Stannat, Wilhelm On the validity of the log-Sobolev inequality for symmetric Fleming-Viot operators. (English) Zbl 1044.60037 Ann. Probab. 28, No. 2, 667-684 (2000). Summary: We prove that Fleming-Viot operators with parent-independent mutation satisfy a logarithmic Sobolev inequality if and only if the set of types is finite. Cited in 2 ReviewsCited in 16 Documents MSC: 60G57 Random measures 35P15 Estimates of eigenvalues in context of PDEs 60K35 Interacting random processes; statistical mechanics type models; percolation theory 46N30 Applications of functional analysis in probability theory and statistics 60J35 Transition functions, generators and resolvents Keywords:Fleming-Viot operators; logarithmic Sobolev inequalities; hypercontractivity × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Aida, S. (1998). Uniform positivity improving property, Sobolev inequalities, and spectral gaps. J. Funct. Anal. 158 152-158. · Zbl 0914.47041 · doi:10.1006/jfan.1998.3286 [2] Bakry, D. (1994). L’hypercontractivité et son utilisation en théorie des semigroupes. Ecole d’Ete de Probabilités de Saint Flour XXII. Lecture Notes in Math. 1581 1-114. Springer, Berlin · Zbl 0856.47026 [3] Davies, E. B. (1989). Heat Kernels and Spectral Theory. Cambridge Univ. Press. · Zbl 0699.35006 [4] Ethier, S. N. and Griffiths, R. C. (1993). The transition function of a Fleming-Viot process. Ann. Probab. 21 1571-1590. · Zbl 0778.60038 · doi:10.1214/aop/1176989131 [5] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes, Characterization and Convergence. Wiley, New York. · Zbl 0592.60049 [6] Ethier, S. N. and Kurtz, T. G. (1993). Fleming-Viot processes in population genetics. SIAM J. Control Optim. 31 345-386. · Zbl 0774.60045 · doi:10.1137/0331019 [7] Fang, K. T., Kotz, S. and Ng, K. W. (1990). Symmetric Multivariate Distributions. Chapman and Hall, London. · Zbl 0699.62048 [8] Gross, L. (1992). Logarithmic Sobolev inequalities and contractivity properties ofsemigroups. Dirichlet Forms. Lecture Notes in Math. 1563 54-88. Springer, Berlin. · Zbl 0812.47037 · doi:10.1007/BFb0074091 [9] Korzeniowski, A. and Stroock, D. W. (1985). An example in the theory ofhypercontractive semigroups. Proc. Amer. Math. Soc. 94 87-90. · Zbl 0577.47043 · doi:10.2307/2044957 [10] Ma,M. and R öckner, M. (1992). Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin. · Zbl 0826.31001 [11] Overbeck, L., R öckner, M. and Schmuland, B. (1995). An analytic approach to Fleming- Viot processes with interactive selection. Ann. Probab. 23 1-36. · Zbl 0833.60053 · doi:10.1214/aop/1176988374 [12] Shimakura, N. (1977). Equations différentielles provenant de la génetique des populations. T ohoka Math. J. 29 287-318. · Zbl 0365.92028 · doi:10.2748/tmj/1178240658 [13] Tavaré, S. (1984). Line-of-descent and genealogical processes, and their applications in population genetics models. Theoret. Population Biol. 26 119-164. · Zbl 0555.92011 · doi:10.1016/0040-5809(84)90027-3 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.