I'm studying the heat kernel of the continuous time simple random walk $X_t$ on $\mathbb{Z}^d$. I know of the carne varopoulos bound for the heat kernel. But I'm lookong for a similiar bound for the derivative (and second order derivative), i.e. for bounds of the type $\lvert p_t(0,x+e)-p_t(0,x)\rvert\leq K t^{-(d+1)/2}e^{-c\lvert x\rvert^2/t}, \quad \forall x\in \mathbb{Z}^d \text{ and unit vectors }e\in \mathbb{Z}^d$
for some constants $c,K>0$. Do bounds like these exist?