Circuit processes and the corresponding one-parameter semigroup of weight operators

Abstract

The relationship between the weight functions representing Markov processes arising as models of diffusion of energy through circuit networks and the corresponding emigroup of contractions is studied. Using the circuit processes approach, a spectral representation is given for the transition matrix function and for the weight functions in term of the directed circuits occurring along the sample paths. Finally, the self-adjointness of any semigroup of weight operators is expressed in terms of the corresponding weight functions.