Theorem 3.3.4 in Davies' Heat Kernels and Spectral Theory begins with ``on-diagonal'' lower bounds for the heat kernel $K$ of $H$, (i.e. $K = e^{-Ht}$), where $H$ is a uniformly elliptic operator acting on $L^{2}(\mathbb{R}^{N})$. That is, Davies has already proved $$K(t,x,y) \geq C t^{-N/2} \qquad \text{ when } \qquad |x-y|^{2} \leq C t$$ Now for arbitrary $x,y \in \mathbb{R}^{N}$ and $t > 0$, he defines a sequence of points $$x_{r} = x + r(y-x)/M$$ where $0 \leq r \leq M$ and $M$ is the smallest integer such that $4(y-x)^{2}/Ct \leq M$. Then he uses the inequality

$$K(t,x, y) \geq \int\cdots\int K(t/M,x,y_{1})K(t/M,y_{1},y_{2})\cdots K(t/M, y_{M-1},y)\,dy_{1}\cdots dy_{M-1}$$

where $y_{r}$ is being integrated over the set $$\{|y_{r} - x_{r}| < 1/4\,C(t/M)^{1/2}\}$$ But this step I do not understand at all. Where does the iterated integral come from??