Let $g\in C_c(\mathbb{R}^n,[0,\infty))$ with $\int g(x)dx=1$. Denote $\Phi$ the Heat kernel given by $\Phi(x,t)=\frac{1}{(4\pi t)^{n/2}}e^{-\frac{|x-y|^2}{4t}}$.
Let $u(x,t)=\int_{\mathbb{R}^n}\Phi(x-y,t)g(y)dy$.
Now I'm claiming: There exists $M>0$, such that
$$\lim_{t\to\infty}\int_{B(0,M\sqrt{t})}u(x,t)dx=\frac{1}{2}$$
Moreover, for every $t>0$ there is $R(t)>0$ (which is not unique), such that
$$\int_{B(0,R(t))}u(x,t)dx=\frac{1}{2}$$
I cannot prove the claim. I think the secound assertion is included in the first one, since for $t\le 1$ the inequalities hold: $$c_1\sqrt{t}\le R(t)\le c_2\sqrt{t}$$
What is the physical meaning of the above two equations. I'm glad about fruitful explanations.
I'm very grateful for an ansatz.