Minimal Q Processes Infinitesimal Operator Depiction

Given a Q matrix,whose elements are finite. Feller has solved the problem about the existence of Q process and constructed one minimal Q process- f(t). Let ( ) denote the Laplace transform of Q process-p(t),i.e.its resolvent operator and A denotes the infinitesimal operator generated by p(t). Known from ref.[2],there are one-one correspondences among p(t),1If(A)and A, as well asParticulary,let ( ) denote the Laplace transform of the minimal Q process- f(t), and A is the infinitesimal operator corresponding to f(t).soThe domain of A is the range of (A), which is independent of A because of the resolvent epuation.The properties and theories of ( ) are described in sec. 1.8 of reference[1], meanwhile, some perfert properties of ( ) are obtained in sec. 1.10.With the aid of these properties and theories ,the main results are calculated as follows:1.The domain D(A):(A,D(A)) (Q,D(Q)).2.D(A) are gained when the Q matrix in zero exit case and in single exit case .the results are described by the domain indepen-dent of A.(i) Given the Q matrix in zero exit case,i.e.m+ = 0,then,(ii)Given the Q matrix in single exit case,i.e.m+=1, I= is the sequence whose elements are different for each other lim Xin (A) = 1 is true for some A, thus it makesall A true, we have3. Expression about the domain of A, which is corresponding to the single boundary birth and death process and the double boundary birth and death process,(l)Let Q be the single boundary birth and death processjsee sec.5.1 of iefeience[1]],(i)Given boundary point entrance or natural, in this case ,m+ = O.then,(ii) Given boundary point regular or exit ,in this case ,ra+ 1.then,(2) Let Q be the double boundary birth and death processjsee sec. 4.1 of reference [1]],(i) Given r1,r2 both the entrance or natural boundary point, in this case ,m+ =0.then,(ii)r1 is the entrance or natural boundary point, r2is the regular or exit boundary point, in this case ,m+ = l.then,(iii)r2 is the entrance or natural boundary point, r1 is the regular or exit boundary point,in this case ,m+ = 1.then,...