Ok, I have to chime in with a logical suggestion after we all have given the childish ones. Have you thought of planting a privacy hedge row? Obviously, you are not going to want to make this neighbor a friend now or in the future and a privacy hedge would most likely do the trick. It would cost a whole lot more to sell your home just because of this issue. You purchased a home which backs up to someone else, and unfortunately, these are issues which arise in such properties. So, if you spend a little, you don't have to get real high hedges, they will grow, just place them in a way the dog can't come onto your property. May even want to consider placing some chicken wire or something on your side. The dog will learn in a short time he can't come into your lawn.
Good luck with your situation and I truly feel sorry that this is making your life miserable. It shouldn't be like that for anyone.
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