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We consider the heat kernel $$ g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ). $$

I have already proved that $$ \begin{align} g(t, x+y) &\le 2^{d/2} g(2t, x) e^{|y|^2/(4t)}, \\ \nabla g(t, x) &= \frac{-x}{2t} g(t, x). \end{align} $$

I'm trying to prove the inequality (2.3) in this paper, i.e.,

$$ |\nabla g(t, x)| \le \frac{2^{d/2}}{\sqrt{t}} g(2t, x). $$

Could you explain how to fix my below failed attempt?


We have $$ |\nabla g(t, x)| = \frac{|x|}{2t} g(t, x) \le \frac{2^{d/2} |x|}{2t} g(2t, x). $$

Then I'm stuck because it's not true that $|x|^2 \le 4t$.

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1 Answer 1

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By your computation, it suffices to show that $$\frac{|x|}{2\sqrt{t}}\leq 2^{d/2}\frac{g(2t,x)}{g(t,x)}=e^{\frac{|x|^2}{4t}}$$ This follows from the fact that $y\leq e^{y^2}$ for $y\geq 0$.

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