Stochastic PDEs with extremal properties
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We consider linear and semilinear stochastic partial differential equations that in some sense can be viewed as being at the "endpoints" of the classical variational theory by Krylov and Rozovskii . In terms of regularity of the coeffcients, the minimal assumption is boundedness and measurability, and a unique L2- valued solution is then readily available. We investigate its further properties, such as higher order integrability, boundedness, and continuity. The other class of equations considered here are the ones whose leading operators do not satisfy the strong coercivity condition, but only a degenerate version of it, and therefore are not covered by the classical theory. We derive solvability in Wmp spaces and also discuss their numerical approximation through finite different schemes.