Let $B$ be a Brownian motion in $\mathbb{R}^{2}$ . Let $\mathbb{P}_{(x,y)}$ denote a probability measure under which $B$ is started at $(x,y)$ . Is it true in general that, for measurable set $A$,
$\lim_{(x,y)\rightarrow(0,0)}\mathbb{P}_{(x,y)}((B_{t})_{t\geq0}\in A)=\mathbb{P}_{(0,0)}((B_{t})_{t\geq0}\in A)$?
I feel like this is just convergence in distribution but Im very uncertain if it is correct or not.
If it is indeed true, is it then part of a more general fact? E.g. is it always true for general stochastic processes. Or is it related to a general theorem or form of convergence?
I'd be extremely grateful for any inputs. Thanks in advance!