Analysis of Stochastic Differential Equations

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The present proposal focusses on some aspects of the analysis of stochastic differential equations that the PI investigated during the last years and that he would like to further develop within the next years.More precisely, the present proposal focusses on three directions of research. In the first direction of research, the PI will study stochastic differential equations driven by fractional Brownian motions. Such equations naturally arise as candidates for the evolution of rough and non-Markovian systems. A better understanding of this theory which is now at its beginnings would certainly provide a deeper understanding of non-Markovian systems that can be observed in different settings, by e.g.financial mathematics, communication networks, turbulence phenomena. In the second direction of research, the PI will study functional inequalities, like gradient bounds for subelliptic heat semigroups. This study could lead to a better understanding of the control of the rate of convergence to equilibrium for subelliptic systems and to a subelliptic generalization of lower Ricci bounds. Finally in the third direction of research, the PI will study subelliptic heat kernels asymptotics on bundles. In the elliptic case, this study provides a striking and fascinating proof of the Atiyah-Singer index theorem. By these methods, the PI would like to study possible index theorems in subelliptic geometry.Randomness is a phenomenon present in everyday life.Predicting traffic flows, communications networks, genetic issues, stock prices on financial markets are examples where stochastic differential equations can be used to model performance. Stochastic differential equations are a mathematical tool describing the evolution in time of a system involving randomness. This project focusses on the theoretical study of such objects and to its applications in different areas within mathematics or applied mathematics.