I am currently trying to prove the following bound on the difference in first exit time distributions:
$$P_x(\tau_D\le t+\delta)-P_x(\tau_D\le t)\le C_D\delta$$
where $x\in D$, $t,\delta>0$, and $D$ is a bounded regular domain. Recalling that $u(x,t)=P_x(\tau_D>t)$ is the solution to the heat equation $\frac{1}{2}\Delta u(x,t)=\frac{\partial}{\partial t} u(x,t)$ with $u(x,t)=0$ for $x\in\partial D$ and $u(x,0)=1$ for $x\in D$, I was informed of the following bound:
$$|u(x,t)-u(y,s)|\le C_D(|t-s|+|x-y|^\alpha)$$
for some $\alpha>0$ (presumably $\alpha\le 1$) and $C_D>0$ a constant dependent only on $D$, but I'm having difficulty finding any sources where it could be (I've looked briefly at some PDE books like Evans, Gilbarg-Treigar, and Friedman), but I'm not sure of where else to look or where it could be. If any of you could let me know of any ideas of where it could be, that would be appreciated.