What is the "distributional derivative" of a Brownian motion?

Question

Let $\emptyset\ne I\subseteq\mathbb R$ be an open interval and $A:C_0^\infty(I)\to\mathbb R$ be a distribution. Then, $$\langle{\rm D}A,\varphi\rangle:=-\langle A,\varphi'\rangle\;\;\;\text{for }\varphi\in C_0^\infty(I)$$ is called distributional derivative of $A$.

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion. How is the distributional derivative of $B$ with respect to $t$ defined?

Answer 1

Presumably distribution derivative of $B$ refers to $\varphi\mapsto\int_0^\infty \varphi(s)\,dB_s=-\int_0^\infty \varphi'(s)B_s\,ds$, on the domain comprising smooth functions supported in compact subsets of $(0,\infty)$.