Convergence of S′-valued processes and space-time random fields☆

Abstract The mapping x → x of D ([0, T ], S′ ( R d )) into S′ ( R d + 1 ) defined by 〈 x ,Φ〉 = ∫ 0 T 〈x(t),Φ(t,·)〉dt, Φ∈S(R d+1 ) is shown to be continuous. This is used to prove that for a tight sequence of processes ( X n ) inn in D ([0, T ], S′ ( R d )), weak convergence of ( X n ) n in S′ ( R d + 1 ) implies weak convergence of ( X n ). When [0, T ] is replaced by [0, ∞) the space S′ ( R d + 1 ) is not appropriate in general, and we investigate the question of what is the smallest nuclear space of distributions containing S′ ( R d + 1 ) that can be used. Our motivation comes from the asymptotic analysis of particle systems.