Multi-dimensional coupled diffusion process (Q1063335)

In this paper, the formal differential expression \[ L=\left( \begin{matrix} Q_ 1\\ 0\end{matrix} \begin{matrix} 0\\ Q_ 2\end{matrix} \right)+\left( \begin{matrix} 0\\ C_{21}\end{matrix} \begin{matrix} C_{12}\\ 0\end{matrix} \right) \] is considered, where \(Q_ 1\), \(Q_ 2\) are general elliptic operators with bounded coefficients \(a_{ij}(x)\), \(b_ i(x)\) and bounded absorption \(C_{11}(x)\), \(C_{22}(x)\), \(x\in {\mathbb{R}}^ n\). It is proved that L generates a \(\hat C\) semigroup, the coupled diffusion semigroup acting on functions with zero limit at infinity. The associated Markov process is constructed and shown to have nice trajectory properties.