Brownian motion and Cauchy distribution

Question

Suppose that $X_t=(B_t,W_t)$ is a $BM^{d+1}$ where $B_t$ is a $d$ dimensional Brownian motion and $W_t$ is a one dimensional Brownian motion. Suppose the process $X_t$ starts at $(0,y),y>0$. Then it is a classical result that $X_{T_y}$ has Cauchy distribution where $T_y$ is the stopping time $T_y=\inf\{t>0: W_t=0\}$. My question is, what happens when $B_t$ doesnt start at 0? Like if $X_t$ starts at $(x,y)$ instead. Does $X_{T_y}$ have the same distribution? More importantly I would like to see a proof (or reference) of the result that $\Phi(x,y)=\int_{R^d} \rho(x,y,z)f(z)dz$ is harmonic where $\rho$ is the density of $X_{T_y}$ and $f\in C_c(R^n)$.