Quasi-ergodicity And Related Problems For Killed Brownian Motion

Ergodic theory and the large deviation principle are important parts of Markov pro-cess theory, and they have strong connections. The Markov processes in our paper gen-erally does not satisfy usual ergodicity condition, so we concentrate on their conditional ergodicity (we call quasi-ergodicity), large deviaitons and related problems.Firstly, killed BM has3kinds of quasi-ergodicity:in distributional sense quasi-ergodicity, fractional quasi-ergodicity and in Birkhoff’s sense mean-ratio quasi-ergodicity. We prove the corresponding quasi-ergodicity theorems, and that they respec-tively have unique limiting distributions:quasi-stationary distribution (qsd), fractional quasi-stationary distribution (fqsd) and mean-ratio quasi-stationary distribution (mrqsd). Fqsd is different from qsd, reflecting certain phase transition. However, fqsd is the same as the mrqsd, which is also the unique stationary distribution of the limiting diffusion process generated by killed BM. These3limiting distributions are different from the Lebesgue measure on the domain, which is the symmetric measure of the correspond-ing L2semigroup. Also we obtain a result of independent interest:for a Feller Markov process, any Yaglom limit is a qsd.Secondly, we prove that quasi-ergodic rate, fractional quasi-ergodic rate and ergodic rate of the limiting diffusion process are characterized by its starting point and the Dirich-let eigenvalues of the Laplace operator, through Dirichlet eigenfunctions. Here the eigen-values may be3rd order or higher. Taking1dimensional killed BM as an example, we demonstrate the above results.Next, we study deviation inequalities for f0t V(Xs)ds. When V belongs to Kato class, we get a deviation inequality. Under further conditions, we obtain the negative expo-nential bound. Especially in the case that V is bounded and continuous, our deviation inequality is asymptotically sharp by the large deviation principle.At last, we study the large deviation principle of the empirical measure. We point out that the unique minimum of the large deviation rate function is the mrqsd, and the minimal value of the rate function is the first Dirichlet eigenvalue. We extend the large deviation principle by proving the local uniform lower bound. We also give a new variational formula for the principal eigenvalue and a strong version of weak law of large numbers. Furthermore, we prove that the limiting difusion process also satisfies the large deviationprinciple, identifying the rate function with its Dirichlet form, which is the same onegoverning conditional large deviation principle of the killed BM.