An extended generator and Schrödinger equations. (English) Zbl 0936.60066

Getoor, R. K.

Electron. J. Probab. 4, Paper No. 19, 23 p. (1999).

The generator of a Borel right process is extended so that it maps functions to smooth measures. This extension may be defined either probabilistically using martingales or analytically in terms of certain kernels on the state space of the process. Then the associated Schrödinger equation with a (signed) measure serving as potential may be interpreted as an equation between measures. In this context general existence and uniqueness theorems for solutions are established. These are then specialized to obtain more concrete results in special situations.

MSC:

60J40	Right processes
60J25	Continuous-time Markov processes on general state spaces
60J35	Transition functions, generators and resolvents
60J45	Probabilistic potential theory

Keywords:

Markov processes; smooth measures