Given a noise power spectral density, how long does it take for the underlying process to deviate by a given amount?

Assume we are given a noise power spectral density $$S(f)$$ for a stochastic process $$x(t)$$.

1. What can be said about how long it takes for the underlying stochastic process to deviate by a given amount $$\delta x$$? As the process is stochastic, surely the answer must be in terms of an average first passage time, or something like that.

2. What can be said about the probability distribution of the process at a time $$t_w$$ after it is measured to be at say $$x_0$$?

• Can you give a formal definition of "deviate"?
– MBaz
Jun 25, 2021 at 22:41
• your question sound like you mean them deterministic – but this is a stochastic process; you can (almost) never give specific prediction; you might be able to give expecations, or even probability densities for deviation, or something, but ultimately, you can never actually use a crystal ball to look into the future. Jun 25, 2021 at 22:50
• @MarcusMüller I doubt Mostafa intends to ask about a deterministic process, because it is clearly written in the question that there is an underlying "stochastic process." Probably the wording is just a little unclear. I think the question is trying ask something like "What is the average time it takes for the process to deviate by a certain amount?" Jun 25, 2021 at 23:50
• @DanielSank that's exactly what I meant with "it sounds like you're asking about deterministic things, but you're considering a stochastic process" :) Jun 25, 2021 at 23:51
• @MarcusMüller My point is that I think Mostafa knows that the process is stochastic and the fact that the question sounds as if it's about deterministic things is just an unfortunate accident of word choice. Keep in mind that the user may not be a native English speaker. Jun 25, 2021 at 23:53

Starting with the first question

What can be said about how long it takes for the underlying stochastic process to deviate by a given amount $$\delta x$$?

we can say something about the variance of the process. As I have previously answered in another related question on the physics site, the variance of a process after time $$\tau$$ is $$\langle x(\tau)^2 \rangle = \tau^2 \int_0^\infty S_{\dot{x}}(\omega) \left( \frac{\sin(\omega \tau / 2)}{\omega \tau / 2} \right)^2 \frac{d \omega}{2\pi}$$ where $$S_{\dot{x}}(\omega)=\omega^2S_x(\omega)$$ is the spectral density of the velocity of the random process$$^{[a]}$$.

Now to the second question

What can be said about the probability distribution of the process at a time $$t$$ after it is measured to be at say $$x_0$$?

You're basically asking for the conditional probability $$P(x, t| x_0, 0)$$, i.e. given that the process was at $$x_0$$ at time $$0$$, what is the probability distribution of the process at time $$t$$? That question cannot be answered from the spectral density alone. As Marcus Muller's answer explains, the spectral density is equivalent to the correlation function, which is less information than the full conditional probability.

However, with some other assumptions, more can be said. For example, if a process is both Markov and Gaussian, then, as proven by Doob, the conditional probability is given by$$^{[b]}$$ $$P(x, t| x_0, 0) = \frac{1}{\sqrt{2 \pi \sigma_t^2}}\exp \left( - \frac{(x - \mu_t)^2}{2 \sigma_t^2}\right)$$ where \begin{align} \mu_t &= \mu + \left(x_0 - \mu \right) \exp(-t / \tau)\\ \sigma_t^2 &= (1 - \exp(-2 t / \tau))\sigma^2 \, . \end{align} Here $$\mu$$ and $$\sigma$$ are the mean and standard deviation of the process and $$\tau$$ is the relaxation time. In fact this theorem guarantees that a Gaussian and Markov process is completely defined by those three parameters.

Of course without the Gaussian and Markov assumptions, much less can be said.

You could ask, for example, what is the mean time it takes for a process to fluctuate by an amount $$dx$$ for the first time. That is called "mean first passage time", and we cannot find it from just spectral density.

For whatever it's worth, one can reasonably easily compute mean first passage time for a discrete time random walk. In fact, there is a delightfully eclectic set of solutions to that problem in this Puzzling Stack Exchange post. There's probably some way to do it for continuous processes too, but right now I do not know how. Perhaps in the Gaussian and Markov case it can be solved...

$$[a]$$: We're assuming that the process is stationary.

$$[b]$$: See this reference (pdf).

How long does it take for the underlying stochastic process to deviate by a given amount of say δx

This really is missing a definition of "deviate", but nevertheless:

This can't be answered; the power spectral density (PSD) does not in itself contain that information. Remember its definition!

It's just the Fourier transform of the autocorrelation function (ACF), so, fine prints about the invertibility of the Fourier transform aside, the information in the PSD and ACF is exactly the same.

The ACF $$\varphi(\tau)$$ is just the expectation $$\varphi(\tau)=\mathbb E\left(X(t)X^*(t-\tau)\right)$$. That's it - it's an expectation value. It doesn't say whether something happens, or with which probability a specific change happens – it just says that for times $$\tau$$ apart, the expected product is the value of $$\varphi(\tau)$$. There's infinitely many different probability density functions that can lead to that same product, and hence to the same ACF, and hence to the PSD.

Again, neither the PSD nor the ACF can answer that; but they can give you an expectation of the product between now and in $$t_w$$, that's literally the ACF's definition.