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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.

We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geometric quantities are derived. Some basic tools from the weighted manifold theory such as general weighted volume comparisons and maximum principles at infinity for diffusion operators are discussed.

In this paper we presents some Liouville type theorems for solutions of di erential inequalities involving the '-Laplacian. Our results, in particular, improve and generalize known results for the Laplacian and the p-Laplacian, and are new even in these cases. Phragmen-Lindelo type results, and a weak form of the Omori-Yau maximum principle are also discussed. 0. Introduction. Let (M h i) be a smooth, connected, non-compact, complete Riemannian manifold of dimension m. We x an origin o, and denote by r(x) the distance function from o, and by B t = fx 2 M : r(x) < tg and @ B t = fx 2 M : r(x) = tg the geodesic ball and sphere of radius

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