**There is a Uniform Hopf Inequality as follow:**


Let $\Omega \subset \mathbb{R}^n$, $n \geq 1$ denote a smoothly bounded domain. Also let $\rho(x)=\mathrm{dist}(x,\partial \Omega)$, the distance function from $\partial \Omega$. Assume that $f ≥ 0$ belongs to $L^∞(Ω)$ and let $u$ denote the solution of
$$
\begin{cases}
-\Delta u = f & \Omega \\
u=0 & \partial \Omega
\end{cases}
$$
There exists $C>0$, independent of $f$, such that:
$$ u(x) \geq C \rho(x) \int\_{\Omega} f(y) \rho(y) \, dy. $$


I want to know that is there a similar type of result for heat equation or fractional one? I will be thanked if someone can give a reference.


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Edit: For the proof of this elliptic result see for example the Proposition A.4.2 in the Appendix of following book: Stable solutions of elliptic partial differential equations by Louis Dupaigne.



Uniform Hopf Inequality
It seems that you are looking for so-called **lower heat kernel bounds**.


* For the heat equation the inequality you are looking for can, for instance, be found in Theorem 1.1 of "Zhang: The Boundary Behavior of Heat Kernels of Dirichlet Laplacians (Journal of Differential Equations, 2002)".
* For the fractional Laplacian the paper "Chen, Kim, Song: Heat kernel estimates for the Dirichlet fractional Laplacian (Journal of the European Mathematical Society, 2010)" seems to be relevant for your question.


