Probabilistic approach to analysis and geometry on metric measure spaces

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We obtained a Jensen's inequality over complete p-uniformly convex spaces. Further we proved the unique existence of the (non-linear) resolvent associated to a coercive proper lower semi continuous function satisfying a weaker notion of p-uniform convexity on a complete metric space and establish the existence of the minimizer of such functions as the large time limit of the non-linear resolvents, which generalizes the pioneering work by J. Jost for harmonic maps into CAT(0)-spaces. The results can be also applied to Lp-Wasserstein space over complete separable p-uniformly convex spaces. As an application, we solve an initial boundary value problem for p-harmonic maps into CAT(0)-spaces in terms of Cheeger type p-Sobolev spaces. On the other hand, we investigated the spectral bounds for symmetric Markov chains with positive n-step coarse Ricci curvature for not only functions but also maps into complete separable 2-uniformly convex spaces with geometric conditions.