Markov property of stationary, discrete, quasi-linear equations (Q1313127)

The equation \[ \begin{cases} -\Delta U(x) + f(U(x)) = A(x) & (x\in \Theta)\\U(x) = 0 & (x \bar\in \Theta)\end{cases} \] is considered, where \(\Delta\) is the discrete Laplace operator, \(\{A(x)\}\) is an i.i.d. \(N(0,1)\) sequence, \(\Theta\) is a finite domain of \(\mathbb{Z}^ d\). It is proven that the existence and uniqueness of the solution holds if \(f\) is monotonic and continuous, and the solution is a Markovian germ field of order 2 if \(f\) is affine.