# Stochastic Partial Differential Equations meets functional calculus

The solution to a stochastic PDE can often be written as a (stochastic) convolution with a semigroup $S(t) = e^{tA}$, where $A$ is a differential operator. After a short but somewhat self-contained introduction on stochastic integrals and PDEs,  I will explain two situations where one can apply functional calculus techniques in the study of the regularity of solutions to SPDEs.

(1): Often one would like to know that the solution to the stochastic PDE has continuous paths.
Assuming a bounded functional calculus of $A$, this can be proved very effectively by writing the semigroup $S(t)$ to a group on a larger space.   This approach also extends to the the setting of Volterra equation, but in this case one needs to apply Naimark's theorem on positive definite functions with values in a Hilbert space.

(2): To prove regularity in space of the solution of a stochastic PDEs, one needs certain square function estimates. The $L^2$-estimates are classical and easy to obtain. $L^p$-results are more complicated to obtain. I will give an overview on several of the recent results on what we now call stochastic maximal $L^p$-regularity.