic spatial network models going beyond the examples considered in the present work.

Duality relations between spatial birth–death processes and diffusions in Hilbert space

Spatially dependent birth–death processes can be modelled by kinetic models such as the BBGKY hierarchy. Diffusion in infinite dimensional systems can be modelled with Brownian motion in Hilbert space. In this work Doi field theoretic formalism is utilised to establish dualities between these classes of processes. This enables path integral methods to calculate expectations of duality functions. These are exemplified with models ranging from stochastic cable signalling to jump-diffusion processes.

The discretely observed immigration-death process: Likelihood inference and spatiotemporal applications

ABSTRACTWe consider a stochastic process, the homogeneous spatial immigration-death (HSID) process, which is a spatial birth-death process with as building blocks (i) an immigration-death (ID) process (a continuous-time Markov chain) and (ii) a probability distribution assigning iid spatial locations to all events. For the ID process, we derive the likelihood function, reduce the likelihood estimation problem to one dimension, and prove consistency and asymptotic normality for the maximum likelihood estimators (MLEs) under a discrete sampling scheme. We additionally prove consistency for the MLEs of HSID processes. In connection to the growth-interaction process, which has a HSID process as basis, we also fit HSID processes to Scots pine data.

Gibbs point process approximation: Total variation bounds using Stein’s method

We obtain upper bounds for the total variation distance between the distributions of two Gibbs point processes in a very general setting. Applications are provided to various well-known processes and settings from spatial statistics and statistical physics, including the comparison of two Lennard-Jones processes, hard core approximation of an area interaction process and the approximation of lattice processes by a continuous Gibbs process. Our proof of the main results is based on Stein's method. We construct an explicit coupling between two spatial birth-death processes to obtain Stein factors, and employ the Georgii-Nguyen-Zessin equation for the total bound.

Transforming Spatial Point Processes into Poisson Processes Using Random Superposition

Most finite spatial point process models specified by a density are locally stable, implying that the Papangelou intensity is bounded by some integrable functionβdefined on the space for the points of the process. It is possible to superpose a locally stable spatial point processXwith a complementary spatial point processYto obtain a Poisson processX⋃Ywith intensity functionβ. Underlying this is a bivariate spatial birth-death process (Xt,Yt) which converges towards the distribution of (X,Y). We study the joint distribution ofXandY, and their marginal and conditional distributions. In particular, we introduce a fast and easy simulation procedure forYconditional onX. This may be used for model checking: given a model for the Papangelou intensity of the original spatial point process, this model is used to generate the complementary process, and the resulting superposition is a Poisson process with intensity functionβif and only if the true Papangelou intensity is used. Whether the superposition is actually such a Poisson process can easily be examined using well-known results and fast simulation procedures for Poisson processes. We illustrate this approach to model checking in the case of a Strauss process.

Transforming Spatial Point Processes into Poisson Processes Using Random Superposition

Most finite spatial point process models specified by a density are locally stable, implying that the Papangelou intensity is bounded by some integrable function β defined on the space for the points of the process. It is possible to superpose a locally stable spatial point process X with a complementary spatial point process Y to obtain a Poisson process X ⋃ Y with intensity function β. Underlying this is a bivariate spatial birth-death process (X t , Y t ) which converges towards the distribution of (X, Y). We study the joint distribution of X and Y, and their marginal and conditional distributions. In particular, we introduce a fast and easy simulation procedure for Y conditional on X. This may be used for model checking: given a model for the Papangelou intensity of the original spatial point process, this model is used to generate the complementary process, and the resulting superposition is a Poisson process with intensity function β if and only if the true Papangelou intensity is used. Whether the superposition is actually such a Poisson process can easily be examined using well-known results and fast simulation procedures for Poisson processes. We illustrate this approach to model checking in the case of a Strauss process.

Spatial birth-death processes with multiple changes and applications to batch service networks and clustering processes

Reversible spatial birth-death processes are studied with simultaneous jumps of multi-components. A relationship is established between (i) a product-form solution, (ii) a partial symmetry condition on the jump rates and (iii) a solution of a deterministic concentration equation. Applications studied are reversible networks of queues with batch services and blocking and clustering processes such as those found in polymerization chemistry. As illustrated by examples, known results are hereby unified and extended. An expectation interpretation of the transition rates is included.

Spatial birth-death processes with multiple changes and applications to batch service networks and clustering processes

Reversible spatial birth-death processes are studied with simultaneous jumps of multi-components. A relationship is established between (i) a product-form solution, (ii) a partial symmetry condition on the jump rates and (iii) a solution of a deterministic concentration equation. Applications studied are reversible networks of queues with batch services and blocking and clustering processes such as those found in polymerization chemistry. As illustrated by examples, known results are hereby unified and extended. An expectation interpretation of the transition rates is included.