Most of my paulownias died from lack of water, and the four or five I still have DO NOT have any seedlings springing up. They are propagated through root stock primarily. Yes, the seeds CAN sprout, but take much longer to grow.
Try digging up some of the root stock; I think that will work. And no, they have NOT become the spreading mess that we are led to believe. They are beautiful and refreshing in the landscape. I DO believe, however, that if they get into waterways they WOULD be a problem. I don't have any waterways on my mountain.
I don't have any waterways in my desert either. lol
We are very pleased with these trees. There are only 4 in my little town that I know of. 3 are ours, 1 is a neighbor's. Strangers knock on our door to find out where they can get one of these amazing trees.
If I had 2,000,000 seeds....hm....