I have observed the historical PE (price / profit) value of a stock and realized that it roughly follows a log normal distribution. However, even when the next earning data point is easily predictable, this distribution cannot be used to predict the distribution of next stock price data point, because, the next PE value depends on the previous one, thus it is not set of repeated independent trials. In other words, for a regular random variable that follows the log normal distribution, the next value can be anything, regardless of the current value. But for the PE value of a stock, it must change gradually.
It is also not exactly a random walk, because for the random walk, the next step's direction is also independent of the previous step. This is not true for the PE, when it is too low, it is much more likely to go up than go down.
So my question is, what kind of random process can properly model the movement of the PE, which changes gradually, but overall, follows a log normal distribution?