Objective Many important properties of stochastic processes are deeply connected with the underlying geometric structure. The crucial quantity in many applications is a lower bound on the Ricci curvature, which yields powerful applications to concentration of measure, isoperimetry, and convergence to equilibrium.Since many important processes are defined in discrete, infinite-dimensional, or singular spaces, major research activity has been devoted to developing a theory of Ricci curvature beyond the classicalRiemannian setting. This led to the powerful theories of Bakry-Émery and Lott-Sturm-Villani, which have been extremely successful in the analysis of geodesic spaces and diffusion processes. Building on our recent work, we will develop a wide research program that allows us to significantly enlarge the scope of these ideas.A) Firstly, we develop a comprehensive theory of curvature-dimension for discrete spaces based on geodesic convexity of entropy functionals along discrete optimal transport. Promising first results suggest that the theory initiated by the PI provides the appropriate framework for obtaining many powerful results from geometric analysis in the discrete setting.B) Secondly, we analyse discrete stochastic dynamics using methods from optimal transport.We focus on non-reversible Markov processes, which requires a significant extension of the existing gradient flow theory, and develop new methods for proving convergence of discrete stochastic dynamics.C) Thirdly, we develop an optimal transport approach to the analysis of quantum Markov processes. We will perform a thorough investigation of noncommutative optimal transport, we aim for geometric and functional inequalities in quantum probability, and apply the results to the analysisof quantum Markov processes.The project extends the scope of optimal transport methods significantly and makes a fundamental contribution to the conceptual understanding of discrete curvature. Fields of science natural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicsapplied mathematicsstatistics and probabilitynatural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equationsnatural scienceschemical sciences Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-2016-STG - ERC Starting Grant Call for proposal ERC-2016-STG See other projects for this call Funding Scheme ERC-STG - Starting Grant Host institution INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA Net EU contribution € 1 074 590,00 Address Am Campus 1 3400 Klosterneuburg Austria See on map Region Ostösterreich Niederösterreich Wiener Umland/Nordteil Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 074 590,00 Beneficiaries (1) Sort alphabetically Sort by Net EU contribution Expand all Collapse all INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA Austria Net EU contribution € 1 074 590,00 Address Am Campus 1 3400 Klosterneuburg See on map Region Ostösterreich Niederösterreich Wiener Umland/Nordteil Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 074 590,00
