Talk:Stochastic dynamical systems
I liked the lecture of this contribution and have a few proposals for improvement.
- Please introduce the transition probability density from a given state x_0 at a time t_0 instead of the probability density for a characterization of sample paths. A transition probability density is necessary for the definition of the correlator. Also it was used, later on, therefore its introduction seems to be reasonable. Its stationary limit is independent on the initial state (Better is to let t-t_0 \to \infty).
-In the section of the OUP: the density is the "stationary" density. Please write stationary.
-In the section of the OUP: Please underline by adding: ...at constant intensity, i.e. \sigma^2 \tau = const.
-In the section for the dichotomic Markov process: Please add that there is a transition to white shot noise (which is a sequence of delta spikes) and from this one can transform to Gaussian white noise later on.
-In the section Ito vs Stratonovich: Practically there are infinite-many interpretations but two are mainly used in the literature, Ito's and Stratonovich's. It depends on the selection of the position of t^\prime during the limit from a sum to an integral in case of the non-differentiable Wiener process.
-Please use g(x,t) instead of b(x,t).
-Please underline: In the following text the Ito calculus is used. also during the Fokker-Planck approach.
-The notion of a(x) and g(x) is sometimes time dependent, some times not, it depends on <\mu> and not. I would hint to use one definition.
-I propose to introduce the discrete part in the master equation only very lately. This part is of importance if discrete jumps of x(t) occur. These jumps relate to maps or spike counts as discrete events. Then an integral part with the nonlocal transitions appear in the balance equation. The W(x,\to x^prime) should be defined outgoing from the transition probability (see van Kampen's book).
-Of importance for the continuous variations of x(t) is that the moments of the increment \delta x in \delta_t either vanish beyond the second (defining a diffusion processes in case of the white noise) or not (Pawula theorem). If not (for example, if the noise is Gaussian colored) the whole Kramers Moyal expansion with infinite many terms and progressive derivatives has to be taken into account.
This is a well-written, concise introduction to stochastic dynamical systems.
I have only minor suggestions for improvements listed below.
1. Wiener-Khintchine theorem is valid for stationary processes. So, please define stationary stochastic processes, preferably before discussion of Wiener-Khintchine theorem and ergodicity.
2. Since the author mentioned Poisson dichotomous noise, a Poisson shot noise will be a reasonable addition. Also, transition to white Gaussian noise will be a useful addition.
3. Since the author addressed Ito/Stratonovich integrals, it will be reasonable to provide two forms of FPE corresponding to two representations of stochastic integrals.
4. Please, use consistent notations for drift and diffusion coefficients in SDEs, FPE and the Chapman-Kolmogorov equation.
5. I suggest addition to book references: M.I. Freidlin, A.D. Wentzell, “Random perturbation of dynamical systems”, Springer.