  In this paper we show how techniques coming from stochastic analysis, such as
stochastic completeness (in the form of the weak maximum principle at
infinity), parabolicity and $L^p$-Liouville type results for the weighted
Laplacian associated to the potential may be used to obtain triviality,
rigidity results, and scalar curvature estimates for gradient Ricci solitons
under $L^p$ conditions on the relevant quantities.
