Continuous semigroup

MathematicaUy, the latter formulation is easier to treat, basicaUy because integral operators are smoother than integro-differential operators (i.e., than the infinitesimal generators Q of the continuous semigroups involved). Though they are singular, they can be smoothed by iteration so as to become uniformly positive. Hence they are uniformly primitive, in all cases which I have studied. 

What we have seen that continuous semigroups can define linear operators and evolution equations for motions in rather abstract spaces. A diffusion process might be defined by the infinitesimal generator ... 

On the other hand, the antiresonances obtained by analytic continuation toward positive values of Rei are associated with exponential decays for negative times. The corresponding expansion defines the backward semigroup ... 

The appropriate extensions of the concepts of essential positivity and primitivity to infinite dimensions are much harder to formulate. This is because the definition of the infinitesimal generator of a one-parameter semigroup is so technical (see [13]). Hence, I shall simply say that a continuous multiplicative process is uniformly primitive when e is uniformly positive for some T > 0. 

At least two kinds of advantages come from the use of the semigroup operator approach. First, a mathematically justified approximation of discontinuous processes by continuous processes can be obtained. Second, not only models for purely temporal processes (e.g. pure chemical reactions) can be given, but stochastic models of spatio-temporal phenomena (e.g. chemical reactions with diffusion) can be defined in a mathematically concise manner. The celebrated monograph on the modern approach to Markov processes is still Dynkin s book (1965), and for chemical applications see Arnold Kotelenez (1981). 

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