Abstract
In Chap. 2, we derived the backward and forward (Fokker–Planck) Kolmogorov equations. The Fokker–Planck equation enables us to calculate the transition probability density, which we can use to calculate the expectation value of observables of a diffusion process. In this chapter, we study various properties of this equation such as existence and uniqueness of solutions, long-time asymptotics, boundary conditions, and spectral properties of the Fokker–Planck operator. We also study in some detail various examples of diffusion processes and of the associated Fokker–Planck equation. We will restrict attention to time-homogeneous diffusion processes, for which the drift and diffusion coefficients do not depend on time.
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Notes
- 1.
In this chapter we will call this equation the Fokker–Planck equation, which is more customary in the physics literature, rather than the forward Kolmogorov equation, which is more customary in the mathematics literature.
- 2.
Of course, a random walk is not a diffusion process. However, as we have already seen, Brownian motion can be defined as the limit of an appropriately rescaled random walk. A similar construction exists for more general diffusion processes.
- 3.
In fact, we prove only that \(\mathcal{L}\) is symmetric. An additional argument is needed to prove that it is self-adjoint. See the comments in Sect. 4.10.
- 4.
The transformation of the generator of a diffusion process to a Schrödinger operator is discussed in detail in Sect. 4.9.
- 5.
These eigenfunctions are the Hermite functions.
- 6.
Of course, the function space \(L^{2}(\rho _{\beta })\) in which we study the eigenvalue problem for \(\mathcal{L}\) does depend on β through ρ β .
- 7.
A complete proof of this result would require a more careful study of the domain of definition of the generator.
- 8.
Note that we can incorporate the normalization constant in the definition of Φ. Alternatively, we can write \(\rho = \frac{1} {Z}e^{-\varPhi }\), \(Z =\int e^{-\varPhi }\,dx\). See, for example, the formula for the stationary distribution of a one-dimensional diffusion process with reflecting boundary conditions, Eq. (4.35). We can write it in the form
$$\displaystyle{p_{s}(x) = \frac{1} {Z}e^{-\varPhi }\quad \mbox{ with}\quad \varPhi =\log (\sigma (x)) -\left (2\int _{\ell}^{x}\frac{b(y)} {\sigma (y)} \,dy\right ).}$$ - 9.
The calculation of the normalization constant requires the calculation of an integral (the partition function) in a high-dimensional space, which might be computationally very expensive.
- 10.
In the statistics literature, this is usually called the Langevin dynamics. We will use this term for the second-order stochastic differential equation that is obtained after adding dissipation and noise to a Hamiltonian system; see Chap. 6. Using the terminology that we will introduce there, the dynamics (4.116) correspond to the overdamped Langevin dynamics.
- 11.
When implementing the MCMC algorithm, we need to discretize the stochastic differential equation and take the discretization and Monte Carlo errors into account. See Sect. 5.2.
- 12.
Two operators A 1, A 2 defined in two Hilbert spaces H 1, H 2 with inner products \(\langle \cdot,\cdot \rangle _{H_{1}},\,\langle \cdot,\cdot \rangle _{H_{2}}\), respectively, are called unitarily equivalent if there exists a unitary transformation U: H 1 ↦ H 2 (i.e., \(\langle Uf,Uh\rangle _{H_{2}} =\langle f,h\rangle _{H_{1}},\;\forall f,\,h \in H_{1}\)) such that
$$\displaystyle{UA_{1}U^{-1} = A_{ 2}.}$$When the operators A 1, A 2 are unbounded, we need to be more careful with their domain of definition.
- 13.
In fact, the Poincaré inequality for Gaussian measures can be proved using the fact that the Hermite polynomials form an orthonormal basis for \(L^{2}(\mathbb{R}^{d},\rho _{\beta })\).
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Pavliotis, G.A. (2014). The Fokker–Planck Equation. 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_4
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