Ergodicity of infinite systems of stochastic equations (Q582697)

The authors consider the infinite system of stochastic Ito equations \[ d\xi^ t_ z=[b_ 0(\xi^ t_ z)+\epsilon b_ 1(\xi^ t_ z,\xi^ t_{z+\ell_ 1},...,\xi^ t_{z+\ell_ N})]dt+dW^ t_ z,\quad \xi^ 0_ z=\eta_ z, \] z is a point of some lattice \(Z^{\nu}\). If the coefficients \(b_ 0, b_ 1\) are sufficiently smooth, bounded and satisfy some conditions, then exist such \(\epsilon_ 0>0\), \(\theta >0\), \(t_ 0>0\), that for any finite subset X of \(Z^{\nu}\), any \(\epsilon\), \(| \epsilon | <\epsilon_ 0\), \(t\geq t_ 0\), \(t'>0\), \(v_ X\in R^{| X|}\), \[ | p^ X(t+t',v_ X)-p^ X(t,v_ X)| <\theta (| X|)e^{-\theta t}, \] where \(\theta (| X|)>0\), and \(p^ X(t,v_ X)\) is a density of the distribution of the process \((\xi^ t_ z\), \(z\in X)\).