Duality Sample Clauses

Duality. Duality is a powerful tool in the study of Markov processes. It has played a funda- mental role in the study of interacting particle systems and in models of population dynamics [19]. For example in the context of the Symmetric Exclusion Process (aka SEP) it has been the crucial tool in order to obtain the complete ergodic theory of this process (see [15], chapters 2 and 8). Two Markov processes {xt : t ≥ 0} and {ξt : t ≥ 0} with state spaces Ω, resp. Ω′ and with generators L, resp. L are called dual to each other if there exist a duality function D : Ω′ × Ω → R such that LD(ξ, →x) = X X(ξ, →x) (0.2.1) where in the lhs of (0.2.1) the operator L is working on the →x variable, and in the rhs the operator L is working on the ξ variable (here we implicitly assume that D(ξ, .) is in the domain of L and D(., →x) is in domain of L ). This relation then lifts to the semigroups (which arise by exponentiation of the generator) and to the processes. This then yields the duality relation between the processes: E→x[D(ξ, →x(t))] = Eˆξ [D(ξ(t), →x)]. (0.2.2) This relation is useful in the case that the {xt : t ≥ 0} is ‘complicated’ and the {ξt : t ≥ 0} is ‘easy’ and the set of dual functions is sufficiently rich. For instance one can think of ξ being discrete objects indexing polynomials in →x. In case that Ω = RN , if the equations for the evolution of correlation functions of degree n for the x process are closed (i.e. there is no polynomial of higher order than n involved), that can be a hint to the existence of the duality property where the dual process will then be a particle system where the number of particles is not increasing.

Duality. In this section we show that the BEP defined above has a dual process which is again a jump process. K To construct the dual we follow a procedure similar to the one of the previous section. Remark that the generator L in (1.6.1) and (1.6.2) can be written in terms of the operators with a ∈ {+, −, 0}, where K+ Ka,(m) = = 1 x2 Σm