In probability theory, a Markov kernel (or stochastic kernel) is a map that plays the role, in the general theory of Markov processes, that the transition matrix does in the theory of Markov processes with a finite state space.
Let , be measurable spaces. A Markov kernel with source and target is a map that associates to each point a probability measure on such that, for every measurable set , the map is measurable with respect to the -algebra .
Let denote the set of all probability measures on the measurable space . If is a Markov kernel with source and target then we can naturally associate to a map defined as follows: given in , we set , for all in .
- Epstein, P.; Howlett, P.; Schulze, M. S. (2003). "Distribution dynamics: Stratification, polarization, and convergence among OECD economies, 1870–1992". Explorations in Economic History 40: 78. doi:10.1016/S0014-4983(02)00023-2.
- Reiss, R. D. (1993). A Course on Point Processes. Springer Series in Statistics. doi:10.1007/978-1-4613-9308-5. ISBN 978-1-4613-9310-8.
- Poletti Laurini, M. R.; Valls Pereira, P. L. (2009). "Conditional stochastic kernel estimation by nonparametric methods". Economics Letters 105 (3): 234. doi:10.1016/j.econlet.2009.08.012.
- §36. Kernels and semigroups of kernels