Analysis of Stochastic Differential Equations

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本提案的重点是PI在过去几年中研究的随机微分方程分析的一些方面，他希望在未来几年内进一步发展。更准确地说，目前的建议侧重于三个研究方向。在第一个研究方向上，PI将研究分数布朗运动驱动的随机微分方程。这样的方程自然会成为粗糙和非马尔可夫系统演化的候选方程。更好地理解这一理论，当然会提供对非马尔可夫系统的更深入的理解，这些系统可以在不同的环境中观察到，例如金融数学、通信网络、湍流现象。在第二个研究方向上，PI将研究泛函不等式，如亚椭圆热半群的梯度界。这一研究有助于更好地理解亚椭圆系统收敛到平衡的速率的控制，以及对下里奇界的亚椭圆推广。最后，在第三个研究方向上，PI将研究束上的亚椭圆热核渐近性。在椭圆的情况下，本研究为Atiyah-Singer指数定理提供了一个引人注目和引人入胜的证明。通过这些方法，PI想要研究亚椭圆几何中可能的指数定理。随机性是日常生活中存在的一种现象。预测交通流量，通信网络，遗传问题，金融市场的股票价格都是随机微分方程可以用来模拟性能的例子。随机微分方程是一种描述随机系统随时间演化的数学工具。该项目侧重于这些对象的理论研究及其在数学或应用数学的不同领域中的应用。

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.