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My question may be a little strange, but I'm wondering how random continuous functions work? First, how is it possible to define a random continuous function and how to investigate it. For example, is such a function differentiable? Or how many preimages does an image point have on average? Or what does a random continuous mapping of a segment into a square look like? How similar is it to a curve filling a square? Sorry if these are all stupid questions, I just have no idea how to research such questions at all.

oclelot335
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  • Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Sep 12 '22 at 10:58
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    In most formulations of random (interpreted in the sense of "most" in some way), the random continuous function is nowhere differentiable. This includes notions based on Wiener measure, Baire category, porosity, and some lesser known notions -- see this answer for an overview of some results of this type. There are many other such results involving images and inverse images, generalized differentiability (e.g. symmetric derivative, and others), and pretty much everything except the kitchen sink. – Dave L. Renfro Sep 12 '22 at 10:59
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    In order to talk about a random function, you need to specify a probability measure on a space of continuous functions $C[0,1]$, just like in order to talk about a random point on $[0,1]$ you need to specify a probability measure on $[0,1]$. A canonical choice is the Wiener measure. In that case a typical random continuous function is nowhere differentiable. – G. Chiusole Sep 12 '22 at 11:00
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    For "random" in the sense of Baire category, see Applications of the Baire category theorem by Jones (1997-1998) and the google search for "typical continuous function". For "random" in the sense of Wiener measure, see this google search. – Dave L. Renfro Sep 12 '22 at 11:04
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    See also this Masters thesis. And earlier I forgot to mention prevalence, another "almost all" notion that definitely doesn't belong in the "lesser known notions" I previously alluded to -- see this google search. – Dave L. Renfro Sep 12 '22 at 11:13

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