Harmonic Liouville theorem

A bounded harmonic function on the complex plane is constant. More generally, an entire harmonic function bounded on either side is constant: write it as $\mathrm{Re}f$ by harmonic function as the real part of a holomorphic function, then apply the Liouville theorem to $e^f$ or $e^{-f}$.

A bounded entire harmonic function satisfies $u(x)={\mathbb{E}}_{x}u(B_t)$ because $u(B)$ is a bounded local martingale. The Gaussian heat kernel has a directional derivative with $L^1$ norm $\sqrt{2/\pi }/\sqrt{t}$. Consequently $|u(x)-u(y)|\le \Vert u{\Vert }_{\infty }\sqrt{2/\pi }|x-y|/\sqrt{t}$; letting $t\to \infty$ proves the harmonic Liouville theorem in every dimension.

A harmonic function on ${\mathbb{R}}^2\setminus \{0\}$ bounded on one side is constant. Shift and change sign to make it nonnegative. Its circular mean has the form $a\log r+b$; positivity for every $r>0$ forces $a=0$. For $|x|=r$, the mean value property for harmonic functions bounds its value by the mean integral over $r/2<|z|<3r/2$, giving at most $8b$. A removable singularity for a bounded harmonic function fills the origin, and the harmonic Liouville theorem finishes the proof. The dimensional distinction matters: $|x{|}^{2-n}$ is a positive nonconstant harmonic function on punctured ${\mathbb{R}}^n$ for $n\ge 3$.

For a bounded harmonic function $f$ on ${\mathbb{R}}^d$, the processes $f(B_t^x)$ and $f(B_t^y)$ are bounded martingales. Couple the Brownian motions from $x$ and $y$ so that they coalesce in finite time almost surely. Thenwhich proves that $f$ is constant in every dimension.

Every nonconstant harmonic function $u:\mathbb{C}\to \mathbb{R}$ is unbounded above and below by the harmonic Liouville theorem. The intermediate value theorem then gives $u(\mathbb{C})=\mathbb{R}$.