Most likely the white stuff is not mold but yeast. I've become a fermenting fiend the last 4 years or so (kraut, pickles, kimchi, tempeh, natto, Effective Microorganisms, dairy kefir, sourdough, water kefir, "ginger bug" soda, ketchup, now chicken feed, etc.). Most ferments involve a symbiotic relationship between yeasts and lactic acid bacteria so the yeasts are beneficial. In my experience, mold can grow, but it usually takes several weeks or more for that to happen. It will depend in large part upon your the climate you live in, more humid environs will be more hospitable to molds. All the molds I'm familiar with smell like mold (I don't know how to explain the aroma in words!) and yeasts smell like yeasts. There can be variations of aromas, but all yeasts I'm familiar with will have a "base note" of yeast smell and the same with molds. There are so many kinds of molds and yeasts I'm sure there are some that don't fit this description.
The white stuff you describe is again, likely yeast and it can show up as a dusting or sometimes even fuzzy. But fuzzy white mold and fuzzy white yeast don't look the same. In any case, stirring it in will kill the mold (if that is what it is) and probably the yeasts and lactic acid bacteria will "eat" the mold. Many folks will do this with their vegetable ferments and others will simply remove the mold, it's a personal preference.
Most of
these photos are of fuzzy white yeast, any of them look familiar?
Most of
these ones are of fuzzy white mold. Notice that usually there are also brown or green or black colors as well and they're generally much fuzzier.
Side note, we used to live in Seattle, which is fairly humid. Once we were helping a friend renovate a house. We found some black-colored mold in the bathroom and immediately thought it was the famous toxic "black mold". We hired one of the top plumbing companies in Seattle to come in and look at it (we also need some drain repairs) and he said in his 30 years of experience in Seattle he has only seen the toxic black mold a couple times and this was not it. He said that as obvious as the extent of the mold was, if it was black mold, we all would have been very sick by now. Not that breathing mold spores of any kind is generally a good idea, but this wasn't the really bad stuff. He said the toxic stuff was not common in Seattle because it wasn't warm enough for it to really thrive, like it is in a humid/hot climate like the Southern US.