MARKOV PROCESSES WITH IDENTICAL BRIDGES

: Let X and Y be time-homogeneous Markov processes with common state space E , and assume that the transition kernels of X and Y admit densities with respect to suitable reference measures. We show that if there is a time t > 0 such that, for each x 2 E , the conditional distribution of ( X s ) 0 (cid:20) s (cid:20) t , given X 0 = x = X t , coincides with the conditional distribution of ( Y s ) 0 (cid:20) s (cid:20) t , given Y 0 = x = Y t , then the in(cid:12)nitesimal generators of X and Y are related by L Y f = − 1 L X ( f ) − (cid:21)f , where is an eigenfunction of L X with eigenvalue (cid:21) 2 R . Under an additional continuity hypothesis, the same conclusion obtains assuming merely that X and Y share a \bridge" law for one triple ( x; t; y ). Our work extends and clari(cid:12)es a recent result of I. Benjamini and S. Lee.


Introduction
Let X = (X t , P x ) and Y = (Y t , Q x ) be non-explosive regular Markov diffusion processes in R. Let P x,y t denote the conditional law of (X s ) 0≤s≤t given X 0 = x, X t = y. Let Q x,y t denote the analogous "bridge" law for Y . Recently, Benjamini & Lee [BL97] proved the following result.
(1.1) Theorem. Suppose that X is standard Brownian motion and that Y is a weak solution of the stochastic differential equation where B is standard Brownian motion and the drift µ is bounded and twice continuously differentiable. If for all x ∈ R and all t > 0, then either (i) µ(x) ≡ k or (ii) µ(x) = k tanh(kx + c), for some real constants k and c.
Our aim in this paper is to generalize this theorem in two ways.
Firstly, we allow X and Y to be general strong Markov processes with values in an abstract state space E. We require that X and Y have dual processes with respect to suitable reference measures, and that X and Y admit transition densities with respect to these reference measures. where λ := k 2 /2. Thus, Theorem (1.1) can be stated as follows: If X is Brownian motion and if Y is "Brownian motion with drift µ," then X and Y have common bridge laws if and only if µ is the logarithmic derivative of a strictly positive eigenfunction of the local infinitesimal generator of X, in which case the laws of X and Y are related by Theorem (1.1) and our extensions of it depend crucially on the existence of a "reference" measure dominating the transition probabilities of X and Y . This fact is amply demonstrated by the work of H.
Föllmer in [F90]. Let E be the Banach space of continuous maps of [0, 1] into R that vanish at 0, and let m denote Wiener measure on the Borel subsets of E. Let X = (X t , P x ) be the associated Brownian motion in E; that is, the E-valued diffusion with transition semigroup given by This semigroup admits no reference measure; indeed P t (x, · ) ⊥ P t (y, · ) unless x − y is an element of the Cameron-Martin space H, consisting of those elements of E that are absolutely continuous and possess a square-integrable derivative. Now given z ∈ E, let Y = (Y t , Q x ) be Brownian motion in E with drift z. By this we mean the E-valued diffusion with transition semigroup is a regular version of the family of conditional distributions Q x ({X s ; 0 ≤ s ≤ t} ∈ · | X t = y), regardless of the choice of z ∈ E. In other words, X and Y have common bridge laws. However, the laws of X and Y are mutually absolutely continuous (as in (1.3)) if and only if z ∈ H.
Before stating our results we describe the context in which we shall be working. Let X = (X t , P x ) now denote a strong Markov process with cadlag paths and infinite lifetime. We assume that the state space E is homeomorphic to a Borel subset of some compact metric space, and that the transition semigroup (P t ) t≥0 of X preserves Borel measurability and is without branch points. In other words, X is a Borel right processes with cadlag paths and infinite lifetime; see [G75, S88]. The process X is realized as the coordinate process X t : ω → ω(t) on the sample space Ω of all cadlag paths from [0, ∞[ to E. The probability measure P x is the law of X under the initial condition X 0 = x. We write (F t ) t≥0 for the natural (uncompleted) filtration of (X t ) t≥0 and (θ t ) t≥0 for the shift operators on Ω: X s •θ t = X s+t .
In addition, we assume the existence of transition densities with respect to a reference measure and (for technical reasons) the existence of a dual process. (The duality hyothesis (1.4) can be replaced by conditions ensuring the existence of a nice Martin exit boundary for the space-time process (X t , r +t) t≥0 ; see [KW65].) Let E denote the Borel σ-algebra on E.
(1.4) Hypothesis. (Duality) There is a σ-finite measure m X on (E, E) and a second E-valued Borel right Markov processX, with cadlag paths and infinite lifetime, such that the semigroup (P t ) ofX is in duality with (P t ) relative to m X : for all t > 0 and all positive E-measurable functions f and g.
(1.6) Hypothesis. (Transition densities) There is an for any bounded E-measurable function f. Furthermore, we assume that the Chapman-Kolmogorov identity holds: (1.9) p t+s (x, y) = E p t (x, z)p s (z, y) m X (dz), ∀ s, t > 0, x, y ∈ E.
Hypothesis (1.6) implies that m X (U ) > 0 for every non-empty finely open subset of E.
When (1.4) is in force, the existence and uniqueness of a (jointly measurable) transition density function p t (x, y) such that (1.7)-(1.9) hold is guaranteed by the apparently weaker condition: P t (x, ·) m X , P t (x, ·) m X for all x ∈ E, t > 0. See, for example, [D80, W86, Y88]. For more discussion of processes with "dual transition densities," see [GS82;§3].
Let Y = (Y t , Q x ) be a second E-valued Borel right Markov process with cadlag paths and infinite lifetime. The process Y is assumed to satisfy all of the conditions imposed on X above. In particular, we can (and do) assume that Y is realized as the coordinate process on Ω. The transition semigroup of Y is denoted (Q t ) t≥0 and we use m Y and q t (x, y) to denote the reference measure and transition density function for Y . (The bridge laws P x,y t and Q x,y t for X and Y will be discussed in more detail in section 2.) In what follows, the prefix "co-" refers to the dual processX (orŶ ).
(1.10) Theorem. Let X and Y be strong Markov processes as described above, satisfying Hypotheses (1.4) There exist a constant λ ∈ R, a Borel finely continuous function ψ : E →]0, ∞[, and a Borel co-finely continuous functionψ : The function ψψ is a Borel version of the Radon-Nikodym derivative dm Y /dm X .
(i) Given functions ψ andψ as in (1.11) and (1.12), the right sides of (1.13) and (1.14) determine the laws of Borel right Markov processes Y * andŶ * on E. It is easy to check that Y * andŶ * are in duality with respect to the measure ψψ · m X , that Hypotheses (1.4) and (1.6) are satisfied, and that Y * (resp.Ŷ * ) has the same bridge laws as X (resp.X).
(ii) As noted earlier, any one-dimensional regular diffusion without absorbing boundaries satisfies Hypotheses (1.4) and (1.6). Such a diffusion is self-dual with respect to its speed measure, which serves as the reference measure. Moreover, the transition density function of such a diffusion is jointly continuous in (x, t, y). See [IM; pp. 149-158].
(1.16) Theorem. Let X and Y be right Markov processes as described before the statement of Theorem (1.10). Suppose, in addition to (1.4) and (1.6), that for each t > 0 the transition density functions p t (x, y) and q t (x, y) are separately continuous in the spatial variables x and y. If there is a triple , then the conclusions (a), (b), and (c) of Theorem (1.10) remain true.
(1.17) Remark. Let us suppose that X is a real-valued regular diffusion on its natural scale, and that its speed measure m X admits a strictly positive density ρ with respect to Lebesgue measure. Let L X denote the local infinitesimal generator of X. Then (1.11) implies L X ψ = λψ, or more explicitly Moreover, (1.13) means that the transition semigroups of X and Y are related by From this it follows that the (local) infinitesimal generators of X and Y are related by where µ := (log ψ) . When X is standard Brownian motion (so that ρ(x) ≡ 2), the right side of (1.18) is the infinitesimal generator of any weak solution of (1.2). By Remark (1.15)(ii), the additional condition imposed in Theorem (1.16) is met in the present situation. Consequently, Theorem (1.16) implies that the conclusion Without some sort of additional condition as in Theorem (1.16), there may be an exceptional set in the conclusions (a)-(c). Recall that a Borel set N ⊂ E is X-polar if and only if P x (X t ∈ N for some t > 0) = 0 for all x ∈ E.
(1.19) Example. The state space in this example will be the real line R.
has the same law as the radial part of a standard 3-dimensional Brownian motion started at (x, 0, 0).) We assume that the probability space on which Z is realized is rich enough to support an independent unit-rate Poisson process (N (t)) t≥0 . The process X is presented (non-canonically) as follows: Both X and Y are Borel right Markov processes satisfying (1.4) and (1.6); indeed, both processes are self-dual with respect to the reference measure m(dx) := x 2 dx. The singleton {0} is a polar set for both processes.
If neither x nor y is equal to 0, then P x,y t = Q x,y t for all t > 0. However, P 0,y t and Q 0,y t are different for all y ∈ R and t > 0, because P 0,y t (X s > 0 for all small s) = Q 0,y t (X s < 0 for all small s) = 1.
The reader will have no trouble finding explicit expressions for the transition densities p t (x, y) and q t (x, y), thereby verifying that for t > 0, y > 0, which is consistent with Theorem (1.16).
This example is typical of what can go wrong when the hypothesis [P x,x t0 = Q x,x t0 , ∀ x] of Theorem (1.10) is weakened to P x0,y0 t0 = Q x0,y0 t0 . In general, under this latter condition, there is a set N ∈ E that is both X-polar and Y -polar and a setN ∈ E that is bothX-polar andŶ -polar, such that the conclusions drawn in Theorem (1.10) remain true provided one substitutes "x ∈ E \ N " for "x ∈ E" and "y ∈ E \N " for "y ∈ E" throughout. (Actually, the functions ψ andψ can be defined so that (1.11) and (1.12) hold on all of E; these functions will be strictly positive on E, but their finiteness can be guaranteed only off N andN, respectively.) Since the proof of this assertions is quite close to that of Theorem (1.10), it is omitted.
After discussing bridge laws in section 2, we turn to the proof of Theorem (1.10) in section 3. Theorem (1.16) is proved in section 4.

Bridges
The discussion in this section is phrased in terms of X, but applies equally to Y . The process X is as described in section 1. All of the material in this section, with the exception of Lemmas (2.8) and (2.9), is drawn from [FPY93], to which we refer the reader for proofs and further discussion.
The following simple lemma shows that in constructing P x,y t it matters not whether we condition P x on the event {X t = x} or on the event {X t− = x}.
In what follows, F t− denotes the σ-algebra generated by {X s , 0 ≤ s < t}.
(2.2) Proposition. Given (x, t, y) ∈ E×]0, ∞[×E there is a unique probability measure P x,y t on (Ω, F t− ) such that for all positive F s -measurable functions F on Ω, for all 0 ≤ s < t. Under P x,y t the coordinate process (X s ) 0≤s<t is a non-homogeneous strong Markov process with transition densities Moreover P x,y t (X 0 = x, X t− = y) = 1. Finally, if F ≥ 0 is F t− -measurable, and g ≥ 0 is a Borel function on E, then (2.5) P x (F · g(X t− )) = E P x,y t (F ) g(y) p t (x, y) m(dy).
Thus (P x,y t ) y∈E is a regular version of the family of conditional probability distributions {P x (· |X t− = y), y ∈ E}; equally so with X t− replaced by X t , because of Lemma (2.1).
The following corollaries of Proposition (2.2) will be used in the sequel.
(2.6) Corollary. The P x,y t -law of the time-reversed process (X (t−s)− ) 0≤s<t isP y,x t , the law of a (y, t, x)bridge for the dual processX.
(2.7) Corollary. For each (F t+ ) stopping time T , a P x,y t regular conditional distribution for (X T +u , 0 ≤ u < t − T ) given F T + on {T < t} is provided by P XT ,y t−T .
Continuity properties are useful in trying to minimize the exceptional sets involved in statements concerning bridge laws. The following simple result will be used in the proof of (1.16).
(2.8) Lemma. Assume that x → p t (x, y) is continuous for each fixed pair (t, y) ∈]0, ∞[×E. Fix 0 < s < t and let G be a bounded F (t−s)− -measurable function on Ω. Then for each y ∈ E, is continuous on E.
Proof. By Corollary (2.7), The ratio on the right side of (2.9) (call it f x (z)) is a probability density with respect to m X (dz), and the mapping x → f x (z) is continuous by hypothesis. It therefore follows from Scheffé's Theorem [B68; p. 224] The backward space-time process associated with X is the (Borel right) process X t (ω, r) := (X t (ω), r − t), realized on the sample space Ω × R equipped with the laws P x ⊗ r . A (universally measurable) function for each y ∈ E, the function is finely continuous with respect to the backward space-time process (X t , r − t) t≥0 .
Proof. Without loss of generality, we assume that 0 < f i ≤ 1 for every i. The expression appearing in (2.11) can be written as the sum of n! terms of the form where (g 1 , g 2 , . . . , g n ) is a permutation of (f 1 , f 2 , . . . , f n ). Let h(x, t) denote the expression in (2.12) multiplied by p t (x, y). Also, leth(z, u) := p u (z, y) · P z,y u (J u ), where For t > 0, the Markov property (2.7) yields The final line in (2.13) exhibits h as a positive linear combination of the space-time excessive functions y) is also space-time excessive, the function appearing in (2.12) is finely continuous as asserted.
3. Proof of (1.10) For typographical convenience, throughout this section we assume (without loss of generality) that t 0 = 2, so the basic hypothesis under which we are working is that Q x,x 2 = P x,x 2 for all x ∈ E.
Proof of (1.10)(a). Given x ∈ E and t ∈]0, 2[, the mutual absolute continuity of P x | Ft and Q x | Ft follows immediately from the hypothesis Q x,x 2 = P x,x 2 because of (2.3). Let us now show that if P x | Ft ∼ Q x | Ft for all x, then P x | F2t ∼ Q x | F2t for all x; an obvious induction will then complete the proof. By an application of the monotone class theorem, given a bounded F 2t -measurable function F , there is a bounded F t ⊗F t -measurable function G such that F (ω) = G(ω, θ t ω) for all ω ∈ Ω. Consequently, and so the equivalence of P x and Q x on F 2t follows from their equivalence on F t , as desired. The dual assertion can be proved in the same way once we notice thatQ x,x 2 =P x,x An important consequence of the equivalence just proved is that X and Y have the same fine topologies, as do their space-time processes. Of course, the same can be said ofX andŶ .
Proof of (1.10)(b). The argument is broken into several steps.
Step 1: m X ∼ m Y . Indeed, because the transition densities are strictly positive and finite by hypothesis, Step 2. For each (x, t) ∈ E×]0, 2[, Q x,y t = P x,y t for m X -a.e. y ∈ E. Fix (x, t) ∈ E×]0, 2[. Then by (2.6) and (2.7), the P x,x 2 -conditional distribution of (X s ) 0≤s<t , given X t− = y, is P x,y t (for m X -a.e. y ∈ E).
Similarly, the Q x,x 2 -conditional distribution of (Y s ) 0≤s<t , given Y t− = y, is Q x,y t (for m Y -a.e. y ∈ E). The assertion therefore follows from the basic hypothesis (Q x,x 2 = P x,x 2 , ∀x) because of Step 1.
for all x ∈ E and all t ∈]0, 2[. By Step 2 and Fubini's for m X ⊗ Leb-a.e. (x, t) ∈ E×]0, 2[. Let I denote the class of processes I of the form where n ∈ N and each f i is a bounded real-valued Borel function on E × [0, ∞[. It is easy to see that for each fixed t > 0, the family {I t : I ∈ I} is measure-determining on (Ω, F t− ). Therefore, it suffices to show for all x ∈ E, t ∈]0, 2[, and I ∈ I. But by Lemma (2.10) and the remark made following the proof of (1.10)(a), the two sides of (3.1) are finely-continuous (with respect to the space-time processes (X t , r − t) t≥0 and (Y t , r−t) t≥0 ) on all of E×]0, ∞[, as functions of (x, t). By the choice of b these functions agree m X ⊗Leba.e. on the (space-time) finely open set E×]0, 2[; consequently, they agree everywhere on E×]0, 2[, because m X ⊗ Leb is a reference measure for the space-time processes.
Step 4. In view of Step 3 there exists b ∈ E such that P x,b 1 = Q x,b 1 for all x ∈ E. This b will remain fixed in the following discussion. Recall from (1.10)(a) that the laws P x and Q x are (locally) mutually absolutely continuous for each x ∈ E. Let Z t denote the Radon-Nikodym derivative dP x | Ft+ /dQ x | Ft+ . Then Z is a strictly positive right-continuous martingale and a multiplicative functional of X; see, for example, [K76; Thm. 5.1]. The term multiplicative refers to the identity Using (2.3) we see that for any x ∈ E, for any F ∈ F s+ , provided 0 < s < 1. Since Z s is F s+ measurable, it follows that for all x ∈ E and 0 < s < 1. Since P x and P x,b 1 are equivalent on F s+ for 0 < s < 1, we see that the fine topology of the latter process is the same as that of (X, r + t) t≥0 because of the mutual absolute continuous on E × [0, 1[. Now from the multiplicativity of Z and the strict positivity of the transition densities of X we deduce that for all x ∈ E and all t, s > 0 such that t + s < 1, there is an m X ⊗ m X -null provided (y, z) / ∈ N (x, t, s). By the preceding discussion, the two sides of (3.3) are space-time finely continuous as functions of (y, s). Moreover, m X ⊗ Leb is a reference measure for (X t , r + t); thus, two space-time finely continuous functions equal m X ⊗ Leb-a.e. must be identical. From this observation and Fubini's theorem it follows that given (x, t) ∈ E×]0, 1[ there is an m X -null set N (x, t) such that (3.3) holds for all (y, s) ∈ E × [0, 1 − t[ and all z / ∈ N (x, t). Taking s = 0 we find that for all y ∈ E, 0 < t < 1, and z / ∈ N (x, t). Thus, defining λ t := − log[ψ t (b)/ψ 0 (b)] and ψ := ψ 0 , we have, for each x ∈ E, for all t ∈]0, 1[, since P x (X t ∈ N ) = 0 for any m X -null set N . The multiplicativity of Z implies first that λ t = λt for some real constant λ, and then that (3.5) holds for all t > 0. This yields (1.13), from which (1.11) follows immediately because Z is a P x -martingale.
The dual assertions (1.12) and (1.14) are proved in the same way, and the fact that ψ andψ correspond to the same "eigenvalue" λ follows easily from (1.5).
Proof of (1.10)(c). Formula (1.13) implies that for each x ∈ E and t > 0, (3.6) q t (x, y) = e −λt 1 ψ(x)ψ(y) p t (x, y), m X -a.e. y ∈ E, because ψψ = dm Y /dm X . For fixed x the two sides of (3.6) are finely continuous (as functions of (y, t) ∈ E×]0, ∞[) with respect to the backward space-time process (X t , r −t) t≥0 . (As before, the equivalence of laws established in (1.10)(a) implies that (X t , r − t) and (Ŷ t , r − t) have the same fine topologies.) Since m X ⊗ Leb is a reference measure for this space-time process, the equality in (3.6) holds for all (y, t) ∈ E×]0, ∞[. The asserted equality of bridges now follows from (1.13) and (2.3).
By hypothesis, the right side of (4.2) is unchanged if P x0,y0 t0 is replaced by Q x0,y0 t0 ; the same is therefore true of the left side, so P x,y0 t1 (F ) = Q x,y0 t1 (F ). The monotone class theorem clinches the matter. The arguments used in the proof of Theorem (1.10) (especially Step 4 of the proof of (1.10)(b)) can now be used to finish the proof. The dual assertion follows in the same way.