Product of harmonic maps is harmonic: a stochastic approach

Let $\phi_j: M_j\rightarrow G$, $j=1,2,\ldots, n$, be harmonic mappings from Riemannian manifolds $M_j$ to a Lie group $G$. Then the product $\phi_1\phi_2 \ldots \phi_n$ is a harmonic mapping between $M_1\times M_2\times \ldots M_n$ and $G$. The proof is a combination of properties of Brownianmotion in manifolds and It\^{o} formulae for stochastic exponential and logarithm of product of semimartingales in Lie groups.