I am trying to solve the following:
Let $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)$ be a two dimensional Brownian motion. Find the distribution of $$\|W_{t}\|=\sqrt{(W_{t}^{1})^2+(W_{t}^{2})^2}.$$
I know by the properties of Brownian motion for any $k \ge 1$ and $0<t_1<\dotsb<t_k$ the random vector $(W_{t_1},\dotsc,W_{t_k})$ is Gaussian with zero mean and covariance matrix $B(t_i,t_j)=\min(t_i,t_j).$ I also know that the sum of square iid Gaussian random variables $N(0,\sigma)$ is exponential with mean $2\sigma^2.$ Can I use this property to infer anything about the $L^2$ norm of a two-dimensional Brownian motion?