Quantum Dynamical Semigroups

Abstract

Feller [1] and Kato [2] proved the existence of a unique minimal semigroup associated with the classical Fokker-Planck equation: (1.1) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaeaada % WcaaqaaiaadsgacaWGWbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOqa % aiaadsgacaWG0baaaiabg2da9maaqafabaGaamiCamaaBaaaleaaca % WGPbGaam4AaaqabaGccqqHPoWvdaWgaaWcbaGaam4AaiaadQgaaeqa % aOGaaiilaiaaywW7caWG0bGaeyyzImRaaGimaiaaywW7caWG3bGaam % yAaiaadshacaWGObGaaGPaVlaadMgacaWGUbGaamyAaiaadshacaWG % PbGaamyyaiaadYgacaaMc8Uaam4yaiaad+gacaWGUbGaamizaiaadM % gacaWG0bGaamyAaiaad+gacaWGUbGaaGPaVlaadchadaWgaaWcbaGa % amyAaiaadQgaaeqaaOGaaiikaiaaicdacaGGPaGaeyypa0JaeqiTdq % 2aaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGRbaabeqdcqGHris5 % aaGccaGL9baaaaa!703E! $$ \left. {\frac{{d{p_{ij}}}}{{dt}} = \sum\limits_k {{p_{ik}}{\Omega _{kj}},\quad t \geqslant 0\quad with\,initial\,condition\,{p_{ij}}(0) = {\delta _{ij}}} } \right\} $$ and subject to the Markov condition: (1.2) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqUbcqqHPo % WvdaWgaaWcbaGaam4AaiaadQgaaeqaaOGaeyyzImRaaGimaiaadAga % caWGVbGaamOCaiaaykW7caWGRbGaeyiyIKRaamOAaiaaykW7caWGHb % GaamOBaiaadsgacaaMc8UaeyOhIu6aaabuaeaacqqHPoWvdaWgaaWc % baGaam4AaiaadQgaaeqaaOGaeyypa0JaeyOeI0IaeuyQdC1aaSbaaS % qaaiaadUgacaWGRbaabeaatCvAUfeBSn0BKvguHDwzZbqegeezVjwz % GyuyUD2CV52zGmfDKbIuaGqbaOGaa8hpaiabg6HiLcWcbaGaamOAai % abgcMi5kaadUgaaeqaniabggHiLdGccaGGUaaaaa!6B15! $$ {\Omega _{kj}} \geqslant 0for\,k \ne j\,and\,\infty \sum\limits_{j \ne k} {{\Omega _{kj}} = - {\Omega _{kk}} < \infty } . $$