Path integral approach to birth-death processes on a lattice (original) (raw)

Moscow Mathematical Journal

Lattice birth-and-death Markov dynamics of particle systems with spins from Z + are constructed as unique solutions to certain stochastic equations. Pathwise uniqueness, strong existence, Markov property and joint uniqueness in law are proven, and a martingale characterization of the process is given. Sufficient conditions for the existence of an invariant distribution are formulated in terms of Lyapunov functions. We apply obtained results to discrete analogs of the Bolker-Pacala-Dieckmann-Law model and an aggregation model.

THE FORMAL THEORY OF BIRTH-AND-DEATH PROCESSES, LATTICE PATH COMBINATORICS AND CONTINUED FRACTIONS

Classic works of Karlin and McGregor and Jones and Magnus have established a general correspondence between continuous-time birth-and-death processes and continued fractions of the Stieltjes–Jacobi type together with their associated orthogonal polynomials. This fundamental correspondence is revisited here in the light of the basic relation between weighted lattice paths and continued fractions otherwise known from combinatorial theory. Given that sample paths of the embedded Markov chain of a birth-and-death process are lattice paths, Laplace transforms of a number of transient characteristics can be obtained systematically in terms of a fundamental continued fraction and its family of convergent polynomials. Applications include the analysis of evolutions in a strip, upcrossing and downcrossing times under flooring and ceiling conditions, as well as time, area, or number of transitions while a geometric condition is satisfied.

Reaction-diffusion on the fully-connected lattice: A+A→ A

2017

Diffusion-coagulation can be simply described by a dynamic where particles perform a random walk on a lattice and coalesce with probability unity when meeting on the same site. Such processes display non-equilibrium properties with strong fluctuations in low dimensions. In this work we study this problem on the fully-connected lattice, an infinite-dimensional system in the thermodynamic limit, for which mean-field behaviour is expected. Exact expressions for the particle density distribution at a given time and survival time distribution for a given number of particles are obtained. In particular we show that the time needed to reach a finite number of surviving particles (vanishing density in the scaling limit) displays strong fluctuations and extreme value statistics, characterized by a universal class of non-Gaussian distributions with singular behaviour.

Reaction–diffusion on the fully-connected lattice: A+ArightarrowAA+A\rightarrow AA+ArightarrowA

Journal of Physics A: Mathematical and Theoretical

Spatial birth-and-death processes with a finite number of particles

Modern Stochastics: Theory and Applications

The aim of this work is to establish essential properties of spatial birth-and-death processes with general birth and death rates on mathbbRmathrmd{\mathbb{R}^{\mathrm{d}}}mathbbRmathrmd. Spatial birth-and-death processes with time dependent rates are obtained as solutions to certain stochastic equations. The existence, uniqueness, uniqueness in law and the strong Markov property of unique solutions are proven when the integral of the birth rate over mathbbRmathrmd{\mathbb{R}^{\mathrm{d}}}mathbbRmathrmd grows not faster than linearly with the number of particles of the system. Martingale properties of the constructed process provide a rigorous connection to the heuristic generator. The pathwise behavior of an aggregation model is also studied. The probability of extinction and the growth rate of the number of particles under condition of nonextinction are estimated.

Path-integral formulation of stochastic processes for exclusive particle systems

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000

We present a systematic formalism to derive a path-integral formulation for hard-core particle systems far from equilibrium. Writing the master equation for a stochastic process of the system in terms of the annihilation and creation operators with mixed commutation relations, we find the Kramers-Moyal coefficients for the corresponding Fokker-Planck equation (FPE), and the stochastic differential equation (SDE) is derived by connecting these coefficients in the FPE to those in the SDE. Finally, the SDE is mapped onto field theory using the path integral, giving the field-theoretic action, which may be analyzed by the renormalization group method. We apply this formalism to a two-species reaction-diffusion system with drift, finding a universal decay exponent for the long-time behavior of the average concentration of particles in arbitrary dimension.

The Reuter-Ledermann representation for birth and death processes

Proceedings of the American Mathematical Society, 1976

The identification of the mass of the integrator at zero is made for the integral representation obtained by Reuter and Ledermann for the transition probabilities of birth and death processes. An ergodic theorem is given as an application of this result.

Lévy walks on lattices as multi-state processes

Journal of Statistical Mechanics: Theory and Experiment, 2015

Continuous-time random walks combining diffusive scattering and ballistic propagation on lattices model a class of Lévy walks. The assumption that transitions in the scattering phase occur with exponentially-distributed waiting times leads to a description of the process in terms of multiple states, whose distributions evolve according to a set of delay differential equations, amenable to analytic treatment. We obtain an exact expression of the mean squared displacement associated with such processes and discuss the emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of initial conditions on the transport coefficients. Of particular interest is the case of rare ballistic propagation, in which case a regime of superdiffusion may lurk underneath one of normal diffusion.

Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum

Journal of Statistical Physics

We study a spatial birth-and-death process on the phase space of locally finite configurations Γ`ˆΓ´over R d. Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator L`pγ´q`1 ε L´, ε ą 0. Here L´describes the environment process on Γ´and L`pγ´q describes the system process on Γ`, where γ´indicates that the corresponding birth-and-death rates depend on another locally finite configuration γ´P Γ´. We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states µ ε t on Γ`ˆΓ´. Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let µ inv be the invariant measure for the environment process on Γ´. In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of µ ε t onto Γ`converges weakly to an evolution of states on Γ`associated with the averaged Markov birth-and-death operator L " ş Γ´L`p γ´qdµ inv pγ´q.

Path integral approach to birth-death processes on a lattice (original) (raw)

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