Weak\(^*\) convergence of iterates of Lasota--Mackey--Tyrcha operators
Let \((X,d)\) be a separable metric space such that closed balls are compact, \(\mu\) a \(\sigma\)-finite measure on Borel sets of \((X,d)\), \({\mathcal D}_\mu= \{f\in L^1(\mu): f\geq 0, \int_X f d\mu= 1\}\).NEWLINENEWLINENEWLINEA linear operator \(P: L^1(\mu)\to L^1(\mu)\) such that \(P({\mathcal D}_\mu)\subset{\mathcal D}_\mu\) is called Markov operator. Markov operator is called a kernel operator if it can be represented as NEWLINE\[NEWLINEPf(x)= \int_X k(x,y) f(y) d\mu(y).NEWLINE\]NEWLINE Kernel operators with specifying properties are called LMT operators. The kernel operator is called (SFS) operator if the mapping \((X,d)\ni Y\mapsto k(x,y)\in({\mathcal D}_\mu, \|\cdot\|)\) is continuous. The substochastic projection onto the sublattice of \(P\)-invariant functions is denoted by \(S\).NEWLINENEWLINENEWLINEThe main results of the paper:NEWLINENEWLINENEWLINE1. If \(P\) is (SFS) operator, \(P^*\) preserves \(C_0(X)\), and \(\Sigma_i(P_C)= \Sigma_d(P_C)\) (\(\Sigma_i\), \(\Sigma_d\) and \(P_C\) are defined), then for every compact set \(K\subset X\) and \(f\in L^1(\mu)\), \(\lim_{n\to\infty} \int_K P^nf d\mu= \int_K Sf d\mu\).NEWLINENEWLINENEWLINE2. Let \(P\) be a LMT operator on \(L^1[0,\infty)\). Then for every compact set \(K\subset [0,\infty)\),NEWLINENEWLINENEWLINE\(\lim_{n\to \infty} \int_K P^n f d\mu= \int_K Sf d\mu\).NEWLINENEWLINENEWLINEMoreover in both cases \(\|(P^nf- Sf)\chi_F\|\to 0\) as \(n\to\infty\) where \(F\) is the minimal set which carries supports of all \(P\)-invariant densities.
scientific article; zbMATH DE number 1605293