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We delve deeper into the study of semimartingale attractors that we recentlyintroduced in Allouba and Langa \cite{AL0}. In this article we focus on secondorder SPDEs of the Allen-Cahn type. After proving existence, uniqueness, anddetailed regularity results for our SPDEs and a corresponding random PDE ofAllen-Cahn type, we prove the existence of semimartingale global attractors forthese equations. We also give some results on the finite dimensional asymptoticbehavior of the solutions. In particular, we show the finite fractal dimensionof this random attractor and give a result on determining modes, both in theforward and the pullback sense.

Semimartingale attractors for Allen-Cahn SPDEs driven by space-time white noise I: Existence and finite dimensional asymptotic behavior