We define stochastic volatility for infinite dimensional processes, using an infinite dimensional Ornstein-Uhlenbeck process. Particular attention is paid to a fractional volatility model. Moreover, we also discuss realized covariation in this context, presenting a law of large numbers and a central limit theorem. The main application we refer to is modelling commodity forward term structures. The presentation is based on joint work with Fabian Harang (BI), Dennis Schroers (Bonn) and Almut Veraart (Imperial).