Chapter 2 Elements of representation theory
3.2 Markov functions of Markov processes
Finding bivariate Markov processes with specified Markov marginals is related to a more general question of when a measurable function of a Markov process is again Markov. Of course in general - it isn’t. There are, however, various results detailing sufficient and necessary conditions ensuring that it is. The most important sufficient criteria are the so-calledDynkin criterionand theintertwining condition. In particular, intertwining plays an important role in the Pitman’s theorem and its extensions.
3.2.1
Dynkin criterion and the Chapman-Kolmogorov equation
LetX = (X(t);t ≥0)be a (discrete or continuous) Markov process, defined on some probability space(Ω,F,P), with a measurable state space(S,B)and an initial distri-
butionµ, i.e. µ(A) =P(X(0)∈A)for anyA∈ B. We define the transition function ofX
as usual
Pt(x,A) =P(X(s+t)∈A|X(s) =x), for allA∈ B,t>0 .
Let (S0,B0) be another measurable statespace and φ : S → S0 a measurable map takingS toS0. Under what conditions is Y := φ(X), a measurable function of X, again Markov? The question has been widely discussed in the literature and the simplest sufficient condition dates back to Dynkin[27].
Theorem 3.5. (Dynkin criterion[27, Thm. 10.13]) Let the setup be as above. Suppose that for any t>0and A∈ B0, and for any x,x0∈S such thatφ(x) =φ(x0)∈S0
Pt(x,φ−1(A)) =Pt(x0,φ−1(A)). (3.12)
Then the process Y = (Y(t);t≥0)is Markov with state space(S0,B0)and tran- sition probabilitieseP defined with respect to the measure
P(Y(t)∈A) =P(X(t)∈φ−1(A)), A∈ B0,t≥0 .
In other words, if the conditional distribution of Ys+t under P depends on Xs
only throughφ(Xs), thenY is Markov for any initial distributionµofX. In particular,
ifX is a discrete-time Markov chain, then (3.12) translates to
P(Yn= yn|Xn−1=xn−1) =P(Yn= yn|φ(Xn−1) = yn−1) for all yn∈S0, xn−1∈S andn≥1.
Note that under Dynkin condition the new process Y is not only Markov with respect to its own filtration, but also with respect to a larger filtration generated by the original processX. Three-dimensional Bessel process, viewed as the radial part of the three-dimensional Brownian motion is an example of Dynkin’s criteria. One can see directly from the form of the generator (3.3) that the marginal processR, which is a function of the bi-variate processZ(0)= (X,R)withφ(x,r) = r, is Markov. Moreover the diffusion and the drift coefficients ofRdo not depend onX and one concludes that in factRis distributed as the BES3-process.
Various papers discussed and extended Dynkin’s result. In [18] Burke and Rosenblatt proved that condition (3.12) is necessary and sufficient in the cases whenX is a stationary reversible chain with finite state space and invariant initial distribution or when X is a continuous-time Markov chain with finite state space and stationary transition probability functions continuous in time. Authors also give a necessary con- dition on the transition probability matrixPof a discrete-timeX in order forφ(X)to be Markov for any measurable transformationφ. A paper[40]by Hachigian and Rosen- blatt considers a stationary and reversible Markov chain living on a more general state space. See also[39], which extends results of[18]to a chain with a denumerable state space.
In [70] Rosenblatt relates the Markov property of the new process φ(X) to whether its first-order transition probability functions satisfy the Chapman-Kolmogorov equation. Recall that a transition probability functionP of a discrete parameter chain
satisfies the Chapman-Kolmogorv equation if
Pn+m(x,A) =X
x0
Pn(x,x0)Pm(x0,A), n,m∈N, (3.13)
or, in caseX is a continuous parameter process,
Pt+s(x,A) =
Z
x0
Pt(x,d x0)Ps(x0,A), s,t>0 (3.14)
for allx ∈S,A∈ B. Transition probability functions of a Markov process necessarily satisfy the Chapman-Kolmolgorov equations. But reverse does not have to hold - one can find a non-Markovian chain whose transition probabilities satisfy (3.13) (or (3.14)). See, for example, a note by Feller[30].
Rosenblatt derives conditions on P, the transition probability functions of X, necessary for the first-order transition probability functionsePofY to satisfy the Chapman-
Kolmogorov equation. He then proceeds to prove that these conditions are also suffi- cient forY to be Markov and in fact imply the Dynkin criteria, making them a special case of the Dynkin’s result. [40]and[39]also discuss the problem from the point of view of the Chapman-Kolmogorov equation.
See also a paper by Kelly[50], where author derives some Dynkin-like sufficient and necessary conditions for a function of a Markov chain to be Markov in the context of discrete-time Markov processes with countable state space.
3.2.2
Intertwining
Suppose nowX = (Xt;t≥0)is a continuous-parameter Markov process with transition functions(Pt;t≥0). Define
Ptf(x) =
Z
S
Pt(x,d x0)f(x0)
for any measurable, bounded function f. Again let φ be a measurable function from S toS0 andY =φ(X). LetΛbe a Markov kernel from S0 toS, that is for any y ∈S0,
function onS0. For any measurable boundedS-valued function f we will writeΛf(y) for the integralR
SΛ(y,d x)f(x). SupposeQt is a probability semigroup on(S
0,B0). We say thatPt and Qt are intertwined with respect to the kernel Λif for all y ∈S0 and A∈ B,t ≥0 Z S Λ(y,d x)Pt(x,A) = Z S0 Qt(y,d y0)Λ(y0,A), t≥0 , or more concisely ΛPt =QtΛ, t ≥0 . (3.15)
We callΛanintertwining kernel.
If X and Y are discrete time Markov chains with n-step transition functions denoted by Pn andQn respectively, then we say P andQ are intertwined with respect to some discrete Markov kernelΛif for all y ∈S0andx ∈S
X x0 Λ(y,x0)Pn(x0,x) = X y0 Qn(y,y0)Λ(y0,x), n≥1 .
Define a kernelΦ on bounded measurableS0-valued functions, denoted bB0, by
Φf = f ◦φ, f ∈bB0.
The following result is due to Pitman and Rogers
Theorem 3.6. ([68, Thm. 2]) Suppose there exists a Markov kernelΛfrom S0 to S such that
ΛΦf = f, ∀f ∈bB0,
and the semigroup defined by Qt= ΛPtΦsatisfies the intertwining relationship(3.15). Let X be a Markov process with initial lawΛ(y,·)for some y∈S0and a transition semigroup
(Pt,t≥0). Then Y :=φ(X)is a Markov process with Y0= y and a transition semigroup
P(Xt∈A|φ(Xs), 0≤s≤t) = Λ(φ(Xt),A) a.s (3.16)
for all t≥0and A∈ B.
The result naturally extends to the set-up of discrete-time Markov chains.
The n-dimensional Bessel process, for n ∈ N, viewed as the radial part, and
so a function, of then-dimensional Brownian motion, with or without drift, provides examples of intertwinings. We are in particular interested in the 3-dimensional case. LetPtµandQµt be the semigroups of the three-dimensional Brownian motion with drift of magnitude|µ| ≥0 and its radial part respectively. Then, for allt≥0[68]
ΛµPtµ=QµtΛµ ,
where, forµ >0,Λµ(r,·)is the von Mises distribution on the sphere of radius r inR3.
Whenµ=0,Λ0:= Λis the uniform distribution on the (surface of the) sphere of radius r inR3. As a consequence it is possible to deduce Theorem 3.2.
Intertwining also plays an important role in the context of Pitman’s theorem. Let Mt := sups≤tXs, where X is the drifting Brownian motion. By showing that the semigroups of the bivariate process(U,M):= (M−X,M)and the process 2M−X = U +M are intertwined, Rogers and Pitman prove Theorem 3.3. One can find more examples of intertwinings involving Brownian motions in[10],[20],[26], and[79].
In conclusion we mention the connection between the Dynkin and the inter- twining conditions. Pitman and Rogers[68]observed that if the intertwining condition holds for a pair of processes (X,Y = φ(X)), then the Dynkin condition must apply to their reverses. Kelly[50], on the other hand, pointed out that, in the context of discrete-time Markov chains, if Dynkin holds for(X,Y =φ(X)), then the intertwining condition holds for the reversed chains.
In the original paper[62]Pitman proved his famous result by first considering its discrete version. He constructed discrete chains approximating the standard Brow- nian motion and the BES3-process and satisfying the same relationship as he hoped to
prove their continuous counterparts to satisfy; taking the diffusion limit then yielded the desired result. We follow the same strategy and first identify a one-parameter family of discrete-time Markov chains with the marginals given by the simple symmetric ran- dom walk (SSRW) and the so called discrete Bessel process, dBES3(see below for defi- nition). We also construct a family of discrete bivariate chains with the marginals given by a drifting random walk and the corresponding discrete Bessel process, dBES3(µ), also defined below. In section 3.7, by applying the appropriate scaling and taking the weak limit, we arrive at the family of diffusions(Z(θ,µ),θ∈[0,∞),µ≥0)of interest.
As a motivation we start by discussing the aforementioned discrete Pitman’s theorem; we identify the result and the associated pair of discrete Markov chains by looking at the dynamics of the randomly growing Young tableaux. There is no dis- crete equivalent of the construction of the BES3 process as the radial part of the three- dimensional Brownian motion in classic probability. However, this analogue can be identified as a coupling of the so called spin process with a quantum random walk in quantum probability. Our objective is to construct a family of discrete bivariate Markov chains linking these two processes in an appropriate sense; this is done in section 3.5.