check whether a convolution has compact support

Question

I am working on a problem which lead me to the following integral, given $h:R \rightarrow R$ is a smooth nonzero function with compact support. For $\epsilon>0$, let$${u_\epsilon(x)} ＝{ \frac{ \intop\nolimits_{-\infty }^{\infty }(x-y) e^{-K(x,y)/\epsilon}dy }{ \intop\nolimits_{-\infty }^{\infty } e^{-K(x,y)/\epsilon}dy }}, \forall x \in R,$$where $K(x,y)=\frac{1}{4}(x-y)^2+\frac{1}{2}h(y) $ for $x,y\in R$. Is each ${u_\epsilon(x)} $ a smooth function with compact support? Must $ \lim_{\epsilon\to 0+} {u_\epsilon(x)}$ exist? If it exist what is the value?