Are white noises always Markovian? Are white noises always Markovian? I am a bit confused about it. As white noise always has a constant power spectrum， its auto correlation function must contain a delta function of time. Thus the correlation time of the noise vanishes. But I don't know whether they can be called Markovian.
 A: Whether a noise process is truly memoryless is very hard to test. Strictly speaking it's impossible to say that it is memoryless because the memory could be longer than the longest time series we can analyze. In practice a lot of low-entropy pseudo-random generators can make synthetic signals that are physically indistinguishable from the "real thing", so it always depends on the application if one can make that assumption and get away with it, or not. 
A: Mathematically, the answer to your question is yes. The dynamics of a physical system that is driven by pure white noise, with constant power spectrum up to arbitrary high frequencies, will be perfectly Markovian. 
However, as CuriousOne points out, it is essentially impossible to verify that a physical noise process is truly Markovian,  because of the finite time or frequency resolution and range we can achieve with our measurements. Furthermore, many of the most common sources of noise are demonstrably not Markovian, e.g. thermal fluctuations at temperature $T$ must have a memory time on the order of $\hbar/k_B T$ in order to satisfy detailed balance. In practice, this often does not matter: as long as the correlation time of the noise is much shorter than the time scales of interest, then the noise can be treated as memory-less to a good approximation. 
