Duality for Stochastic Processes

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In the past couple of weeks, we’ve launched a new junior probability seminar in Oxford. (If you’re reading this and would like to come, please drop me an email!) The first session featured Daniel Straulino talking about his work on the Spatial Lambda Fleming-Viot process, a model for the evolution of populations allowing for selection and geometry. A lively discussion of duality broke out halfway through, following which it seemed reasonable to devote another session to this topic. Luke Miller presented some classical and less classical examples of the theory this afternoon. I found all of this interesting and relevant, and thought it would be worth writing some things down, and tying it in with one of the courses on this subject that we attended at ALEA in Luminy last October.

The majority of this article is based on Luke’s talk. Errors, omissions and over-simplifications are of course my own.

The setup is that we have two stochastic processes X_t\subset R, Y_t\subset S. For now we make no assertion about whether the two state spaces R and S are the same or related, and we make no comment on the dependence relationship between X and Y. Let P_x,Q_y be the respective probability measures, representing starting from x and y respectively. Then given a bivariate, measurable function h(.,.) on R x S, such that:

E^P_x h(X_t,y)=E^Q_y h(x,Y_t),\quad \forall x,y\quad\forall t,

then we say X and Y are dual with respect to h.

The interpretation should be that X is a process forwards in time, and Y is a process backwards in time. So X_t, Y_0 represent the present, while X_0, Y_t represent the past, which is the initial time for original process X. The fact that the result holds for all times t allows us to carry the equality through a derivative, to obtain an equality of generators:

\mathcal{G}^X h(x,y)=\mathcal{G}^Y h(x,y),\quad \forall x,y.

On the LHS, the generator acts on x, while on the RHS it acts on y. Although it still isn’t obvious (at least to me) when a pair of processes might have this property, especially for an arbitrary function, this seems the more natural definition to think about.

Note that this does indeed require a specific function h. There were murmurings in our meeting about the possibility of a two processes having a strong duality property, where this held for all h in some broad class of test functions. On more reflection, which may nonetheless be completely wrong, this seems unlikely to happen very often, except in some obviously degenerate cases, such as h constant. If this holds, then as the set of expectations of a class of functions of a random variable determines the distribution, we find that the instantaneous behaviour of Y is equal in distribution to the instantaneous behaviour of X when started from fixed (x,y). It seems unlikely that you might get many examples of this that are not deterministic or independent (eg two Brownian motions, or other space-, time-homogeneous Markov process).

Anyway, a canonical example of this is the Wright-Fisher diffusion, which provides a simple model for a population which evolves in discrete-time generations. We assume that there are two types in the population: {A,a} seems to be the standard notation. Children choose their type randomly from the distribution of types in the previous generation. In other words, if there are N individuals at all times, and X_k is the number of type A individuals, then:

X_{k+1} | X_k \stackrel{d}{=} \mathrm{Bin}(N, \frac{X_k}{N}).

It is not hard to see that in a diffusion limit as the number of individuals tends to infinity, the proportion of type A individuals is a martingale, and so the generator for this process will not depend on f’. In fact by checking a Taylor series, we can show that:

\mathcal{G}_{WF}f(x)=\frac{1}{2} x(1-x)f''(x),

for all f subject to mild regularity conditions. In particular, we can show that for f_n(x)=x^n, we have:

\mathcal{G}_{WF} f_n(x)=\binom{n}{2}(f_{n-1}(x)-f_n(x))

after some rearranging. This looks like the generator of a jump process, indeed a jump process where all the increments are -1. This suggests there might be a coalescent as the dual process, and indeed it turns out that Kingman’s coalescent, where any pair of blocks coalesce at uniform rate, is the dual. We have the relation in expectation:

\mathbb{E}_x[X_t^n]= \mathbb{E}_n[x^{N_t}],

where the latter term is the moment generating function of the number of blocks at time t of Kingman’s coalescent started from n blocks.

In particular, we can control the moments of the Wright-Fisher diffusion using the mgf of the Kingman’s coalescent, which might well be easier to work with.

That’s all very elegant, but we were talking about why this might be useful in a broader setting. In the context of this question, there seems to be an obstacle towards applying this idea above more generally. This is an underlying idea in population genetics models that as well as the forward process, there is also a so-called ancestral process looking backwards in time, detailing how time t individuals are related historically. It would be convenient if this process, which we might expect to be some nice coalescent, was the dual of the forward process.

But this seems to be a problem, as duals are a function of the duality function, so do we have uniqueness? It would not be very satisfying if there were several coalescents processes that could all be the dual of the forward process. Though some worked examples suggest this might not happen, because a dual and its duality function has to satisfy too many constaints, there seems no a priori reason why not. It seems that the strength of the results you can derive from the duality relation is only as strong as the duality relation itself. This is not necessarily a problem from the point of view of applications, so long as the duality function is something it might actually be useful to work with.

It’s getting late and this text is getting long, so I shall postpone a discussion of duality for interacting particle systems until tomorrow. In summary, by restricting to a finite state space, we can allow ourselves the option of a more algebraic approach, from which some direct uses of duality can be read off. I will also mention a non-technical but I feel helpful way to view many examples of duality in interacting particle systems as deterministic forward and backwards walks through a random environment, in what might be considering an extreme example of coupling.

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