The derivative of a stochastic process is defined in a formal way:
Definition: A process ξ(t) is mean-square differentiable at t if, and only if, there exists a random variable, denoted by ξ0(t), such that the limit
lim h→0 * ξ(t+h)−ξ(t) h −ξ 0(t) 2+ = 0, (7.7) exists.
The definition, as usual, is difficult to use in practice. As a more useful corollary, one can show that Eq. 7.7 is satisfied if, and only if, the autocorrelation function B(t1, t2) = hξ(t1)ξ(t2)i is differentiable at the
pointt=t1=t2,i.e. the limit
lim h1, h2→0 1 h1h2 [B(t−h1, t−h2)−B(t, t−h2)−B(t−h1, t) +B(t, t)], (7.8) must exist. If the process ξ(t) is wide-sense stationary, then B(t, t−
τ) =B(τ) is a function of the time-difference only and the condition for differentiability is simplified: The limit
lim
h→0
1
h2 [B(h)−2B(0) +B(−h)], (7.9)
must exist. Moreover, the autocorrelation function of the derivativeξ0(t) is given by,
Bξ0(t)(t1, t2) =
∂2
∂t1∂t2
B(t1, t2),
or, for a stationary process,
Bξ0(t)(τ) =−
d2
dτ2B(τ).
With these conditions in hand, it is straightforward to show that the Wiener process and the Ornstein-Uhlenbeck processare not differentiable
(precisely Doob’s objection), but we shall also show that this does not matter in the least for the modeling of physical systems.
7.3.1
Wiener process is
not
differentiable
Recall from Chapter 6 (cf. Eq. 6.31 on page 134) that in the absence of a damping force, the Langevin equation reduces to the equation charac- terizing the Wiener processW(t),
dW
dt =η(t).
We will now show that W(t) is not differentiable – and so the equation above, as it stands, is meaningless. In Chapter 6 (Eq. 6.32 on page 135), we derived the autocorrelation function for the Wiener process,
hW(t1)W(t2)i= min(t1, t2).
Clearly, the autocorrelation function indicates that the process W(t) is
nonstationary, so we must use Eq. 7.8 to check its differentiability: lim h1, h2→0 1 h1h2 [min(t−h1, t−h2)−(t−h2)−(t−h1) +t] = lim h1, h2→0 1 h1h2 [min(t−h1, t−h2) +h2+h1−t].
The limit does not exist, and we must conclude that the Wiener process isnot differentiable (although one can easily show that it is mean-square
continuous).
The Ornstein-Uhlenbeck process is stationary at steady-state, and the autocorrelation function,
B(τ) = Γ 2τc
e−|τ|/τc
is a function of the time-difference τ – We can therefore use Eq. 7.9 to check its differentiability. One finds that the Ornstein-Uhlenbeck process is likewisenot differentiable (Excercise 3a). That was Doob’s point in his criticism of the work of Ornstein and Uhlenbeck (quoted on page 20).
BUT (and this is critically important)
We shall now show that if the forcing function F(t) is not strictly delta- correlated, if the process has a non-zero correlation time (however small), then the differential equation,
dy
is well-defined and y(t) is differentiable. For sake of example, suppose F(t) is an Ornstein-Uhlenbeck process. As shown above, the steady-state correlation function for the Ornstein-Uhlenbeck process is the exponen- tial,
BF(τ) =
Γ 2τc
e−|τ|/τc,
where we have written the correlation time explicitly, and made the pre- factor proportional to 1/τc to clarify the correspondence between the
present example and the Wiener process studied above. Notice, in the limit of vanishing correlation time,
lim
τc→0 1 2τc
e−|τ|/τc→δ(τ).
From Chapter 2 (Eq. 2.19 on page 42), the spectral density of the processy(t) (as characterized by Eq. 7.10) is simply,
Syy(ω) = Γ ω2+γ2 (1/τc)2 ω2+ (1/τ c)2 , (7.11)
from which the autocorrelation function follows,
By(τ) = Γ 2γ e−γ|τ|−γτ ce−|τ|/τc 1−γ2τ2 c . (7.12)
Taking the limit (cf. Eq. 7.9), lim h→0 By(h)−2By(0) +By(−h) h2 =− 1 2τc Γ 1 +γτc . (7.13)
The limit exists, so we conclude that the processy(t) defined by the dif- ferential equation (7.10) is differentiable. Obviously, as τc →0, the limit
above becomes undefined, butfor any non-zero correlation time, however small, the derivative of y(t)is well-defined and can be manipulated using the rules of ordinary calculus. Incidently, Ornstein and Uhlenbeck never explicitly state that their forcing function is delta-correlated, merely that it is very narrow (G. E. Uhlenbeck and L. S. Ornstein (1930) Physical Review 36: 823–841):
[W]e will naturally make the following assumptions. . . There will be correlation between the values of [the random forcing
functionA(t)] at different timest1 andt2 only when|t1−t2|
is very small. More explicitly we shall suppose that:
hA(t1)A(t2)i=φ(t1−t2),
whereφ(x) is a function with a very sharp maximum atx= 0. As shown above, under these assumptions the derivation of Ornstein and Uhlenbeck is perfectly correct and no questions of inconsistency arise. This led Wang and Uhlenbeck to write as a footnote in a later publication (M. C. Wang and G. E. Uhlenbeck (1945)Reviews of Modern Physics17: 323–342),
The authors are aware of the fact that in the mathematical literature (especially in papers by N. Wiener, J. L. Doob, and others; cf. for instance Doob, Ann. Math. 43, 351 (1942), also for further references) the notion of a random (or stochas- tic) process has been defined in a much more refined way. This allows for instance to determine in certain cases the probabil- ity that a random function y(t) is of bounded variation, or continuous or differentiable, etc. However, it seems to us that these investigations have not helped in the solution of prob- lems of direct physical interest, and we will, therefore, not try to give an account of them.