Probability Approximations and Beyond pp 75-99 | Cite as

# Basic Estimates of Stability Rate for One-Dimensional Diffusions

## Abstract

In the context of one-dimensional diffusions, we present basic estimates (having the same lower and upper bounds with a factor of 4 only) for four Poincaré-type (or Hardy-type) inequalities. The derivations of two estimates have been open problems for quite some time. The bounds provide exponentially ergodic or decay rates. We refine the bounds and illustrate them with typical examples.

## Keywords

Neumann Boundary Elliptic Operator Principal Eigenvalue Exponential Convergence Dual Operator## Notes

### Acknowledgments

Research supported in part by the Creative Research Group Fund of the National Natural Science Foundation of China (No. 10721091), by the “985” project from the Ministry of Education in China. The author is fortunate to have been invited by Professor Louis Chen three times with financial support to visit Singapore. He is deep appreciative of his continuous encouragement and friendship in the past 30 years. Sections 6.2–6.4 of the paper are based on the talks presented in “Workshop on Stochastic Differential Equations and Applications” (December, 2009, Shanghai), “Chinese-German Meeting on Stochastic Analysis and Related Fields” (May, 2010, Beijing), and “From Markov Processes to Brownian Motion and Beyond —An International Conference in Memory of Kai-Lai Chung” (June, 2010, Beijing). The author acknowledges the organizers of the conferences: Professors Xue-Rong Mao; Zhi-Ming Ma and Michael Rökner; and the Organization Committee headed by Zhi-Ming Ma (Elton P. Hsu and Dayue Chen, in particular), for their kind invitation and financial support.

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