Is there a theory of computation that takes failure and decay of the computation substrate into account? There are obvious differences between a Turing machine and a real computer. Not only is the latter finite in size, it is also prone to failures and it is made from decaying matter. The kind of framework I have in mind is this (but please excuse the handwavy vagueness): In some sense, (quasi)particles that constitute matter can be thought to perform computation: They have state that persists at least for some time (properties such as position, spin, charge, momentum) and they can change their state in feedback loops by interacting with other particles. The more elementary particles locally affect each other, the more complex the state and feedback loops can be. What kinds of computation we can perform on such a substrate depends on the configurations and rules that govern the particles, in particular, how many feedback loops they allow to simultaneously affect a local arrangement of particles. In this sense I'm looking for a framework that is analogous to the Chomsky hierarchy, i.e. in which the complexity of the system is successively extended to eventually build up to a notion of general computation. Is there something similar to this in the literature?
I'm not 100% sure what the question is about, but the title seems to ask about computation that allows failure. There is a lot of work on noisy (erroneous) computation in the sense that I think you are asking about. I don't have time to give a complete overview, but here are some pointers that may be of interest.


Distributed systems. This is probably by far the most developed field because the very definition of a distributed system is based on the possibility of (partial) faults. There are elaborate theories of DS depending on the chosen fault model, e.g. fail/stop or byzantine failure. See also the work on failure detectors. 
Cryptography. This may seem a bit bizarre, but I think much of the field of cryptography can be seen as a form of extreme failure, with the adversary trying to undermine (in a general sense) the working of the system. You can think of the usual polynomially bound attacker as an case of failure model. This lead to things like secure multi-party computation as an algorithmic tool in the face of extreme adversity.
π-calculus: see (1) for a generalisation of Shannon-style information theory to interacting processes. I think the error model in (1) is unrealistic, because it does not cater for the situation where an error converts a free into a bound name or vice versa.
Turing machines: see (2, 3, 4).
Cellular automata: various researchers have added failure models, sorry no references at the moment, maybe look at work by  P. Gacs.
λ-calculus: see (6).
Assembly language: see (5, 7).
Quantum computation: is all about error correction, I'm not qualified to say more on this.
An increasingly important subfield of computation is the use of probabilistic process calculi to model chemical reactions. Have a look at Luca Cardelli's recent work.


Note also that this line of thinking is very old: J. von Neumann wrote about this in 1956 [8]. Maybe even more impressive C.-A. Petri invented Petri-nets in August 1939 at the age of 13 for the purpose of describing chemical processes.




M. Ying, Pi-Calculus With Noisy Channels.
I. Capuni, P. Gacs, A Turing Machine Resisting Isolated Bursts Of Faults.
I. Capuni, Fault-tolerant Turing Machine.
E. Asarin, P. Collins, Noisy Turing Machines.
J. Chang, G. A. Reis, D. I. August, Automatic Instruction-Level Software-Only Recovery.
D. Walker, L. Mackey, J. Ligatti, G. A. Reis, D. I. August, Static Typing for a Faulty Lambda Calculus.
F. Perry, L. Mackey, G. Reis, J. Ligatti, D. I. August, D. Walker, Fault-tolerant Typed Assembly Language.
J. von Neumann, Lectures on Probabilistic Logics and the Synthesis of Reliable Organisms From Unreliable Components.