Brownian motion started at x

Assume the Axiom of Choice. Let $d\ge 1$ be a finite integer and let $B$ be a standard $d$-dimensional Brownian motion on a probability space $(\mathrm{\Omega },\mathcal{F},P)$ $d$-dimensional Brownian motion, as supplied by Existence and scaling of $d$-dimensional Brownian motion. For $x\in {\mathbb{R}}^d$, choose the measurable probability-one event on which $B_0=0$ and the path $t\mapsto B_t$ is continuous, and put $\widehat{B}=B$ on that event and ${\widehat{B}}_t=0$ for every $t$ off it. Then put $B_t^x:=x+{\widehat{B}}_t,t\ge 0.$ The process $B^x$ is the standard $d$-dimensional Brownian motion started at $x$, and its law $P_x:=\text{the law of the random element }\omega \mapsto (x+{\widehat{B}}_t(\omega ){)}_{t\ge 0}$ is called the shifted Brownian law at $x$. Here the target is the canonical continuous path space $C([0,\mathrm{\infty }),{\mathbb{R}}^d)$ with its compact-open Borel sigma-algebra. The finite-dimensional version of Borel sigma-algebra of continuous path space is generated by coordinates (applied coordinatewise) says that this Borel sigma-algebra is generated by the evaluations $f\mapsto f(t)$. Thus the displayed map is a random element, and $P_x$ is a probability measure The law of a random element is a probability measure. We write $P_x(A)=P((x+{\widehat{B}}_t{)}_{t\ge 0}\in A)$.

The following are part of the definition and are used later in this form.

1. Initial value and path space. Every path in the image is continuous and starts at $x$. In particular $P_x(f(0)=x)=1$. The normalization changes $B$ only on a null event and therefore changes none of its finite-dimensional distributions.
2. Increments. For $0\le s\le t$ one has the pathwise identity $B_t^x-B_s^x={\widehat{B}}_t-{\widehat{B}}_s$. Consequently, under $P_x$ the increments are independent with laws $N_d(0,(t-s)I_d)$ $d$-dimensional Brownian motion; in particular $B^x$ is again a standard Brownian motion up to its initial value $x$.
3. Translation of hitting times. Let $C\subseteq {\mathbb{R}}^d$ be closed and let $T_C(f):=\inf \{t\ge 0:f(t)\in C\}$ (with $\inf \mathrm{\varnothing }:=+\mathrm{\infty }$) be the first hitting functional of $C$, evaluated on path space. Then pathwise $T_C(B^x)=T_{C-x}(\widehat{B}),C-x:=\{z-x:z\in C\}$ because $x+{\widehat{B}}_t\in C$ if and only if ${\widehat{B}}_t\in C-x$. The functional $T_C$ is Borel on continuous path space: for finite $t$, $\{T_C\le t\}$ is the closed set of paths whose compact restriction to $[0,t]$ meets $C$. In particular $P_x(T_C\le t)=P(T_{C-x}\le t)$ for every $t\ge 0$, and for $d=1$ the one-point case reads $P_x(T_a<\mathrm{\infty })=P(T_{a-x}<\mathrm{\infty })$.
4. These are the only shifted laws used below. The one-dimensional items use $d=1$, the planar items use $d=2$, and $P_0$ is the law of $B$ itself. No statement below treats $P_x$ as a kernel in $x$ or as a regular conditional distribution.

Durrett, Section 7.5, and Sousi, Section 6.1, use the notation $P_x$ for Brownian motion started at $x$ without minting a separate definition. The definition above fixes that notation on canonical continuous path space, so that closed-set hitting-time events are Borel events of the shifted law, and records the translation identity that every later use consumes.