The standard Brownian motion $B_t$ starts in $0$. Of course, we can add a drift $x$ so that $B_t +x$ starts in $x$ instead. I read in many papers the terms "Brownian motion with initial distribution $\mu$".
Clearly, the standard Brownian motion $B_t$ is the Brownian motion with initial distribution $\delta_0$ and $B_t +x$ is the Brownian motion with initial distribution $\delta_x$.
But what does "Brownian motion with initial distribution $\mu$" means when $\mu$ is not a Dirac measure ? What is the definition ? I am tempted to say $X_t$ is a "Brownian motion with initial distribution $\mu$" if $X_t=B_t + X$ and $X\sim \mu$, $X$ independent of $B_t$. Any insight ?