Nelson's proof of Liouville's theorem

This doesn't answer any of the three specific questions asked, but addresses an implicit question: "Why did the editor accept it?"

In 1961, the Proceedings of the AMS established a section called "Mathematical Pearls" devoted to, I quote:

The purpose of this department is to publish very short papers of an unusually elegant and polished character, for which normally there is no other outlet.

In the issue in which Nelson's proof appears, that section starts on page 991 and continues to the end, including 7 papers in all, none of which exceeds 2 pages. In a different issue you can find this paper which also contains the quoted disclaimer above.

This is to answer question (2): what are the standard proofs before 1961?

According to Barry Simon (Harmonic Analysis [volume 3 of his "Comprehensive course"], p197) the standard citations for "Liouville" theorem pre 1961 are this paper of Maxime Bôcher from 1903 and this paper of Emile Picard from 1924.

Bôcher proves the Liouville's theorem with a one-sided bound in a footnote to the following theorem:

The function $u$ being harmonic when $r>R$, it either becomes both positively and negatively infinite for different ways of going to infinity, or it approaches one and the same finite limit for every method by which the point P recedes to infinity.

This was proven by taking first the Kelvin transform (which Lord Kelvin proved in 1847 [Extraits de deux Lettres adressées à M. Liouville; this is published in Liouville Journal, a.k.a. J. Math. Pure. Appl]) which makes the problem one of an isolated singularity at the origin, and by a theorem Bôcher proved in the pages before, this means that function has the form $$ u=\frac{c}{r^{n-2}}+v, $$ where $v$ is harmonic at the origin. The Liouville's theorem then follows immediately by applying the mean value property to a large circle.

Picard proves that a positive harmonic function $u$ on $\mathbb{R}^3$ is constant by Harnack's estimates $$ c_R u(0) \leq u(x) \leq C_R u(0), $$ where $$ c_R=\min_{y\in\partial B_R(0)}P_y(x),\quad C_R=\max_{y\in\partial B_R(0)}P_y(x) $$ are explicit expressions that tend to 1 as $R$ goes to infinity and $P_y(x)$ is the Poisson kernel in the disc $B_R(0)$. The estimates follow readily from the representation $$u(x)=\frac{1}{4\pi R^2}\int_{y\in\partial B_R(0)} P_y(x)u(y)dy.$$ In dimension 3, they first appeared in Poincare (1890); Harnack (1887) did the two-dimensional case. In fact, a Nelson-type argument gives similar bounds with worse constants, which might be an explanation as to why it was neglected.

Picard gives no attribution to any of the results in his paper (none of which were actually his), just saying "these are theorems I prove in my course for a long time". So, it is reasonable to assume that he knew about the paper of Bôcher. On the other hand, Bôcher's proof seems to require an additional argument to make it rigorous (Sard's lemma does the job, but it was unknown until 1939), so Picard's paper might be indeed the first reference for a complete and explicit proof of Liouville's theorem.

The difference between Picard's and the "modern" proof is not huge (from the modern perspective). Both are using some sort of "mean value" property of harmonic functions. But Nelson's proof using the simplest version of the mean value property by allowing the domains of integration to differ leads to a proof that can be written down without symbols; I don't think the same can be done easily with Picard's version.

This proof was new to me when I read it:-) The standard proof, which I teach, and which is given in most books uses Harnack's inequality, which follows from Poisson's formula for the ball, or Poisson's formula directly. If I were the editor or a referee, I would accept this paper.