What is the big-O (worst-case upper bound) for time and space requirement of the different Chomsky classes? Everybody knows the Chomsky-hierarchy for describing formal languages and big-O notation for describing time and space complexity of a function.

We know, that each class in the Chomsky-hierarchy corresponds to an atomata, that can recognize any language in the corresponding class.

The question is: What is the worst-case time and space complexity of these automatas given any language of the above class?

What I found out already:

Chomksy-3 class, the regular languages can be recognized by an FSM of n-states where n is a finite number. IMO this makes it require O(n) space and O(l) time, where l is the length of the input.

Chomsky-2 class, the context free languages can be recognised by a Non-deterministic pushdown automaton. CYK algorithm run $n^3 * G$ where n is the length of the input and G is the number of rules in the grammar in Chomsky Normal form.

But how much space needed? Do we have better bounds? 

Chomsky-1 class, the context sensitive languages can be recognised by a Linear-bounded non-deterministic Turing machine.

This class seem to depend on the length of the context to be recognised. Maybe in exponential time.

Chomksy-0 class, the recursively enumerable languages can be recognised by Turing machine.

Can anything stated for this class? If not, why? 
Maybe the $P \stackrel{?}{=} NP$ problem is related.

Are the above statements true?

Please give citations to your answer!
Regular languages can be accepted in linear time and constant space.

Valiant's algorithm parses arbitrary context-free languages in time $O(n^\omega)$, where $\omega$ is the matrix multiplication constant. Deterministic context-free languages (which are those used in practice) can be parsed in linear time.

Context-sensitive languages are identical to the class $\mathsf{NSPACE}(n)$. I'm not aware of any non-trivial time upper bound.

Recursively enumerable languages cannot be parsed in general, since $\mathsf{R} \neq \mathsf{RE}$.

The $\mathsf{P}\stackrel?=\mathsf{NP}$ question is completely unrelated. Unfortunately, the undergraduate curriculum usually spends much more time on the old-fashioned Chomsky hierarchy compared to more modern hierarchies afforded by computational complexity theory.