This is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best...This is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best constant in the L2-Poincare inequality and describes the decay rate of the corresponding diffusion process. We present some variational formulas for the mixed principal eigenvalues of the operators. As applications of these formulas, we obtain case by case explicit estimates, a criterion for positivity, and an approximating procedure for the eigenvalue.展开更多
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant No. 11131003）, The Specialized Research Fund for the Doctoral Program of Higher Education （Grant No. 20100003110005）, the ＇985＇ project from the Ministry of Education in China, and the Fundamental Research Funds for the Central Universities.
文摘This is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best constant in the L2-Poincare inequality and describes the decay rate of the corresponding diffusion process. We present some variational formulas for the mixed principal eigenvalues of the operators. As applications of these formulas, we obtain case by case explicit estimates, a criterion for positivity, and an approximating procedure for the eigenvalue.
