Representations of Markov Processes as Multiparameter Time Changes

Let Y₁, Y₂,⋯ be independent Markov processes. Solutions of equations of the form Zᵢ(t) = Yᵢ(∫ᵗ₀βᵢ(Z(s))ds), where βᵢ(z) 0, are considered. In particular it is shown that, under certain conditions, the solution of this "random time change problem" is equivalent to the solution of a corresponding martingale problem. These results give representations of a large class of diffusion processes as solutions of X(t) = X(0) + ∑Nᵢ₌₁αᵢWᵢ(∫ᵗ₀βᵢ(X(s))ds) where αᵢ ∈ Rᵈ and the Wᵢ are independent Brownian motions. A converse to a theorem of Knight on multiple time changes of continuous martingales is given, as well as a proof (along the lines of Holley and Stroock) of Liggett's existence and uniqueness theorems for infinite particle systems.