Levy process is freqently cited recently. When we come to multidimensional Levy process, the components are usually assumed to be independent. Are there any example on how to construct a Levy process with some certain dependent structure? I know that with the help Kolmogrov consistent theorem we can complete the target principally, but no example available now.

Levy process is frequently cited recently. When we come to multidimensional Levy process, the components are usually assumed to be independent. Are there any examples on how to construct a Levy process with some certain dependent structure? I know that with the help of Kolmogorov consistent theorem we can complete the target principally, but no example is available now.