Given a field ${B(x)}_{xinmathbf{Z}^d}$ of independent standard Brownian motions, indexed by $mathbf{Z}^d$, the generator of a suitable Markov process on $mathbf{Z}^d,,,mathcal{G},$ and sufficiently nice function $sigma:[0,infty)to[0,infty),$ we cons
ider the influence of the parameter $lambda$ on the behavior of the system, begin{align*} rm{d} u_t(x) = & (mathcal{G}u_t)(x),rm{d} t + lambdasigma(u_t(x))rm{d} B_t(x) qquad[t>0, xinmathbf{Z}^d], &u_0(x)=c_0delta_0(x). end{align*} We show that for any $lambda>0$ in dimensions one and two the total mass $sum_{xinmathbf{Z}^d}u_t(x)to 0$ as $ttoinfty$ while for dimensions greater than two there is a phase transition point $lambda_cin(0,infty)$ such that for $lambda>lambda_c,, sum_{mathbf{Z}^d}u_t(x)to 0$ as $ttoinfty$ while for $lambda<lambda_c,,sum_{mathbf{Z}^d}u_t(x) otto 0$ as $ttoinfty.$
