## On a Generalized Transfer Operator

Please always quote using this URN: urn:nbn:de:0297-zib-43162

- We introduce a generalized operator for arbitrary stochastic processes by using a pre-kernel, which is a generalization of the Markov kernel. For deterministic processes, such an operator is already known as the Frobenius-Perron operator, which is defined for a large class of measures. For Markov processes, there exists transfer operators being only well defined for stationary measures in $L^2$. Our novel generalized transfer operator is well defined for arbitrary stochastic processes, in particular also for deterministic ones. We can show that this operator is acting on $L^1$. For stationary measures, this operator is also an endomorphism of $L^2$ and, therefore, allows for a mathematical analysis in Hilbert spaces.

Author: | Adam Nielsen, Konstantin FackeldeyORCiD, Marcus Weber |
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Document Type: | ZIB-Report |

Tag: | Perron Frobenius Generalization; Pre Kernel; Transfer Operator |

MSC-Classification: | 37-XX DYNAMICAL SYSTEMS AND ERGODIC THEORY [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX] |

47-XX OPERATOR THEORY / 47Nxx Miscellaneous applications of operator theory [See also 46Nxx] / 47N30 Applications in probability theory and statistics | |

60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Gxx Stochastic processes | |

CCS-Classification: | G. Mathematics of Computing / G.3 PROBABILITY AND STATISTICS / Stochastic processes (NEW) |

PACS-Classification: | 30.00.00 ATOMIC AND MOLECULAR PHYSICS / 31.00.00 Electronic structure of atoms and molecules: theory |

Date of first Publication: | 2013/11/26 |

Series (Serial Number): | ZIB-Report (13-74) |

ISSN: | 1438-0064 |

Licence (German): | Creative Commons - Namensnennung |