π day in classical Wiener space

It's π day so I thought I might do a math post.

The other day I was doing laundry and I thought to myself, "if I finish this whole pile, my laundry will really only be done for a moment. It won't stay done even past bedtime."

This caused me to flash back to Physics I class in high school. The teacher gave the definition of an object being "at rest" whenever the first derivative of position was zero, i.e., dx/dt = 0. But this meant that a rock, thrown perfectly vertically from the surface of the earth, is "at rest" at the top of its flight. Right at the apex of the parabola you would get on an x vs. t graph.

This upset my classmate Jeff, since the rest state was only achieved instantaneously. He argued that there was a fundamental difference between a rock at rest because it was lying on the ground, and one at "rest" because it had reached the apex of its vertical flight. At the teacher's provocation he made his own definition of rest precise. I'm not sure I remember it but I think it was more or less the idea that an object was at rest at time t if there was an open interval (t−ε,t+ε) such that dx/dt was zero on the entire interval.

But this made me think of the interesting function f (x)=e^-1/x. This function has the property that f(0) = 0 and f'(0)=0 but f(x) ≠ 0 when x ≠ 0. Therefore it might be said to be at rest in my physics teacher's definition but not in Jeff's. But on the other hand, this f occupies a kind of middle ground: the flatness it experiences at 0 is much more profound that the flatness of the parabola traced by the thrown rock at its apex. The function is infinitely differentiable, but the derivatives of all orders are zero at x=0: f ⁽ⁿ⁾(0) = 0 ∀n. Indeed, this function is not analytic at zero, i.e., its Taylor series expansion has zero radius of convergence there (and is therefore useless). This is the classic example of a non-analytic smooth function. The zero function z certainly satisifes the trait of having z ⁽ⁿ⁾(x) = 0 ∀n, and indeed this holds on an open interval containing any x; so we see that f is kind of at the boundary of functions absolutely at rest and those instantaneously at rest. If the Taylor series of a function f is identically 0 at a point x, it is very hard to argue that we are dealing with motion rather than rest.

All of this brought to mind an interesting digression in Miles Reid's Undergraduate Algebraic Geometry (which I confess that I have only skimmed), in which he notes:

[O]ne can reasonably consider the following rings of functions on U [an open interval]

• C⁰(U) = all continuous functions f: U→ℝ;
• C^∞(U) = all smooth functions (that is, differentiable to any order);
• C^ω(U) = all analytic functions (that is, convergent power series);
• ℝ[x] = the polynomial ring, viewed as polynomial functions on U.

There are of course inclusions ℝ[x] ⊂ C^ω(U) ⊂ C^∞(U) ⊂ C⁰(U).

These rings of functions correspond to some of the important categories of geometry: C⁰(U) to the topological category, C^∞(U) to the differentiable category (differentiable manifolds), C^ω(U) to real analytic geometry, and ℝ[x] to algebraic geometry. The point I want to make here is that each of these inclusions represents an absolutely huge gap, and that this leads to the main characteristics of geometry in the different categories. Although it's not stressed very much in school and first year university calculus, any reasonable way of measuring C⁰(U) will reveal that the differentiable functions have measure 0 in the continuous functions (so if you pick a continuous function at random then with probability 1 it will be nowhere differentiable, like Brownian motion).

What a fascinating thought! Constructing a function which is continuous on a whole interval but nowhere differentiable on that same interval requires a fair bit of effort. The effort required creates the false sense that such functions must be rare. But Reid asserts that they are insanely common in the wild landscape of function space. I wondered whether Wikipedia had enough information to supply some information about this and I was somewhat pleasantly surprised that it did. The answer is found in classical Wiener space (why didn't I think to look there?)

Right in the the article I just linked there's a section on the density of nowhere-differentiable functions. It turns out, though, that measure in function space is more complicated than on ℝⁿ; I was surprised to learn that "there is no infinite-dimensional Lebesgue measure," and that in fact classical Wiener space was invented to create a notion of measure that worked for spaces of continuous functions. A great deal of the previous, I wish to stress, is way over my head. I can't understand the definition of Wiener space at all; I checked out of real analysis too early. But I do find it fascinating when exotic things seem to dominate familiar things in number the way Brownian functions seem to do with continuous functions.