# Under what assumptions does $\langle\eta(t_1)\eta(t_2)\rangle=\int d\eta_1d\eta_2\eta_1\eta_2p(\eta_2,t_2;\eta_1,t_1)$ become $\delta(t_1-t_2)$?

In the Langevin model, if we make the assumption that the random force $$\eta(t)$$ acting on the Brownian particle is a stationary, Markovian, and gaussian process, does it automatically ensure that the autocorrelator $$\langle\eta(t_1)\eta(t_2)\rangle\propto \delta(t_1-t_2)$$?

The reason I am asking this is that deviation from delta-correlated autocorrelator is often regarded as a signature of non-Markovianity.

Since there is an explicit formula for the autocorrelator, I want to check under what assumptions, this becomes delta-correlated (and whether those assumptions have anything to do with Markovianity). To that end, we recall that for a stationary, Gaussian process the relevant probability density function to calculate $$\langle\eta(t_1)\eta(t_2)\rangle$$ is given by

$$p(\eta_2;t_2;\eta_1,t_1)=C\exp\left[-\sum\limits_{j=1}^{2}\sum\limits_{k=1}^{2}\alpha_{jk}(\eta_j-\bar{\eta})(\eta_k-\bar{\eta})\right]\tag{1}$$

where $$C$$ and $$\alpha_{jk}$$ depend only on $$(t_1-t_2)$$. Therefore, $$\langle \eta(t_1)\eta(t_2)\rangle=\int d\eta_1\int d\eta_2~~ \eta_1\eta_2p(\eta_2,t_2;\eta_1,t_1).\tag{2}$$

But using $$(2)$$ in $$(1)$$ does not seem to lead to a $$\delta(t_1-t_2)$$ dependence. I know that when the process is Markovian, we have, $$p(\eta_2,t_2;\eta_1,t_1)=p(\eta_2,t_2|\eta_1,t_1)p(\eta_1,t_1)$$ but I am not sure if that is going to help here. I have used the symbol '$$|$$' to denote conditional probability.

Delta-correlated process and Markovian process is not necessarily the same thing (although the terminology may vary - my discussion is based on the theory of random processes, but, e.g., in the theory of master equations Markovian is understood somewhat differently).

Markovian process
Markovian process is often said to be the process where the probability of an event depends only on the state of the process in the previous time moment. This is not literally the preceding time instant, but the moment where the process was last measured.

On a more technical level, let us consider a process $$x(t)$$ that was measured at times $$t_n, t_{n-1}, ... , t_2, t_1$$. The joint probability density that at these times the process took values $$x_n, x_{n-1}, ..., x_2, x_1$$ is $$p(x_n, t_n; ... ;x_2, t_2; x_1, t_1) = p(x_n, t_n| x_{n-1}, t_{n-1}; ... ;x_2, t_2; x_1, t_1)\times p(x_{n-1}, t_{n-1}| x_{n-2}, t_{n-2};... ;x_2, t_2; x_1, t_1)\times p(x_2, t_2| x_1, t_1)\times p(x_1, t_1)$$ Note that so far we have made no assumptions, but only used the definition of the conditional probability density. In particular, if the process is measured only at two time instants: $$p(x_2, t_2; x_1, t_1)=p(x_2, t_2| x_1, t_1)p(x_1, t_1)$$ - this is just a definition.

For a Markovian process the conditional probability densities in the equation above depend only on the latest time instant: $$p(x_n, t_n; ... ;x_2, t_2; x_1, t_1) = p(x_n, t_n| x_{n-1}, t_{n-1})\times p(x_{n-1}, t_{n-1}| x_{n-2}, t_{n-2})\times ...\times p(x_2, t_2| x_1, t_1)p(x_1, t_1)$$

Gaussian process
For a Gaussian process the probability density above is given by a multivariate Gaussian/normal distribution $$p(x_n, t_n; ... ;x_2, t_2; x_1, t_1) = \frac{1}{\sqrt{(2\pi)^n\det{\Sigma}}} e^{-\frac{1}{2}\sum_{i,j=1}^n\left(x_i-\bar{x(t_i)}\right)\left(\Sigma^{-1}\right)_{ij}\left(x_j-\bar{x(t_j)}\right)}$$ If the number of measurements goes to infinity, the distribution is replaced by a functional $$p\left[x(t)\right] \propto \exp\left[-\frac{1}{2}\int dt_1, dt_2 \left(x(t_2)-\bar{x(t_2)}\right)\left(\Sigma^{-1}\right)(t_2,t_1)\left(x(t_1)-\bar{x(t_1)}\right)\right]$$

White noise
White noise is a process with uniform spectrum, i.e., with a constant Fourier transform of its covariance function. In real space it means that $$\langle x(t_2)x(t_1)\rangle -\langle x(t_2)\rangle\langle x(t_1)\rangle =\sigma_x^2\delta(t_2-t_1)$$

Remark As I noted in the beginning, the term Markovian is sometimes used with somewhat different meaning. In statistical literature, the deviations of the correlation function from delta-function are usually referred to as colored noise (as opposed to white noise). In this case the process is no more diffusive and one cannot use usual Fokker-Planck equation - instead one has to resort to its non-local version, such as Kolmogorov-Feller equation for a jump process. This has parallel developments in terms of Langevin equation, which becomes non-local (an integral term appears).

• I am aware of the Markov process definition that you have written down. But if you look at this paper Quantum Langevin equation by G. W. Ford, J. T. Lewis, and R. F. O’Connell , it seems to say that in the quantum case, deviation of the autocorrelator from $\delta(t_1-t_2)$ is a sign of departure from Markovianity. – SRS Jun 14 at 7:51
• I have expanded the answer. As I noted in the beginning, the definition of Markovian is sometimes misused - I noticed it when dealing with people working in quantum optics. – Roger Vadim Jun 14 at 7:55
• @SRS it seems that there is linguistic ambiguity regarding the Markovian process as I described, and Markov approximation which implies no memory of previous events - essentially a delta-correlated process. Deviations from either can be referred to as non-Markovian. – Roger Vadim Jun 14 at 8:12

To me, the white noise is an idealization that is defined by three properties: $$<\eta(t)>=0$$, $$<\eta(t)\eta(t')>=\delta(t-t')$$ and $$\eta(t)$$ is Gaussian. It is Markovian because of these properties. Physically, the white noise is a random sequence of negative and positive pulses and every pulse is independent from the previous one.

All of this is consistent with your equation (1) since you get a Dirac delta from a normal distribution making the standard deviation going to zero while keeping the area under the curve constant.