... Can I train the dog to bite trespassers on the lawn??

(I'm kidding, I'm just here to rant a little).
Someone went entirely out of their way to come up onto our private front door sidewalk in order to cut through the half foot deep snowy yard all the way to the public sidewalk on the other side of the street when there were entirely shoveled sidewalks all the way there... Which were the sidewalks they used to get to our yard in the first place.

People keep cutting through and it's making me pretty mad. I just don't know what to do about it, I've always lived in the country and now we're on a corner lot in a suburb. Hopefully having a dog will give us an alarm bell and I'll be able to catch them in the act and ask them to use the nice, clean, wide open, safe, hazard free sidewalk.