Just been reading with curiosity on the Paulownia tree. In the past I have grown alot of these and yes it does depend on the variety. The tomentosa is listed on the invasive species list and it does drop it seeds everywhere. They have lots of names, Princess tree, Kiri tree, empress tree which are all the tomentosa variety. There are several varieties and the elongata is certainly not invasive as I several acres that will be soon harvested for the lumber and have never had a problem with them. Yes they are extremely fast and I have 12"-15" diameter trees right now after 7 years. My acres look so beautiful in springtime with all the flowers which I will miss for a growing season when they are harvested. Hope this helps you a little further. I am certainly not an angry grower of Paulownia. Doing your homework is important.
Jason