Let $u(t,x)$ be the fundemental solution to the heat equation $u_t = \frac{1}{2}u_{xx}$ with initial condition $u(0,\cdot)$. That is, $u(t,x) = \int_{\mathbb{R}}p_{t}(x-y)u(0,y)\mathrm{d}y$ where $p_t(\cdot)$ is the heat kernel defined by $p_t(x)=e^{\frac{-x^2}{2t}}\frac{1}{\sqrt{2\pi t}}$. Assume that $u(0,\cdot)$ is a bounded continuous function (you may assume finite limits at $\pm$infinity if it helps). I have a conjecture that for every compact set $B$ of the real line, we have that:
$$\lim_{t \to \infty}u(t,x) = \lim_{L\to\infty}\frac{1}{2L}\int_{-L}^L u(0,y)\mathrm{d}y \quad \text{ uniformly in $x\in B$}$$
I wasn't able to prove this. If anyone knows of a reference for this, a proof (or a counterexample), I will appereciate it. Thanks!