Abstract
When the white noise in a stochastic differential equation is approximated by a smoother process, then in the limit as we remove the regularization, we obtain the Stratonovich stochastic equation. This is usually called the Wong–Zakai theorem. In this section, we derive the limiting Stratonovich SDE for a particular class of regularization of the white noise process using singular perturbation theory for Markov processes. In particular, we consider colored noise, which we model as a Gaussian stationary diffusion process, i.e., the Ornstein–Uhlenbeck process.
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Notes
- 1.
- 2.
We should also add a function ψ 1(x, t), which belongs to the null space of \(\mathcal{L}_{0}\). However, as for the one-dimensional problem, we can check that this function does not affect the limiting backward Kolmogorov equation.
- 3.
For example, the Taylor expansion, which is the main tool for obtaining higher-order numerical methods for ODEs, has to be replaced by the stochastic Taylor expansion, which is based on Itô’s formula.
- 4.
Maximizing the likelihood function is, of course, equivalent to maximizing the log likelihood function.
- 5.
Note, however, that this trajectory is not stationary, since the initial conditions are not distributed according to the invariant distribution \(\frac{1} {Z}e^{-\beta V (x)}\). We could, in principle, sample from this distribution using the MCMC methodology that was mentioned in Sect. 4.10.
- 6.
We have that in law,
$$\displaystyle{ \int _{0}^{T}f(s)\,dW(s) = W\left (\int _{ 0}^{T}f^{2}(s)\,ds\right ). }$$(5.80)Generalizations of this formula are discussed in Sect. 5.5.
- 7.
This means, of course, that we are not really looking at the dynamics but only at the dependence of the stationary state on the bifurcation parameter.
- 8.
Alternatively, we can show that U(x) = x 2 is a Lyapunov function; see Exercise 15.
- 9.
- 10.
In bounded domains. For evolution PDEs in unbounded domains, the amplitude equation is a also a PDE, the Ginzburg–Landau equation. See [39] for details.
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Pavliotis, G.A. (2014). Modeling with Stochastic Differential Equations. In: Stochastic Processes and Applications. Texts in Applied Mathematics, vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1323-7_5
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