Martínez, Servet; San Martín, Jaime Classification of killed one-dimensional diffusions. (English) Zbl 1045.60083 Ann. Probab. 32, No. 1A, 530-552 (2004). Summary: We show necessary and sufficient conditions for \(R\)-recurrence and \(R\)-positivity of one-dimensional diffusions killed at the origin. These conditions are stated in terms of the bottom eigenvalue function. Cited in 11 Documents MSC: 60J60 Diffusion processes 60F99 Limit theorems in probability theory Keywords:killed diffusions; \(R\)-positive; \(R\)-transient × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Collet, P., Martínez, S. and San Martín, J. (1995). Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption. Ann. Probab. 23 1300–1314. JSTOR: · Zbl 0867.60046 · doi:10.1214/aop/1176988185 [2] Ferrari, P. A., Kesten, H. and Martínez, S. (1996). \(R\)-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata. Ann. Appl. Probab. 6 577–616. · Zbl 0855.60061 · doi:10.1214/aoap/1034968146 [3] Karatzas, I. and Shreve, S. (1988). Brownian Motion and Stochastic Calculus . Springer, New York. · Zbl 0638.60065 [4] Kesten, H. (1976). Existence and uniqueness of countable one-dimensional random fields. Ann. Probab. 4 557–569. · Zbl 0367.60080 · doi:10.1214/aop/1176996027 [5] Mandl, P. (1961). Spectral theory of semi-groups connected with diffusion processes and its applications. Czech. Math. J. 11 558–569. · Zbl 0115.13503 [6] Martínez, S. and San Martín, J. (2001). Rates of decay and \(h\)-processes for one dimensional diffusions conditioned on nonabsorption. J. Theoret. Probab. 14 199–212. · Zbl 0981.60073 · doi:10.1023/A:1007881317492 [7] Seneta, E. (1981). Nonnegative Matrices and Markov Chains . Springer, New York. · Zbl 0471.60001 [8] Vere-Jones, D. (1967). Ergodic properties of nonnegative matrices I. Pacific J. Math. 22 361–386. · Zbl 0171.15503 · doi:10.2140/pjm.1967.22.361 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.