Thinking along sample paths and path integrals

Richard Feynman's paper on the formulation of quantum mechanics by thinking along sample paths (Feynman 1948) is still worth reading, even if one does not know a lot about quantum mechanics. From this idea (and Kac's use in the solution of the standard diffusion equation) the notion of path integrals developed, and there are lots of instances of the application of path integrals today (two of my favorites are Schulman (1981) and Friedlin and Wentzell (1984)). Formore about Feynman and Kac, see Gleick (1992) and Kac's autobiography (Kac 1985).

Independent increments, Ito and Stratonovich

Our treatment of Brownian motion has been to look from time t to the end of the interval t + dt. This is called the Ito calculus, named after the great Japanese mathematician Kiyosi Ito (whose daughter happens to be a faculty member in Department of Linguistics at UCSC). Ito's work has recently been reviewed, in very mathematical form, by Stroock (2003). This is a hard book to read, but the preface is something that everyone can understand. If we think about not the end of the interval, but imagine that the process is truly continuous, then there are changes throughout the interval. This view is called Stratonovich calculus and the effect is to change the form of the diffusion coefficient (see, for example Stratonovich (1963), Wong and Zakai (1965), Wong (1971), and van Kampen (1981a, b)). Engineers sometimes call Stratonovich calculus the Wong-Zakai correction. Karlin and Taylor (1981) give a readable introduction to the different stochastic calculi. Hakoyama et a/. (2000) consider a problem in risk of the extinction of populations in which the environmental fluctuations obey Stratonovich calculus and the demographic fluctuations obey Ito calculus; see also Hakoyama and Iwasa (2000).

Fluctuation and dissipation

One of Einstein's great contributions in his 1905 paper was to show how fluctuation could be connected to the Maxwell-Boltzmann distribution; this result has connections to nineteenth-century mechanistic materialism and the general phenomenon of diffusion (Wheatley and Agutter 1996). The crowning achievement in this area belongs to Uhlenbeck and Ornstein (1930) who showed how to fully connect fluctuation and dissipation (the resistance or friction term in the Ornstein-Uhlenbeck process). Uhlenbeck was a particularly interesting person, who had an enormous effect on twentieth-century physics. When my UC Davis colleague Joel Keizer began his development of non-equilibrium thermodynamics (summarized in Keizer (1987)), Uhlenbeck acted as referee and it took Kac as interpreter of Joel's ideas to convince Uhlenbeck of their validity.

Red, white, and blue noise

There are examples of biological systems with noise that has a spectrum which is far from white (Cohen 1995, White et al. 1996, Vasseur and Yodzis 2004). For example, slowly varying environments (as in the North Pacific ocean; see Hare and Francis (1995), Mantua et al. (1997)) will have a spectrum that is very red, so that low frequencies are represented more strongly. Some diseases exhibit high frequency fluctuations, so that their spectra are bluer. These can be generated, in discrete time, from a model of the form Y(t + 1)= aY(t) + V1 - a2Z(t + 1), where Y(t) is the environmental noise at time t, a is a parameter with range — 1 < a < 1 and Z(t) is a normally distributed random variable. When a is positive, low frequency components dominate (the spectrum is red) and when a is negative the high frequency components dominate. Furthermore, because biological responses are generally nonlinear, they can filter the environmental noise (for examples, see Petchey et al. (1997), Petchey (2000) or Laakso et al. (2003)).

Stochastic differential equations and stochastic integrals

There is an enormous literature on stochastic differential equation and stochastic integrals. The mathematical levels range from pretty applied, as here, to highly abstract and theoretical. Two older but solid introductions to the material are Arnold (1973) and Gardiner (1983). Another good starting point is Karlin and Taylor (1981), who have a 240 page chapter on diffusion processes. A general discussion of numerical methods for stochastic differential equations is found in Higham (2001). Exact numerical methods for the Ornstein-Uhlenbeck process and its integral are discussed by Gillespie (1996).

Applications in ecology

As we discussed in Chapter 2, diffusion processes arise in a natural way in the study of organismal movement and dispersal (Turchin 1998) and the various connections given there can now take on deeper meaning.

For example, if we were to let (X, Y) denote the position of an animal, we could now write stochastic differential equations to characterize the increments in Xand Y. Stochastic differential equations can also be used to describe the dynamics of populations (see, for example, Nisbet and Gurney (1982), Engen et a/. (2002), Lande et a/. (2003), and Saether and Engen (2004)). Costantino and Desharnais (1991) use diffusion models to characterize the population dynamics of flour beetles. If N(t) denotes adult numbers at time t, they work with models of the form dN = N(t)[be— cN(t) — ^]dt + oN(t)dw where the parameters have a natural interpretation. In this case, the stationary distribution of population size is a gamma density (also see Peters et a/. (1989)), thus connecting us to material in Chapter 3.

Applications in population genetics

Diffusion processes underlie an entire approach to population genetics, with many entrance points to the literature. Some of my favorites are Crow and Kimura (1970), Kimura and Ohta (1971), and Gillespie (1991, 1998). For applications in population genetics, we usually work with the forward equation to specify the evolution of a gene frequency from an initial starting distribution of the frequency, or with the backward equation to describe the time until fixation of an allele.

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