What's the distributional derivative of a Banach space valued almost surely continuous stochastic process?

Question

Let

• $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\lambda$ be the Lebesgue measure on $[0,\infty)$
• $(H,\left\|\;\cdot\;\right\|)$ be a Banach space over the field $\mathbb F\in\left\{\mathbb R,\mathbb C\right\}$
• $(X_t)_{t\ge 0}$ be a $H$-valued almost surely continuous stochastic process on $(\Omega,\mathcal A,\operatorname P)$

Let $\omega\in\Omega$ with $X(\omega)\in C^0([0,\infty);H)$. Then $$X_{[a,b]}(\omega)\text{ is compact in }(H,\left\|\;\cdot\;\right\|)\;\;\;\text{for all }0\le a\le b<\infty\;$$ Especially, $X(\omega)\in\mathcal L_{\text{loc}}^p(\lambda;H)$ and hence $$X(\omega)\varphi\in\mathcal L^p(\lambda;H)\;\;\;\text{for all }\varphi\in C_c^\infty([0,\infty);H)$$ for all $p\in [1,\infty)$. Thus, we can view $X(\omega)$ as being a distribution $$C_c^\infty([0,\infty);H)\to H\;,\;\;\;\varphi\mapsto\int X(\omega)\varphi\;{\rm d}\lambda\;.\tag 1$$

Is there anything I'm missing? Are some of my conclusions wrong? And what's the distributional derivative of $X$?

[I've read the Wikipedia article about the distributional derivative, but I don't know how we need to translate the definition to the setting described here].

EDIT: The question arose as I saw people talking about the distributional derivative of a (cylindrical) Brownian motion on a Hilbert space without giving a definition of what they mean. As @charlestoncrabb pointed out, my attempt in $(1)$ seems to be wrong, since the product $X(\omega)\varphi$ is not defined. Maybe we need to let $H$ be a Hilbert space with inner product $\langle\;\cdot\;,\;\cdot\;\rangle$ and replace $X(\omega)\varphi$ by $\langle X(\omega),\varphi\rangle$. By the same argumentation as before, the integral would exist. But honestly, I'm only guessing. Until now, I failed to generalize the usual notion to this scenario, but since people are using it, there must be some notion of distributional derivative of $X$.

Answer 1

The main issue I see here is that since $X,\varphi$ are $H-$valued, how do we define the product $X \varphi$? It seems you need additional structure to even get past this point (i.e., $H$ needs to be a Banach algebra). Assuming an algebra structure, how are you defining the space $\mathcal{L}^p(\lambda;H)$? In particular, what is the norm of this space? Finally, for a distributional derivative to make sense (by which I mean, align in some way with the usual usage), we need integration by parts to hold: i.e., "$Y$ is the distributional derivative of $X$ if": $$\int \nabla\varphi(\omega)X(\omega)d\lambda(\omega)=-\int\varphi(\omega)Y(\omega)d\lambda(\omega)$$ for all $\varphi\in C^\infty_c(\lambda;H)$ (noting here $\nabla$ refers to the Fréchet derivative). Again, for this to make sense, all of the prior issues need to make sense as well.