I am interested in the scaling of


$$F(x\_1,x\_4)=\int\_{\mathbb R^2} e^{-\vert x\_1 -x\_2 \vert -\varepsilon \vert x\_2 -x\_3 \vert- \vert x\_3 -x\_4 \vert } \ dx\_2 dx\_3 $$


In particular, I suspect that


$$F(x\_1,x\_4) \le C \varepsilon^{-n} e^{-{\varepsilon}\vert x\_1 -x\_4\vert}$$
for some universal $C>0$ and $n \ge 0$.


But this is really only based on pure heuristic and I do not know which $n$ could be optimal here.



Scaling of double convolution
$\newcommand\ep\varepsilon$$F(x\_1,x\_4)$ is $8/\ep$ times a value of the convolution of two copies of a pdf with maximum value $1/2$ and a pdf with maximum value $\ep/2$. So,
$$F(x\_1,x\_4)\le(8/\ep)\min(1/2,1/2,\ep/2)=4\min(1/\ep,1)$$
for all real $x\_1,x\_4$.




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The straightforward integration gives
$$F(x\_1,x\_4)=2\frac{\ep e^{-\left| x\right| } \left(\ep^2 (\left| x\right| +1)-\left| x\right| -3\right)+2
 e^{-\ep \left| x\right| }}{\left(1-\ep^2\right)^2}$$
for $\ep\in(0,1)\cup(1,\infty)$, with
$$F(x\_1,x\_4)=\frac{1}{2} e^{-|x|} \left(x^2+3|x|+3\right)$$
for $\ep=1$, where $x:=x\_4-x\_1$, for all real $x\_1,x\_4$.


In particular, for each $\ep\_\*\in(0,1)$ and all $\ep\in(0,\ep\_\*]$,
$$F(x\_1,x\_4)\le C(\ep\_\*)e^{-\ep\left|x\_4-x\_1\right| }$$
for some real $C(\ep\_\*)>0$ depending only on $\ep\_\*$ and all real $x\_1,x\_4$.


