Price fluctuation of a dividend paying stock

I am trying to validate hypothesis of price fluctuation of a dividend paying stock. Most stock pay dividends quarterly. I assume the dividend is paid to who holds the stack on the day if dividend payment, not who held it prior to this date? Therefore, taking aise regular market price fluctuation I would envision that the price of dividend paying stock should rise gradually till the date the dividend is paid and then drop the day after dividend is paid. And then this quarterly cycle should continue.