Transition distribution of a stochastic process

Question

I am currently reading papers about diffusion models (such as https://arxiv.org/abs/2101.09258, bottom of page 2), where we are given a stochastic process $x(t)$ (which is a diffusion process, i.e. of the form d$x=f(x,t)\text{d}t+g(t)\text{d}w$ for some Brownian motion $w$), and came across the term '$p_{0t}(x'|x)$, the transition distribution from $x(0)$ to $x(t)$'. Now I'm not sure I'm understanding the term correctly, and wanted to make sure my explanation is correct. Let's say $x:\Omega\times [0,T]\to\mathbb{R}$, and $\mathcal{A}_{x(0)}$ is the $\sigma$ -field on $\mathbb{R}$ induced by $x(0)$. Is $p_{0t}(x'|x)$ a map from $\mathcal{A}_{x(0)}$ to the set of probability distributions on $\mathbb{R}$ then, i.e. one that maps $A\in\mathcal{A}_{x(0)}$ to $\mathbb{P}(x(t)\in \cdot\ |x(0)\in A)$?