Vector-valued stochastic delay equations—A weak solution and its Markovian representation

A class of stochastic delay equations in a type 2 umd Banach space EE driven by the cylindrical Wiener process is studied. We investigate two concepts of solutions: weak and generalised strong, and give conditions under which they are equivalent. We present an evolution equation approach in a Banach space Ep:=E×Lp(−1,0;E)Ep:=E×Lp(−1,0;E) proving that the solutions can be reformulated as EpEp-valued Markov processes. Based on the Markovian representation we prove the existence and continuity of the solutions. The results are applied to stochastic reaction–diffusion equations with delays.