Gaussian bound from a decreasing net growth rate

Statement

Let $g\in C^1([0,L])$ , $L>0$ , satisfy $g(L/2)=0$ and

-C/L\le g'(v)\le-c/L\qquad(0<c\le C).

For the envelope $P(v)=\exp(\int_{L/2}^v g(s)\,ds)$ ,

\exp\left(-\frac{C(v-L/2)^2}{2L}\right) \le P(v)\le \exp\left(-\frac{c(v-L/2)^2}{2L}\right)\le1.

Proof and endpoint decay

The function $h=\log P$ satisfies $h(L/2)=h'(L/2)=0$ and $-C/L\le h''\le-c/L$ . Integrating twice, or applying Taylor's theorem with integral remainder, gives the bounds on either side of the midpoint. In particular, $P(0),P(L)\le e^{-cL/8}$ . A cutoff varying only in endpoint regions a fixed proportion away from the midpoint acts where $P\le e^{-c' L}$ , for a fixed $c'>0$ .