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We prove optimal regularity estimates in Sobolev spaces in time and space forsolutions to stochastic porous medium equations. The noise term considered hereis multiplicative, white in time and coloured in space. The coefficients areassumed to be H\"older continuous and the cases of smooth coefficients of atmost linear growth as well as $\sqrt{u}$ are covered by our assumptions. Theregularity obtained is consistent with the optimal regularity derived for thedeterministic porous medium equation in [Gess 2020] and [Gess, Sauer, Tadmor2020] and the presence of the temporal white noise. The proof relies on asignificant adaptation of velocity averaging techniques from their usual $L^1$context to the natural $L^2$ setting of the stochastic case. We introduce a newmixed kinetic/mild representation of solutions to quasilinear SPDE and use$L^2$ based a priori bounds to treat the stochastic term.

Optimal regularity in time and space for stochastic porous medium equations