I have a purple loosestrife, and I have never had any seeds sprout, it has never came up by runners or anything-not invasive at all. It blooms like crazy all summer. I wonder if it is something different and not the same kind as the white. I don't think the foliage looks quite the same. ? I don't notice any fragrance on it either. It does grow really fast and fill in, but not invasive. I notice the butterflies like it.
