A submartingale type inequality with applicatinos to stochastic evolution equations

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Abstract
On a fixed time interval [0,T] we consider where M is a square integrable martingale with values in a Hilbert space K,Φ s (ω) suitable operators from K to H, another Hilbert space, and U t s is a mild evolution operator on H. Under the assumption that there is a β≧0 such that we prove a submartingale type inequality for that integral. As applicatins we show that it has a continuous version, if M has, or is cadlage if M is calage, and we prove that it is stable under some additional assumption on U t s Finally we ontain an existence and uniquenss theorem for an SPDE.