 A bounded drift cannot be uniformly strictly dissipative on the real line

If $|b|\le B$, then $2(x-y)(b(x)-b(y))+(\sigma (x)-\sigma (y){)}^2\ge -4B|x-y|$. 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.