Differential forms on Dirichlet spaces and Bakry-\'Emery estimates on   metric graphs
  We develop a general framework on Dirichlet spaces to prove a weak form of the Bakry-\'Emery estimate and study its consequences. This estimate may be satisfied in situations, like metric graphs, where generalized notions of Ricci curvature lower bounds are not available. 
Fabrice Baudoin, Daniel J. Kelleher
math.PR math.DG math.MG