Is there a theory of computation that takes failure and decay of the computation substrate into account?
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.