This is the abstract of the paper
  "On the Infinite Product of $C_0$-Semigroups"
by W. Arendt and A. Driouich and O. El-Mennaoui.

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Abstract: Given a family $(e^{tA_k})_{t\ge 0}$ $(k \in N)$
of commuting contraction semigroups, we investigate when the infinite
product $\prod\limits^\infty_{k=1} e^{tA_k}$ converges and defines a
$C_0$-semigroup. A particular case is the heat semigroup in infinite
dimension introduced by Cannarsa - Da Prato.