A bounded drift cannot be uniformly strictly dissipative on the real line
= A bounded drift cannot be uniformly strictly dissipative on the real line
{title2=$|b|\leq B\quad\Longrightarrow\quad\text{no global }k>0\text{ contraction}$}
If $|b|\leq B$, then $2(x-y)(b(x)-b(y))+(\sigma(x)-\sigma(y))^2\geq-4B|x-y|$. It cannot also be at most $-k|x-y|^2$ for every pair on the real line when $k>0$. Global strict dissipativity thus requires relaxing global boundedness of the drift; linear confining drifts give natural compatible examples.