Continuum limit for some growth models. II. (Q1872226)

[For part I, see Stochastic Processes Appl. 101, No. 1, 1--41 (2002; Zbl 1075.82011).] The paper continues investigation of the classs of growth model introduced in the previous paper. The configuration space \(\Gamma \) consists of functions \(h:\mathbb{Z}^{d}\rightarrow \mathbb{Z}\) satisfying certain growth condition. There are two sequences of independent Poisson clocks \( p^{\pm }(i,t)\), \(i\in \mathbb{Z}^{d}\), with rates \(\lambda ^{\pm }.\) The function \(h\) increases (decreases) by one unit at site \(i\) when the clock \( p^{+}(i,t)\) \ (resp. \(p^{-}(i,t))\) rings and the resulting configuration is still in \(\Gamma ,\) otherwise the change in \(h\) is supressed. The rescaled process \(\varepsilon h([ \frac{x}{\varepsilon }] ,\frac{t}{ \varepsilon })\) is expected to converge to a solution of the Hamilton-Jacobi equation \(u_{t}+H(u_{x})=0.\) This was established in the previous paper in the case where \(\lambda ^{+}\) or \(\lambda ^{-}\) is zero, employing the monotonicity property of the process \(h.\) In this paper a new approach is initiated when both \(\lambda ^{+}\) and \(\lambda ^{-}\) are nonzero and \(d=1\). For higher dimensions it is shown that any limiting process satisfies a Hamilton-Jacobi equation for a suitable (possibly random) Hamiltonian \(H.\)