Algebraic structures on typed decorated rooted trees

Abstract: Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation processon stochastic PDEs. We here study the algebraic structures on these objects: multiple prelie algebrasand related operads (generalizing a result by Chapoton and Livernet), noncommutative and cocommutative Hopf algebras (generalizing Grossman and Larson's construction),commutative and noncocommutative Hopf algebras (generalizing Connes and Kreimer's construction),bialgebras in cointeraction (generalizing Calaque, Ebrahimi-Fard and Manchon's result). We also define families of morphisms and in particular we prove that any Connes-Kreimer Hopf algebraof typed and decorated trees is isomorphic to a Connes-Kreimer Hopf algebra of non--typed and decoratedtrees (the set of decorations of vertices being bigger), through a contraction process,and finally obtain the Bruned-Hairer-Zambotti construction as a subquotient.