Abstract
Stochastic processes are defined and grouped into different classes, their basic properties are listed and compared. The Chapman–Kolmogorov equation is introduced, transformed into a differential version, and used to classify the three major types of processes: (i) drift and (ii) diffusion with continuous sample paths, and (iii) jump processes which are essentially discontinuous. In pure form these prototypes are described by Liouville equations, stochastic diffusion equations, and master equations, respectively. The most popular and most frequently used continuous equation is the Fokker–Planck (FP) equation, which describes the evolution of a probability density by drift and diffusion. The pendant to FP equations on the discontinuous side are master equations which deal only with jump processes and represent the appropriate tool for modeling processes described by discrete variables. For technical reasons they are often difficult to handle unless population sizes are relatively small. Particular emphasis is laid on modeling conventional and anomalous diffusion processes. Stochastic differential equations (SDEs) model processes at the level of random variables by solving ordinary differential equations upon which a diffusion process, called a Wiener process, is superimposed. Ensembles of individual trajectories of SDEs are equivalent to time dependent probability densities described by Fokker–Planck equations.
With four parameters I can fit an elephant
and with five I can make him wiggle his trunk.
Enrico Fermi quoting John von Neumann 1953 [119].
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- 1.
Identical conditions means that all parameters are the same except for the random fluctuations. In computer simulations this is achieved by keeping everything precisely the same except the seeds for the pseudorandom number generator.
- 2.
By general we mean here methods that are widely applicable and not tailored specifically for deriving stochastic solutions for a single case or a small number of cases.
- 3.
The Russian mathematician Andrey Markov (1856–1922) was one of the founders of Russian probability theory and pioneered the concept of memory-free processes, which are named after him. He expressed more precisely the assumptions that were made by Albert Einstein [133] and Marian von Smoluchowski [559] in their derivation of the diffusion process.
- 4.
For the moment we need not specify whether \(\mathcal{X}(t)\) is a simple random variable or a random vector \(\boldsymbol{\mathcal{X}}(t) ={\bigl ( \mathcal{X}_{k}(t); k = 1,\mathop{\ldots },M\bigr )}\), so we drop the index k determining the individual component. Later on, for example in chemical kinetics where the distinction between different (chemical) species becomes necessary, we shall make clear the sense in which \(\mathcal{X}(t)\) is used, i.e., random variable or random vector.
- 5.
Here we shall use the notion of phase space in a loose way to mean an abstract space that is sufficient for the characterization of the system and for the description of its temporal development. For example, in a reaction involving n chemical species, the phase space will be a Cartesian space spanned by n axes for n independent concentrations. In classical mechanics and in statistical mechanics, the phase space is precisely defined as a—usually Cartesian—space spanned by the 3n spatial coordinates and the 3n coordinates of the linear momenta of an n-particle system.
- 6.
- 7.
In order to leave the subscript free to indicate discrete times or different chemical species, we use the somewhat clumsy superscript notation \(\mathcal{X}^{(i)}\) or x (i) (\(i = 1,\mathop{\ldots },N\)), to specify individual trajectories, and we use the physical numbering of times t 0 → t n .
- 8.
- 9.
For convenience we change the numbering of times here and apply the notation of (3.3 ′).
- 10.
The term càdlàg is an acronym from French which stands for continue à droite, limites à gauche. The English expression is right continuous with left limits (RCLL). It is a common property of step functions in probability theory (Sect. 1.6.2). We shall reconsider the càdlàg property in the context of sampling trajectories (Sect. 4.2.1).
- 11.
The Russian mathematician Andrey Markov (1856–1922) was one of the founders of Russian probability theory and pioneered the concept of memory-free processes which is named after him. Among other contributions he expressed the assumptions that were made by Albert Einstein [133] and Marian von Smoluchowski [559] in their derivation of the diffusion process in more precise terms.
- 12.
The notion of autocovariance reflects the fact that the process is correlated with itself at another time, while cross-covariance implies the correlation of two different processes (for the relation between autocorrelation and autocovariance, see Sect. 3.1.6).
- 13.
The notation used for time dependent variables is explained in Fig. 3.4. For convenience and readability, we write x for \(z + \Delta z\).
- 14.
Later on we shall discuss the limit of the random walk for vanishing step size in more detail and call it a Wiener process (Sect. 3.2.2.2).
- 15.
- 16.
Here, we need not yet specify whether the sample space is discrete as in P n (t), or continuous as in P(x, t), and we indicate this by the notation P(n, t). However, we shall specify the variables in Sect. 3.2.1.
- 17.
The derivation is already contained in the first edition of Gardiner’s Handbook of Stochastic Methods [193], and it was Crispin Gardiner who coined the term differential Chapman–Kolmogorov equation.
- 18.
It is important to note a useful trick in the derivation: by substituting the 1, the time order is reversed in the integral.
- 19.
The notation ∥ ⋅ ∥ refers to a suitable vector norm, here the L 1 norm given by ∥ y ∥ = ∑ k | y k | . In the one-dimensional case, we would just use the absolute value | y | .
- 20.
Differentiation with respect to x has to be done with respect to the components x i . Note that u vanishes through integration.
- 21.
A positive definite matrix has exclusively positive eigenvalues λ k > 0, whereas a positive semidefinite matrix has nonnegative eigenvalues λ k ≥ 0.
- 22.
The idea of the Liouville equation was first discussed by Josiah Willard Gibbs [202].
- 23.
Phase space is an abstract space, which is particularly useful for visualizing particle motion. The six independent coordinates of particle S k are the position coordinates q k = (q k1, q k2, q k3) and the (linear) momentum coordinates p k = (p k1, p k2, p k3). In Cartesian coordinates, they are q k = (x k , y k , z k ) and p k = m k v k , where v = (v x , v y , v z ) is the velocity vector.
- 24.
For simplicity, we write p(x, t) instead of the conditional probability p(x, t | x 0, t 0) whenever the initial condition (x 0, t 0) refers to the sharp density p(x, t 0) = δ(x −x 0).
- 25.
We distinguish the two formally identical equations (3.55) and (3.56), because the interpretation is different: the former describes the evolution of a probability distribution with the conservation relation ∫dw p(w, t) = 1, whereas the latter deals with a concentration profile, which satisfies \(\int \mathrm{d}x\,c(x,t) = c_{\mathrm{tot}}\) corresponding to mass conservation. In the case of the heat equation, the conserved quantity is total heat. It is worth considering dimensions here. The coefficient 1∕2 in (3.55) has the dimensions [t−1] of a reciprocal time, while the diffusion coefficient has dimensions [l2t−1], and the commonly used unit is [cm2/s].
- 26.
Integration by parts is a standard integration method in calculus. It is encapsulated in the formula
$$\displaystyle{ \int _{a}^{b}u(x)v'(x)\,\mathrm{d}x = u(x)v(x)\Big\vert _{a}^{b} -\int _{a}^{b}u'(x)v(x)\,\mathrm{d}x\phantom{0}. }$$Characteristic functions are especially well suited to partial integration, because exponential functions v(x) = exp(isx) can be easily integrated, and probability densities u(x) = p(x, t) as well as their first derivatives u(x) = ∂ p(x, t)∕∂ x vanish in the limits x → ±∞.
- 27.
- 28.
For a system in 3D space, the wave vector in reciprocal space is denoted by k, and its length | k | = k is called the wave number.
- 29.
An autoregressive process of order n is denoted by AR(n). The order n implies that n values of the stochastic variables at previous times are required to calculate the current value. An extension of the autoregressive model is the autoregressive moving average (ARMA) model.
- 30.
The variance of the Wiener process diverges, i.e., \(\lim _{t\rightarrow \infty }\mathrm{var}{\bigl (\mathcal{W}(t)\bigr )} = \infty\). The same is true for the Poisson process and the random walk, which are discussed in the next two sections.
- 31.
From here on, unless otherwise stated, we shall consider cases in which the limits lim | x−z | → 0 W(x | z, t) and lim | x−z | → 0 W(z | x, t) of the transition probabilities are finite and the principal value integral can be replaced by a conventional integral. Riemann–Stieltjes integration converts the integral into a sum, and since we are dealing exclusively with discrete events, we use an index on the probability P n (t).
- 32.
The notation δ ij denotes the Kronecker delta , named after the German mathematician Leopold Kronecker , which means
$$\displaystyle{\delta _{ij} = \left \{\begin{array}{@{}l@{\quad }l@{}} 1\,,\ \ \quad &\mathrm{if}\ \ i = j\,,\\ 0\,,\ \ \quad &\mathrm{if }\ \ i\neq j\,. \end{array} \right.}$$It is the discrete analogue of the Dirac delta function.
- 33.
In the literature both expressions, waiting time and arrival time, are common. An inter-arrival time is a waiting time.
- 34.
The litter size is defined as the mean number of offspring produced by an animal in a single birth.
- 35.
Exceptions with only one transition are the lowest and the highest state, n = n min and n = n max, which are the boundaries of the system. In biology, the notation w n + ≡ λ n and w n − ≡ μ n for death and birth rates is common.
- 36.
An excellent tutorial on this subject by Bahram Houchmandzadeh can be found at http://www.houchmandzadeh.net/cours/Master_Eq/master.pdf. Retrieved 2 May 2014.
- 37.
In general these equations hold also for summations from 0 to + ∞ if the corresponding physically meaningless probabilities are set equal to zero by definition: \(P_{n}(t) = 0\,,\;\forall \,n \in \mathbb{Z}_{<0}\).
- 38.
The probability of extinction from state Σ i is the probability of proceeding one step down multiplied by the probability of extinction from state Σ i−1 plus the probability of going one step up times the probability of becoming extinct from Σ i+1.
- 39.
- 40.
The most straightforward way to take the limit is to introduce a scaling assumption , using a variable γ such that Δ x = γ Δ x 0 and \(\Delta t =\gamma ^{2}\Delta t_{0}\). Then we have \(\varDelta \,x^{2}/2\Delta t =\varDelta \, x_{0}^{2}/2\Delta t_{0} = D\) and the limit γ → 0 is trivial.
- 41.
If the jump lengths and waiting times were coupled, we would have to deal with φ(ξ, τ) = φ(ξ | τ)ψ(τ) = φ(τ | ξ)f(ξ). Coupling between space and time could arise, for example, from the fact that it is impossible to jump a certain distance within a time span shorter than some minimum time.
- 42.
As in the previous examples, we assume that the random walk is symmetric and started at the origin. Then the expectation value of the location of the particle stays at the origin and we have \(\left <\xi \right> = 0\), \(\left <\xi \right>^{2} = 0\), and hence \(\mathrm{var}(\xi ) = \left <\xi ^{2}\right>\).
- 43.
For Lévy processes in general it will be necessary to replace the integral by a principal value integral because they may lead to a singularity at the origin, i.e., lim z → x w(x − z) = ∞, and this prohibits conventional integration.
- 44.
The discreteness of u would require here a Stieltjes integral with dW(u) but the trick with a Dirac delta function allows one to use the differential expression w(u)du.
- 45.
The Laurent series is an extension of the Taylor series to negative powers of (z − z 0), named in honor of the French mathematician Pierre Alphonse Laurent .
- 46.
Although the value α = 2 leads to divergence in the regular derivation, applying α = 2, β = 0, and γ = 1∕2 yields the probability density of the normal diffusion process.
- 47.
We use the relations lim α → 1 Γ(−α) = ±∞, but lim α → 1 Γ(−α)cos(π α∕2) = −π∕2, which are easy to check.
- 48.
The absolute value of the wave number | k | is sometimes used in all expressions, which is necessary when complex k values are admitted, or in the multidimensional case where k is the wave vector. Here we use real k values in one dimension and we need | k | only to express a cusp at k = 0.
- 49.
Because of this similarity we called the α = 2 Pareto process a Lev́y walk.
- 50.
In order to avoid confusion we shall reserve the variable y(τ) and y(0) = y 0 for backward computation.
- 51.
Boundaries are also called barriers in the literature and the notions are taken to be synonymous. We shall use here exclusively the word boundary. The term barrier will be reserved for obstacles to motion inside the domain of the random variable.
- 52.
We remark that the relation between the diffusion functions in the Fokker–Planck equation and the SDE, B(x, t) and b(x, t), is more subtle, and involves a square root according to (3.42), as will be discussed later on here in this section.
- 53.
In a telescopic sum, all terms except the first and the last summand cancel.
- 54.
To derive this relation, we use the fact that the stochastic variables of the Wiener process at different times are uncorrelated, i.e., \(\left <\mathcal{W}(t_{i})\mathcal{W}(t_{j})\right> = 0\), and the variance is \(\mathrm{var}{\bigl (\mathcal{W}(t_{i})\bigr )} = \left <\mathcal{W}(t_{i})^{2}\right> -\left <\mathcal{W}(t_{i})\right>^{2} = \left <\mathcal{W}(t_{i})^{2}\right> - (t_{i} - t_{0})^{2}\). We simplify the expressions in the derivation and write from here on \(\mathcal{W}_{i}\) for \(\mathcal{W}(t_{i})\) and \(\Delta \mathcal{W}_{i}\) for \(\Delta \mathcal{W}(t_{i})\).
- 55.
Every deterministic process is nonanticipating: in order to calculate the value G(t + dt) of a function t → G(t), no value G(τ) with τ > t is required.
- 56.
A function assigns a value to the argument of the function, x 0 → f(x 0), whereas a functional relates a function to the value of a function, f → f(x 0).
- 57.
In order to distinguish the two versions of stochastic integrals we use the symbol \(\int _{t_{0}}^{t}\) for the Itō integral and \(\mbox{ s}\!\!\!\!\int _{t_{0}}^{t}\) for the Stratonovich integral [277, p. 86]. The distinction from ordinary integrals is automatically provided by the differential \(\mathrm{d}\mathcal{W}(t)\).
- 58.
We use here the sanserif font for the diffusion matrix in order to distinguish it from the conventional diffusion matrix B = B t B in the Fokker–Planck equation.
- 59.
In order to prove the conjecture, one makes use of the fact that all cumulants κ n with n > 2 vanish (see Sect. 2.3.3). The reader is encouraged to complete the proof.
- 60.
A Poissonian distribution depends on a single parameter \({\bigl (\alpha (\tau ),\tau \bigr )}\equiv \alpha (\tau )\tau\). For simplicity, we shall write α instead of α(τ) from now on.
- 61.
We use here α{(n(t))} for the propensity function in order to show the relationship with a Poisson process. In chemical kinetics (Chap. 4), it will be denoted by χ{(n(t))}.
- 62.
Under certain rather rare circumstances, modeling reactions with time dependent reaction rate parameters may be advantageous.
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Schuster, P. (2016). Stochastic Processes. In: Stochasticity in Processes. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-39502-9_3
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