Pics would help! Their ruts are undermining your fence, don't they have the hillside planted?, Since they don't seem so nice I think I would dig out(when no rain is expected) under or to your side of fence a trench & put in a 6 in. wide by at least a foot deep of cement the whole length of that fence section. To lessen cost perhaps you could put some large rocks maybe broken concrete hunks that you could have standing & even deeper than 1 ft & then fill in center with concrete to stabilize whole thing. It would really have to rain to take that out, I would like to see how deep the ruts are. Seems like there should be some provision of law that doesn't allow that much runoff. Might try putting this over on the landscaping forum as they have some experts over there & they get more people coming to that forum. This 1 doesn't have many posts. If you do nothing you will loose soil & have to replace it on your side, that can get expensive. They are responsible for filling it in on their side of property line. Take pics in case you have to take them to court or just so you don't get blamed for their soil taking off down the hill or drive. They may need to install a drain to the street or something. Are there others with same problem? How are they handling it? If the water is going down a paved or concrete drive it will eventually weaken it. Good Luck!

you need retaining wall on the hill, it is avoid a soil solution from erosion, if you can try retaining product like geotextile or geogrid

Here is a link that might be useful: geotextile