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Existence of the zero range process and a deposition model with superlinear growth rates. (English) Zbl 1138.60340
This paper gives a rigorous construction of the bricklayer’s process with exponential jump rates and establishes its ergodicity. This process was introduced and studied by M. Balasz in [J. Statist. Phys. 105, 511–524 (2001; Zbl 1017.82035)] and [J. Statist. Phys. 117, 77–98 (2004; Zbl 1017.82035)]. The bircklayer’s process can be described either by the height process \((\underline h(t); t\geq 0)\) or by the increment process \((\underline \omega(t); t\geq 0)\) where for \(i\in\mathbb Z\), \( \omega_i = h_{i-1} - h_i\). One can set \(h_0(0)=0\). Given a configuration \(\underline h\) in \(\Omega:=\mathbb Z^\mathbb Z\) and \(i\in \mathbb Z\), let \(\underline h^{(i)}\) be the configuration \[ (\underline h^{(i)})_j =\begin{cases} h_j & {\text{ if}}\, j\neq i\cr h_j +1 & {\text{ if}}\, j=i\end{cases} \] Formally, the jump \(\underline h\to \underline h^{(i)}\) of the height process occurs independently for each site \(i\) with rate \(r(\omega_i) + r(\omega_{i+1})\). The assumptions on the rate function \(r\) are that it is strictly increasing on \(\mathbb N\) with \(\lim_{z\to \infty } r(z) =+\infty\) and it has exponential growth, that is, there is a constant \(\beta> 0\) such that \(r(z) < e^{\beta z}\) for all \(z\in\mathbb N\). The jump rates for negative \(z\) are determined by \(r(z)r(1-z) = 1\). Under these conditions, there is a family of i.i.d. invariant distributions which are ergodic and extremal. It is shown that the appropriate state space \(\tilde \Omega\subset \mathbb Z^\mathbb Z\) is the space of configurations whose asymptotic slopes are Cesarò bounded in some sense. When the initial increments \(\underline \omega\) is in \(\tilde \Omega\), then \( \underline \omega(\cdot)\) and \( \underline h (\cdot)\) are both Markov processes with cadlag paths. The height process is first constructed as a monotone non-decreasing limit of systems with finitely many sites. The construction of the infinite systems uses conditional couplings to compare processes with different initial configurations and attractivity with bounds on the probability that a large block of adjacent sites all experience a jump in a given time interval. Analytic properties of the semigroup and the generator are also obtained.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
Full Text: DOI arXiv
[1] Andjel, E. D. (1982). Invariant measures for the zero range process. Ann. Probab. 10 525–547. · Zbl 0492.60096
[2] Balázs, M. (2001). Microscopic shape of shocks in a domain growth model. J. Statist. Phys. 105 511–524. · Zbl 1017.82035
[3] Balázs, M. (2003). Growth fluctuations in a class of deposition models. Ann. Inst. H. Poincaré Probab. Statist. 39 639–685. · Zbl 1029.60075
[4] Balázs, M. (2003). Stochastic bounds on the zero range processes with superlinear jump rates. Period. Math. Hungar. 47 17–27. · Zbl 1045.60101
[5] Balázs, M. (2004). Multiple shocks in bricklayers’ model. J. Statist. Phys. 117 77–98. · Zbl 1108.82031
[6] Booth, L. (2002). Random spatial structures and sums. Ph.D. thesis, Utrecht Univ. · Zbl 1040.35122
[7] Liggett, T. M. (1973). An infinite particle system with zero range interactions. Ann. Probab. 1 240–253. · Zbl 0264.60083
[8] Liggett, T. M. (1985). Interacting Particle Systems . Springer, New York. · Zbl 0559.60078
[9] Norris, J. R. (1997). Markov Chains . Cambridge Univ. Press. · Zbl 0938.60058
[10] Quant, C. (2002). On the construction and stationary distributions of some spatial queueing and particle systems. Ph.D. thesis, Utrecht Univ.
[11] Rezakhanlou, F. (1995). Microscopic structure of shocks in one conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 119–153. · Zbl 0836.76046
[12] Rosenblatt, M. (1971). Markov Processes. Structure and Asymptotic Behavior . Springer, New York. · Zbl 0236.60002
[13] Royden, H. L. (1988). Real Analysis , 3rd ed. Macmillan Publishing Company, New York. · Zbl 0704.26006
[14] Sethuraman, S. (2001). On extremal measures for conservative particle systems. Ann. Inst. H. Poincaré Probab. Statist. 37 139–154. · Zbl 0981.60098
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