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Unformatted text preview: Chapter 12 Generators of Markov Processes This lecture is concerned with the infinitessimal generator of a Markov process, and the sense in which we are able to write the evo- lution operators of a homogeneous Markov process as exponentials of their generator. Take our favorite continuous-time homogeneous Markov process, and con- sider its semi-group of time-evolution operators K t . They obey the relationship K t + s = K t K s . That is, multiplication of the operators corresponds to addition of their parameters, and vice versa. This is reminiscent of the exponential func- tions on the reals, where, for any k ∈ R , k ( t + s ) = k t k s . In the discrete-parameter case, in fact, K t = ( K 1 ) t , where integer powers of operators are defined in the obvious way, through iterated composition, i.e., K 2 f = K ◦ ( Kf ). It would be nice if we could extend this analogy to continuous-parameter Markov pro- cesses. One approach which suggests itself is to notice that, for any k , there’s another real number g such that k t = e tg , and that e tg has a nice representation involving integer powers of g : e tg = ∞ i =0 ( tg ) i i ! The strategy this suggests is to look for some other operator G such that K t = e tG ≡ ∞ i =0 t i G i i ! Such an operator G is called the generator of the process, and the purpose of this section is to work out the conditions under which this analogy can be carried through. In the exponential function case, we notice that g can be extracted by taking the derivative at zero: d dt e tg t =0 = g . This suggests the following definition. 64 CHAPTER 12. GENERATORS 65 Definition 119 (Infinitessimal Generator) Let K t be a continuous-parameter semi-group of homogeneous Markov operators. Say that a function f ∈ L 1 be- longs to Dom( G ) if the limit lim h ↓ K h f- K f h ≡ Gf (12.1) exists in an L 1-norm sense, i.e., there exists some element of L 1 , which we shall...
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- Spring '06
- Derivative, semi-group Kt