In studying $X$ (Banach space) valued stochastic processes, I tend to see two different norms used: $$ \sup_{t\leq T} \mathbb{E}[\|u(t)\|_{X}^p]^{1/p} $$ and $$ \mathbb{E}[\sup_{t\leq T} \|u(t)\|_X^p]^{1/p} $$ for $p\geq 1$.

I am looking to understand the differences/similarities between these two, and the associated sets of processes for which the norms are finite. I am mostly concerned with this in the context of SPDE.

For instance, the first norm above makes a substantial amount of sense when constructing mild solutions to SPDE via the Banach fixed point method. But, in some ways, the latter norm/set of proccesses is more natural (from an applied perspective) in that the solutions of the problem are random $C(0,T; X)$ valued functions.

In studying $X$ (Banach space) valued stochastic processes, I tend to see two different norms used: $$ \sup_{t\leq T} \mathbb{E}[\|u(t)\|_{X}^p]^{1/p} $$ and $$ \mathbb{E}[\sup_{t\leq T} \|u(t)\|_X^p]^{1/p} $$ for $p\geq 1$.

I am looking to understand the differences/similarities between these two, and the associated sets of processes for which the norms are finite. I am mostly concerned with this in the context of SPDE.

For instance, the first norm above makes a substantial amount of sense when constructing mild solutions to SPDE via the Banach fixed point method. But, in some ways, the latter norm/set of processes is more natural (from an applied perspective) in that the solutions of the problem are random $C(0,T; X)$ valued functions.