Dirichlet Forms and Stochastic Analysis of Symmetric Markov Processes

狄里克莱型理论是L^2-理论，而马尔可夫过程理论在某种意义上是L^1-理论。为了弥补这一空白，我们研究了广义Feynman-Kac(Schroedinger)半群的增长界的L p无关性。证明L p-独立性的一个关键思想是利用Donsker-Varadhan大偏差理论中的论点。L p-无关性使我们能够控制对称马尔可夫过程的L-∞性质，实际上，我们可以用L^2-谱的下界来表示Feynman-Kac泛函的可积性和薛定谔热核的Gauss型两侧估计稳定的充要条件。

The theory of Dirichlet forms is an L^2-theory, while the theory of Markov processes is, in a sense, an L^1-theory. To bridge this gap, we study the L^p-independence of growth bounds of Markov semigroups, more generally, of generalized Feynman-Kac (Schroedinger) semigroups. A key idea for the proof of the L^p-independence is to employ arguments in the Donsker-Varadhan large deviation theory. The L^p-independence enables us to control L^∞-properties of the symmetric Markov process ; in fact, we can state, in terms of the bottom of L^2-spectrum, a necessary and sufficient conditions for the integrability of Feynman-Kac functionals and for the stability of Gaussian both side estimates of Schroedinger heat kernels.