  We present a multidimensional extension of Kellerer's theorem on the
existence of mimicking Markov martingales for peacocks, a term derived from the
French for stochastic processes increasing in convex order. For a
continuous-time peacock in arbitrary dimension, after Gaussian regularization,
we show that there exists a strongly Markovian mimicking martingale It\^o
diffusion. A novel compactness result for martingale diffusions is a key tool
in our proof. Moreover, we provide counterexamples to show, in dimension $d
\geq 2$, that uniqueness may not hold, and that some regularization is
necessary to guarantee existence of a mimicking Markov martingale.
