Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\) be a \(\mathrm{BM}^{2}\) and \(f(x+i y)=u(x, y)+i v(x, y), x, y \in \mathbb{R}\), an analytic function. If \(u_{x}^{2}+u_{y}^{2}=1\), then \(\left(u\left(b_{t}, \beta_{t}ight), v\left(b_{t}, \beta_{t}ight)ight), t \geqslant 0\), is a \(\mathrm{BM}^{2}\).