  In this article we prove that nonlinear resolvents of infinitesimal
generators on bounded and convex subdomains of $\C^n$ are decreasing Loewner
chains. Furthermore, we consider the problem of the existence of nonlinear
resolvents on unbounded convex domains in $\C$. In the case of the upper
half-plane, we obtain a complete solution by using that nonlinear resolvents of
certain generators correspond to semigroups of probability measures with
respect to free convolution.
