Vector-valued ergodic theorems for multiparameter additive processes

Let X be a reflexive Banach space and (Ω,Σ,μ) be a σ-finite measure space. Let d ≥ 1 be an integer and T={T(u):u=(${\displaystyle u_1}$, ... ,${\displaystyle u_d)}$, ${\displaystyle u_i}$ ≥ 0, 1 ≤ i ≤ d } be a strongly measurable d-parameter semigroup of linear contractions on ${\displaystyle L_1}$((Ω,Σ,μ);X). We assume that to each T(u) there corresponds a positive linear contraction P(u) defined on ${\displaystyle L_1}$((Ω,Σ,μ);ℝ) with the property that ∥ T(u)f(ω)∥ ≤ P(u)∥f(·)∥(ω) almost everywhere on Ω for all f ∈ ${\displaystyle L_1}$((Ω,Σ,μ);X). We then prove stochastic and pointwise ergodic theorems for a d-parameter bounded additive process F in ${\displaystyle L_1}$((Ω,Σ,μ);X) with respect to the semigroup T.