while i dont have experience with bikes in snow i do do a decent amount of ATVs in the dunes and it seems to be similar (chains/parts need lubrication, lots of sand/salt around)
#1 thing is you do not want to use oil and grease is even worse, what happens is with grease especially all the sand/salt sticks to it and then you are basically making sure you are grinding those into your chain and gears
i have used...i think it was lithium white with fairly good results, a more wax based lube helps like this
Maxima Chain Wax - RevZilla
a lot of guys dont lube their chains for dune riding, that way the sand just falls straight off, i lube with a more wax based but go very light, seems to work well