I'm trying to figure out if the Heat Kernel is an approximation to identity.
I can't see if the Heat Kernel verifies this: $$|K_d(x)|\leq A d/ |x|^2 \forall x \in \mathbb{R}, \forall d > 0 $$ where the heat kernel equation is $$K_d(x) = (4 \pi d)^{-1/2} e^{\frac{-x^2}{4d}}. $$
Any idea?
------Edit------
My first attempt was to prove that there was a positive constant $A$, independent from $d$ such that $$\frac{|x|^2}{d} K_d \leq A $$ but I failed because as $d \to 0$ that function seems to peak to $\infty$.
Your question seems to be whether $(4\pi d)^{-1/2} e^{-\frac{x^2}{4d}} \le \frac{Ad}{|x|^2}$. Stripped of a bit of decoration this is the question whether $e^{-\frac{x^2}{4d}} \le \frac{A d^{3/2}}{|x|^2}$. What happens in case $|x|=1$ and $d \to \infty$?