Paper on mathematical analysis using tools from other (almost unrelated) field: how to introduce them?
An appendix, maybe
In many other fields it is typical to first choose a journal and only then start writing. In mathematics this is often not done. But here it might be relevant to consider your target audience and thereby target journal.
If the problem is from analysis and you want to publish it in a journal devoted to analysis (or in any non-stochastics journal), an appendix with the stochastics might be appropriate. In any case, formatting the paper so that it can be read without the material in the appendix is a worthwhile task. It also makes it simple to react to referee comments concerning the usefulness of the appendix, if you get such: you can keep it, integrate it back into the main text or remove it with only a little trouble.
If you think the paper is of (more) interest to people into Malliavin calculus, or would be easier to publish in a journal close to stochastics, then maybe leaving out the basics is desirable. But remember to motivate the problem sufficiently.