Excursions Above the Minimum for Diffusions *

P.J. Fitzsimmons May 13, 1998

1. Introduction

Let X = (X_{t})_{t≥0} be a regular diffusion process on an interval E ⊂ IR. Let H_{t} := min_{0≤u≤t} X_{u} denote the past minimum process of X and consider the excursions of X above its past minimum level: If [a, b] is a maximal interval of constancy of t ↦ H_{t}, then (X_{t} : a ≤ t ≤ b) is the “excursion above the minimum” starting at time a and level y = H_{a}. These excursions, when indexed by the level at which they begin, can be regarded (collectively) as a point process. The independent increments property of the first-passage process of X implies that this point process is Poisonnian in nature, albeit non-homogeneous in intensity. Moreover, intuition tells us that the distribution of an excursion above the minimum (X_{t} : a ≤ t ≤ b) should be governed by the Itô excursion law corresponding to excursions above the fixed level y = H_{a}(ω).

Our first task is to render precise the ruminations of the preceding paragraph. This is accomplished in sections 2 and 3 by applying Maisonneuve’s theory of exit systems [10] to a suitable auxiliary process (\overline{X}_{t}) associated with X. The basic result, stated in section 2, affirms the existence of a “Lévy system” for the point process of excursions of X above its past minimum.

In sections 4, 5, and 6 we discuss several applications of the Lévy system constructed in section 3; these applications concern path decompositions of X involving the minimum process H. Such decompositions, and related results, have been found by various authors (see [6, 9, 11, 12, 14, 15, 16, 17, 18]), most often in the special case where X is Brownian motion. The possibility of using Lévy systems to give a unified treatment of path decompositions is, of course, not surprising. In an excellent synthesis [13] Pitman has shown how the existence of a Lévy system for a point process attached to a Markov process leads naturally to various path decompositions of the Markov process.

In section 4 we obtain a general version of Williams’ decomposition of a diffusion at its global minimum. A “local” version of Williams’ decomposition can be found in section 5. In section 6 we give a new proof of a result of Vervaat [17], which states that a Brownian bridge, when split at its minimum and suitably “ rearranged” becomes a (scaled) Brownian

* This is a lightly edited form of a manuscript written in the spring of 1985. excursion. Indeed, we produce an inversion of Vervaat’s transformation, showing how a Brownian excursion may be split and rearranged to yield Brownian bridge.

2. Notation and the basic result

Let X = (Ω, F, F_{t}, θ_{t}, X_{t}, P^{x}) be a canonically defined regular diffusion on an interval E ⊂ IR. Here Ω denotes the space of paths ω: [0, +∞[→ E ∪ {Δ} which are absorbed in the cemetery point Δ ∉ E at time ζ(ω), and which are continuous on [0, ζ(ω)[. For t ≥ 0, X_{t}(ω) = ω(t), and θ_{t}ω denotes the path u ↦ ω(u + t). The σ-fields F and F_{t} (t ≥ 0) are the usual Markovian completions of F^{∘} = σ{X_{u} : u ≥ 0} and F_{t}^{∘} = σ{X_{u} : 0 ≤ u ≤ t} respectively. The law P^{x} on (Ω, F^{∘}) corresponds to X started at x ∈ E. We shall also make use of the killing operators (k_{t}) defined for t ≥ 0 by

k_{t}ω(u) = \{ \begin{matrix}ω(u), & u < t, \\ Δ, & u ≥ t. \end{matrix}

Let A = inf E, B = sup E, and write E^{∘} =]A, B[. We assume throughout the paper that A ∉ E, and that B ∈ E if and only if B is a regular boundary point which is not a trap for X. In particular, these assumptions imply that the transition kernels of X are absolutely continuous with respect to the speed measure m (recalled below). See §4.11 of Itô-McKean [8].

Let s (resp. m, resp. k) denote a scale function (resp. speed measure, resp. killing measure) for X. Recall from [8] that the generator G of X has the form

(2.1)  Gu(x) · m(dx) = du^{+}(x) − u(x) · k(dx),  x ∈ E^{∘},

for u ∈ D(G), the domain of G. Here and elsewhere u^{+} denotes the scale derivative:

u^{+}(x) = lim_{y↓x} \frac{u(y) − u(x)}{s(y) − s(x)}.

Let (U^{α} : α > 0) denote the resolvent family of X. Subsequent calculations require an explicit expression for the density of U^{α}(x, dy) with respect to m(dy). Recall from [8] that for each α > 0 there are strictly positive, linearly independent solutions g_{1}^{α} and g_{2}^{α} of

(2.2)  Gg(x) = αg(x),  x ∈ E^{∘};

g_{1}^{α} (resp. g_{2}^{α}) is an increasing (resp. decreasing) solution of (2.2) which also satisfies the appropriate boundary condition at A (resp. B). Both g_{1}^{α} and g_{2}^{α} are uniquely determined up to a positive multiple. We sometimes drop the superscript α, writing simply g_{1} and g_{2}. Since g_{1} and g_{2} are linearly independent solutions of (2.2), the Wronskian W = g_{1}^{+}g_{2}−g_{2}^{+}g_{1} is constant. The resolvent U^{α} is given by

(2.3)  U^{α}f(x) = U^{α}(x, f) = ∫_{E} u^{α}(x, y)f(y) m(dy),

where

(2.4)  u^{α}(x, y) = u^{α}(y, x) = g_{1}^{α}(x)g_{2}^{α}(y)/W,  x ≤ y.

See §4.11 of [8] and note that in (2.3) the mass m({B}) is the “stickiness” coefficient occurring in the boundary condition at B for elements of D(G).

A jointly continuous version (L_{t}^{y} : t ≥ 0, y ∈ E) of local time for X may be chosen, and normalized to be occupation density relative to m, so that

(2.5)  P^{x} ∫_{0}^{∞} e^{−αt} dL_{t}^{y} = u^{α}(x, y).

Fixing a level y ∈ E, the local time (L_{t}^{y} : t ≥ 0) is related to the Itô excursion law [7], for excursions from level y, as follows. Let G(y) denote the (random) set of left-hand endpoints (in ]0, ζ[) of intervals contiguous to the level set {t > 0 : X_{t} = y}. Define the hitting time T_{y} by

T_{y} = inf{t > 0 : X_{t} = y} (inf ∅ = +∞).

The Itô excursion law n_{y} is determined by the identity

(2.6)  P^{x} ∑_{u∈G(y)} Z_{u} F^{∘}k_{T_{y}}^{∘}θ_{u} = P^{x} (∫_{0}^{∞} Z_{u} dL_{u}^{y}) · n_{y}(F),

where x ∈ E, F ∈ pF^{∘}, and Z ≥ 0 is an (F_{t})-optional process. Under n_{y} the coordinate process (X_{t} : t > 0) is strongly Markovian with semigroup (Q_{t}^{y}) given by

(2.7)  Q_{t}^{y}(x, f) = P^{x}(f^{∘}X_{t}; t < T_{y}).

The entrance law n_{y}(X_{t} ∈ dz) is determined by the corresponding Laplace transform

(2.8)  W^{α}f(y) = W^{α}(y, f) = n_{y} ∫_{0}^{ζ} e^{−αt}f^{∘}X_{t} dt.

Conversely, n_{y} is the unique σ-finite measure on (Ω, F^{∘}) which is carried by {ζ > 0} and under which (X_{t} : t > 0) is Markovian with semigroup (2.7) and entrance law (2.8).

Let V_{y}^{α} denote the resolvent of the semigroup (Q_{t}^{y}). Taking Z_{u} = e^{−αu}, F = ∫_{0}^{ζ} e^{−αt}f(X_{t}) dt in (2.6), and using (2.5), we obtain the important identity

(2.9)  U^{α}f(x) = V_{y}^{α}f(x) + u^{α}(x, y)[m({y})f(y) + W^{α}f(y)].

We also recall from §4.6 of [8] that the distribution of T_{y} is given by

(2.10)  P^{x}(e^{−αT_{y}}) = \{ \begin{matrix}g_{1}^{α}(x)/g_{1}^{α}(y), & x ≤ y, \\ g_{2}^{α}(x)/g_{2}^{α}(y), & x ≥ y. \end{matrix}

Finally, the point process of excursions above the minimum is defined as follows. For t ≥ 0 set

H_{t}(ω) = \{ \begin{matrix}min_{0≤u≤t} X_{u}(ω) & if t < ζ(ω), \\ −∞ & if t ≥ ζ(ω); \end{matrix}
M(ω) = {u > 0 : X_{u}(ω) = H_{u}(ω)};
R_{t}(ω) = inf{u > 0 : u + t ∈ M(ω)};
G(ω) = {u > 0 : u < ζ(w), R_{u−}(ω) = 0 < R_{u}(ω)}.

Thus G is the random set of left-hand endpoints of intervals contiguous to the random set M. For u ∈ G we have the excursion e^{u} defined by

e_{t}^{u} = \{ \begin{matrix}X_{u+t}, & 0 ≤ t < R_{u}, \\ Δ, & t ≥ R_{u}. \end{matrix}

The point process Π = (e^{u} : u ∈ G) admits a Lévy system as follows. Define a continuous increasing adapted process C = (C_{t} : t ≥ 0) by

C_{t} = \{ \begin{matrix}s(H_{0}) − s(H_{t}), & if t < ζ, \\ C_{ζ−} & if t ≥ ζ. \end{matrix}

(2.11) Theorem. For Z ≥ 0 and (F_{t})-optional, and F ∈ pF^{∘},

(2.12)  \begin{aligned}P^{x} ∑_{u∈G} Z_{u} F (e^{u}) & = P^{x} ∫_{0}^{∞} Z_{u} n_{X_{u}}^{↑}(F) dC_{u} \\ & = P^{x} ∫_{A}^{x} Z_{T_{y}}1_{\{T_{y}<+∞\}}n_{y}^{↑}(F) ds(y), \end{aligned}

where n_{y}^{↑} denotes the restriction of n_{y} to {ω : ω(t) > y, ∀t ∈]0, ζ(ω)[}.

(2.13) Remark. The second equality in (2.12) follows from the first by the change of variable u = T_{y}. The equality of the first and third terms in (2.12) amounts to the statement that the time-changed point process (e^{T_{y}} : R_{T_{y}−} < R_{T_{y}}, A < y < x) is a stopped Poisson point process under P^{x}, with (non-homogeneous) intensity ds(y)n_{y}^{↑}(dω), stopped at the first level y for which T_{y} = +∞. See [14] for this result in the case of Brownian motion, with or without drift. The general result (2.11) was suggested by §4.10 of [8].

3. Proof of Theorem (2.11)

Maisonneuve’s theory of exit systems [10] provides a Lévy system description of the point process of excursions induced by a closed, optional, homogeneous random set. Unfortunately the set M introduced in §2 is not (θ_{t})-homogeneous; however the theory of [10] can be brought to bear once we note that M is homogeneous as a functional of the strong Markov process (X_{t}, H_{t}), t ≥ 0. This key observation is due to Millar [12] and has been formalized by Getoor in [4]. In the terminology of [4], the process H is a “min-functional”: H_{t+u} = H_{t} ∧ H_{u}^{∘}θ_{t}. This property ensures that \overline{X} := (X, H) is Markovian, as a simple computation shows.

Following [4] we first construct a convenient realization of \overline{X}. Let \overline{Ω} = Ω×(E∪\{−∞\}), \overline{E} = \{(x, a) ∈ E × E : a ≤ x\}, and for (ω, a) ∈ \overline{Ω} set

\begin{aligned}\overline{X}_{t}(ω, a) & = (X_{t}(ω), a ∧ H_{t}(ω)), \\ \overline{θ}_{t}(ω, a) & = (θ_{t}(ω), a ∧ H_{t}(ω)). \end{aligned}

Clearly \overline{X}_{t}^{∘}\overline{θ}_{u} = \overline{X}_{t+u}, \overline{θ}_{t}^{∘}\overline{θ}_{u} = \overline{θ}_{t+u}. Moreover, M can be realized over \overline{X} as

(3.1)  \overline{M}(ω, a) = \{t > 0 : \overline{X}_{t}(ω, a) ∈ D\},

where D = \{(x, x) : x ∈ E\}. Let \overline{F}^{∘} = σ\{\overline{X}_{u} : u ≥ 0\}, \overline{F}_{t}^{∘} = σ\{\overline{X}_{u} : 0 ≤ u ≤ t\}, and for (x, a) ∈ \overline{E} let \overline{P}^{x,a} = P^{x} ⊗ ϵ_{a}. The usual Markovian completion of the filtration (\overline{F}_{t}^{∘}) relative to the laws (\overline{P}^{x,a} : (x, a) ∈ \overline{E}) is denoted by (\overline{F}_{t}). Clearly \overline{P}^{x,a}(\overline{X}_{0} = (x, a)) = 1 so that \overline{X} has no branch points. Appealing to §2 of [4] we have the following

(3.2) Lemma. (i) \overline{X} = (\overline{Ω}, \overline{F}, \overline{F}_{t}, \overline{θ}_{t}, \overline{X}_{t}, \overline{P}^{x,a}) is a right-continuous, strong Markov process with state space \overline{E} and cemetery \overline{Δ} = (Δ, −∞). The semigroup of \overline{X} maps Borel functions to Borel functions, so that \overline{X} is even a Borel right process.

(ii) Let π: (ω, a) → ω denote the projection of \overline{Ω} onto Ω. If Z is an (F_{t})-optional process, then Z^{∘}π is (\overline{F}_{t})-optional.

Now \overline{M} is an (\overline{F}_{t})-optional, (\overline{θ}_{t})-homogeneous set, and each section \overline{M}(ω, a) is closed in ]0, \overline{ζ}(ω, a)[. Set \overline{R} = inf \overline{M}, so that \overline{R} is an exact terminal time of \overline{X} with reg(\overline{R}) = \{(x, a) ∈ \overline{E} : \overline{P}^{x,a}(\overline{R} = 0) = 1\} = D. This last fact follows from the regularity of X and the identity

\overline{P}^{x,a}(\overline{R} = T_{a}^{∘}π) = 1, (x, a) ∈ \overline{E}.

Let \overline{G} denote the set of left-hand endpoints of intervals contiguous to \overline{M}. The properties of the Maisonneuve exit system (^{∗}\overline{P}^{x,a}, \overline{K}) for \overline{M} are summarized in the next proposition. In what follows, \overline{E}^{∗} and \overline{F}^{∗} denote the universal completions of \overline{E} (the Borel sets in \overline{E}) and \overline{F}^{∘} respectively.

(3.3) Proposition. [Maisonneuve] There exists a continuous additive functional (CAF), \overline{K}, of \overline{X} with a finite 1-potential, and a kernel ^{∗}\overline{P}^{x,a} from (\overline{E}, \overline{E}^{∗}) to (\overline{Ω}, \overline{F}^{∗}) such that

(3.4)  \overline{P}^{x,a} ∑_{u∈\overline{G}} \overline{Z}_{u}\overline{F}_{u}^{∘}\overline{θ}_{u} = \overline{P}^{x,a} ∫_{0}^{∞} \overline{Z}_{u} ^{∗}\overline{P}^{\overline{X}^{u}}(\overline{F}_{u}) d\overline{K}_{u},

whenever \overline{Z} ≥ 0 is (\overline{F}_{t})-optional and (u, \overline{ω}) ↦ \overline{F}_{u}(\overline{ω}) is a B_{[0,+∞[}⊗\overline{F}^{∗}-measurable, positive function. The CAF \overline{K} is carried by D. For each (x, a) ∈ \overline{E}, ^{∗}\overline{P}^{x,a} is a σ-finite measure on (\overline{Ω}, \overline{F}^{∗}) under which the coordinate process is strongly Markovian with the same transition semigroup as \overline{X}.

(3.5) Remarks. The version of (^{∗}\overline{P}^{x,a}, \overline{K}) cited in (3.3) is a variant of that constructed in [10]; the difference stems from the possibility that \overline{P}^{x,a}(\overline{ζ} < +∞) may be positive. The fact that \overline{K} is continuous (and so carried by D = reg(\overline{R})) follows from the construction in [10], since \overline{M} = \{t > 0 : \overline{X}_{t} ∈ D\} and D is finely perfect (with respect to \overline{X}). Renormalizing the kernel ^{∗}\overline{P}^{x,a} if necessary, we can and do assume that ^{∗}\overline{P}^{y,y}(1 − e^{−\overline{R}}) = 1 for all y ∈ E.

Our plan is to prove Theorem (2.11) by identifying ^{∗}\overline{P}^{x,a} and \overline{K} explicitly, thereby deducing (2.12) from (3.4). First note that by taking x = y in (2.9) and using (2.4) we have

(3.6)  W^{α}f(y) = ∫_{]A,y[}[g_{1}^{α}(z)/g_{1}^{α}(y)]f(z) m(dz) + ∫_{]y,B]}[g_{2}^{α}(z)/g_{2}^{α}(y)]f(z) m(dz),

where y ∈ E, α > 0, and f ≥ 0 is Borel measurable on E.

To identify \overline{K} we define a second CAF of \overline{X}, \overline{C}, by the formula

\overline{C}_{t}(ω, a) = \{ \begin{matrix}s(a ∧ H_{0}(ω)) − s(a ∧ H_{t}(ω)) & if t < \overline{ζ}(ω, a), \\ \overline{C}_{\overline{ζ}−}(ω, a) & if t ≥ \overline{ζ}(ω, a); \end{matrix}

and notice that \overline{C}_{t}(ω, X_{0}(ω)) = C_{t}(ω). Clearly the fine support of \overline{C} is D.

For x ∈ E put ψ(x) = W 1_{]x,B]}(x).

(3.7) Proposition. The CAFs \overline{K} and ∫_{0}^{t} ψ(X_{u}) d\overline{C}_{u} are equivalent.

Proof. By [1; IV(2.13)] it suffices to check that the CAFs in question have the same finite 1-potential (over \overline{X}). An argument of Vervaat [17] shows that P^{x}(t ∈ M) = 0 for all x ∈ E and all t > 0. Consequently, \overline{P}^{x,a}(t ∈ \overline{M}) = 0 for all (x, a) ∈ \overline{E} and all t > 0. By Fubini’s theorem, ∫_{0}^{∞} e^{−t}1\overline{M}(t) dt = 0 a.s. \overline{P}^{x,a} for all (x, a) ∈ \overline{E}. Thus taking \overline{Z}_{u}(\overline{ω}) = e^{−u}, \overline{F}_{u}(\overline{ω}) = 1 − exp(−\overline{R}(\overline{ω}) ∧ \overline{ζ}(\overline{ω})) in (3.4), we may compute

(3.8)  \begin{aligned}\overline{P}^{x,a} ∫_{0}^{∞} e^{−u} d\overline{K}_{u} & = \overline{P}^{x,a} ∑_{u∈\overline{G}} e^{−u} (∫_{0}^{\overline{R}∧\overline{ζ}} e^{−t})^{∘}\overline{θ}_{u} \\ & = \overline{P}^{x,a} ∫ \overline{ζ} ; \overline{R}∧\overline{ζ} e^{−u} du \\ & = \overline{P}^{x,a}(e^{−\overline{R}∧\overline{ζ}} − e^{−\overline{ζ}}) \\ & = P^{x}(e^{−T_{a}∧ζ} − e^{−ζ}) \\ & = P^{x} ∫_{0}^{ζ} e^{−t} dt − P^{x} ∫_{0}^{T_{a}∧ζ} e^{−t} dt \\ & = U^{1}1(x) − V_{a}^{1}1(x) \\ & = \frac{u^{1}(x, a)}{u^{1}(a, a)} U^{1}1(a), \end{aligned}

where the last equality follows easily from (2.9). On the other hand, our hypothesis regarding the boundary A implies that g_{1}^{1}(A+)/g_{2}^{1}(A+) = 0 (see [8; §4.6]). Thus

(3.9)  \begin{aligned}\overline{P}^{x,a} ∫_{0}^{∞} e^{−t}ψ(X_{t}) d\overline{C}_{t} & = P^{x} ∫_{T_{a}}^{∞} e^{−t}ψ(X_{t}) dC_{t} \\ & = P^{x} ∫_{A}^{a} e^{−T_{y}}ψ(y) ds(y) \\ & = ∫_{A}^{a}[g_{2}^{1}(x)/g_{2}^{1}(y)]ψ(y) ds(y). \end{aligned}

In (3.9) we have used the change of variables t = T_{y} to obtain the second equality, and (2.1) to obtain the third. Now from the definition of the Wronskian W we see that d(g_{1}/g_{2}) = W · [g_{2}]^{−2}ds. Using this fact and the expression for ψ provided by (3.6) we may continue the computation begun in (3.9) with

(3.10)  \begin{matrix}= ∫_{A}^{a}[g_{2}(x)/g_{2}(y)] ∫_{]y,B]}[g_{2}(z)/g_{2}(y)] m(dz) ds(y) \\ = ∫_{A}^{a} ∫_{]y,B]}[g_{2}(x)g_{2}(z)/W] m(dz) d(g_{1}/g_{2})(y) \\ = ∫_{E} ∫_{A}^{z∧a} d(g_{1}/g_{2})(y)[g_{2}(x)g_{2}(z)/W] m(dz) \\ = ∫_{E}[g_{1}(z ∧ a)/g_{2}(z ∧ a)] · [g_{2}(x)g_{2}(z)/W] m(dz) \\ = [u^{1}(x, a)/u^{1}(a, a)]U^{1}1(a). \end{matrix}

The last equality in (3.10) follows from (2.3) and (2.4). In view of (3.8)–(3.10), we see that \overline{K} and ∫_{0}^{t} ψ(X_{s}) c\overline{C}_{s} have the same finite 1-potential and so the proposition is proved.

For y ∈ E define a measure \overline{Q}^{y} on (\overline{Ω}, \overline{F}^{∘}) by \overline{Q}^{y}(F) = ^{∗}\overline{P}^{y,y}(F^{∘}\overline{k}\overline{R}), where \overline{k}_{t} is the killing operator on \overline{Ω}. Since ^{∗}\overline{P}^{y,y}(\overline{R} ≠ T_{y}^{∘}π) = 0, the first coordinate of \overline{X}, namely (X_{t} : t > 0), is Markovian under \overline{Q}^{y}, with (Q_{t}^{y}) as semigroup. Indeed, we claim that ψ(y)π(\overline{Q}^{y}) = n_{y}^{↑}, at least for ds-a.e. y ∈ E. To verify this claim it suffices to compare the associated entrance laws.

(3.11) Lemma. Let f be a bounded positive Borel function on E. Then for ds-a.e. y ∈ E we have

(3.12)  ψ(y)\overline{Q}^{y} ∫ \overline{ζ} ; 0 e^{−αt}f(X_{t}) dt = W^{α}f(y),  ∀α > 0.

Proof. Fix f as in the statement of the lemma and also fix α > 0. For y ∈ E write

γ(y) = \overline{Q}^{y} ∫ \overline{ζ} ; 0 e^{−αt}f(X_{t}) dt.

As noted in the proof of (3.7), we have ∫_{0}^{∞} 1\overline{M}(t) dt = 0, \overline{P}^{x,a}-a.s. for all (x, a) ∈ \overline{E}. Thus, for x ∈ E,

(3.13)  \begin{aligned}U^{α}f(x) & = \overline{P}^{x,x} ∑_{u∈\overline{G}} e^{−αu} (∫ \overline{R} ; 0 e^{−αt}f(X_{t}) dt)^{∘}\overline{θ}_{u} \\ & = \overline{P}^{x,x} ∫_{0}^{∞} e^{−αu} ^{∗}\overline{P}^{\overline{X}^{u}} (∫ \overline{R} ; 0 e^{−αt}f(X_{t}) dt) d\overline{K}_{u} \\ & = \overline{P}^{x,x} ∫_{0}^{∞} e^{−αu}γ(X_{u})ψ(X_{u}) d\overline{C}_{u} \\ & = P^{x} ∫_{A}^{x} e^{−αT_{y}}γ(y)ψ(y) ds(y) \\ & = ∫_{A}^{x}[g_{2}^{α}(x)/g_{2}^{α}(y)]γ(y)ψ(y) ds(y). \end{aligned}

On the other hand, by (2.3) and (2.4), we have

(3.14) U^{α}f(x) = ∫_{]A,x]}[g_{1}^{α}(x)g_{2}^{α}(y)/W]f(y) m(dy) + ∫_{]x,B]}[g_{1}^{α}(y)g_{2}^{α}(x)/W]f(y) m(dy).

If we equate the last line displayed in (3.13) with the right side of (3.14), divide the resulting identity by g_{2}^{α}(x), and then differentiate in x, we obtain

[∫_{]x,B]}[g_{2}^{α}(y)/g_{2}^{α}(x)]f(y) m(dy)] ds(x) = γ(x)ψ(x) ds(x)

as measures on E, and the lemmas follows.

(3.15) Corollary. For ds-a.e. y ∈ E, ψ(y)π(\overline{Q}^{y}) = n_{y}^{↑} as measures on (Ω, F^{∘}).

Proof. As noted earlier, both ψ(y)π(\overline{Q}^{y}) and n_{y}^{↑} make the coordinate process on (Ω, F^{∘}) into a Markov process with transition semigroup (Q_{t}^{y}). It follows from Lemma (3.11) that these measures have the same one-dimensional distributions (and consequently the same finite dimensional distributions) for ds-a.e. y. Since F^{∘} = σ(X_{u} : u ≥ 0) is countably generated, the corollary follows.

Proof of Theorem (2.11). Let Z ≥ 0 be (F_{t})-optional and let F ≥ 0 be F^{∘}-measurable. By (3.2)(ii), the process Z^{∘}π is (\overline{F}_{t})-optional. We may now use (3.4), (3.7), and (3.15) to compute

P^{x} ∑_{u∈G} Z_{u}F (e^{u}) = \overline{P}^{x,x} ∑_{u∈\overline{G}} Z_{u}^{∘}πF (π^{∘}\overline{k}\overline{R}^{∘}\overline{θ}_{u})
= \overline{P}^{x,x} ∫_{0}^{∞} Z_{u}^{∘}π ^{∗}\overline{P}^{\overline{X}^{u}}(F (π^{∘}\overline{k}\overline{R})) d\overline{K}_{u}
= \overline{P}^{x,x} ∫_{0}^{∞} Z_{u}^{∘}π \overline{Q}^{\overline{X}^{u}}(F (π^{∘}\overline{k}\overline{R})) ψ(X_{u}) d\overline{C}_{u}
= P^{x} ∫_{0}^{∞} Z_{u} n_{X_{u}}^{↑}(F) dC_{u}.

The proof of Theorem (2.11) is complete.

4. Williams’ decomposition

In this section we use the Lévy system (2.12) to obtain a new proof (of a general version) of Williams’ decomposition [18] of a diffusion at its global minimum. A more “computational” proof of Williams’ theorem, based on the same idea used in the present paper, may be found in [3].

For simplicity we assume that γ := H_{ζ−} satisfies P^{x}(γ > A) = 1 for all x ∈ E. We also assume that ρ := inf{t > 0 : X_{t} = γ} satisfies P^{x}(ρ < ζ) = 1 for all x ∈ E. Then ρ is the unique time at which X takes its global minimum value γ (cf. [17]). Note that for x ≥ y (both in E),

(4.1)  P^{x}(γ > y) = P^{x}(T_{y} = +∞).

Define a function r on E by

r(x) = \{ \begin{matrix}P^{x}(T_{x_{0}} < +∞), & x ≥ x_{0}, \\ [P^{x_{0}}(T_{x} < +∞)]^{−1}, & x < x_{0}, \end{matrix}

where x_{0} ∈ E^{∘} is fixed but arbitrary. Clearly r is strictly positive and decreasing. Arguing as in [8; pp. 128–129] one may check that r is the unique positive decreasing solution of Gr ≡ 0 on E^{∘} which satisfies r(x_{0}) = 1 and the boundary condition at B. Note that

(4.2)  P^{x}(T_{y} < +∞) = r(x)/r(y),  x > y.

Before proceeding to the decomposition theorem we need a preliminary result.

(4.3) Lemma. For y ∈ E^{∘} let S_{y} = inf{t > 0 : X_{t−} = y}. Then

n_{y}^{↑}(S_{y} = +∞) = −\frac{r^{+}(y)}{r(y)}, ∀y ∈ E^{∘}.

(Recall that r^{+} = d^{+}r/ds^{+}.)

Proof. Let q be an increasing solution of Gq ≡ 0 on E^{∘} such that q is linearly independent of r. We assume that q is normalized so that the Wronskian q^{+}r − r^{+}q is identically 1. Fix a < b both in E^{∘} and let v_{ab} denote the potential density (relative to m) of X killed at time T_{a} ∧ T_{b}. One checks that for x ≤ y,

v_{ab}(x, y) = v_{ab}(y, x) = \frac{D(a, x)D(y, b)}{D(a, b)},

where D(x, y) is the determinant

| \begin{matrix}q(y) & q(x) \\ r(y) & r(x) \end{matrix} | .

Note that D(x, y) > 0 if x < y. Now let y ∈]a, b[ and use (2.6) to compute

\begin{aligned}P^{y}(T_{b} < T_{a}) & = P^{y} ∑_{u∈G(y)} 1_{\{u<T_{a}∧T_{b}\}}1_{\{T_{b}<T_{y}\}}^{∘}θ_{u} \\ & = P^{y}(L_{T_{a}∧T_{b}}^{y}) n_{y}(T_{b} < ζ). \end{aligned}

But clearly P^{y}(T_{b} < T_{a}) = D(a, y)/D(a, b) while P^{y}(L_{T_{a}∧T_{b}}^{y}) = v_{ab}(y, y), so that

n_{y}(T_{b} < ζ) = [D(a, y)/D(a, b)]/v_{ab}(y, y) = D(y, b)^{−1}.

Finally,

n_{y}^{↑}(S_{y} = +∞) = lim_{x↓y} n_{y}^{↑}(T_{x} < +∞, S_{y} = +∞)

= lim_{x↓y} n_{y}^{↑}(T_{x} < +∞) P^{x}(T_{y} = +∞)

= lim_{x↓y} n_{y}(T_{x} < ζ)[1 − r(x)/r(y)]

= lim_{x↓y} [\frac{1}{r(y)} · \frac{r(y) − r(x)}{s(x) − s(y)} · \frac{s(x) − s(y)}{D(y, x)}]

= −r(y)^{−1} · r^{+}(y),

since lim_{x↓y}[s(x) − s(y)]/D(y, x) is the reciprocal of the Wronskian q^{+}r − r^{+}q ≡ 1.

Now define probability laws on (Ω, F^{∘}) by

(4.4)  P_{y}^{x↓}(F) = P^{x}(F^{∘}k_{T_{y}}|T_{y} < +∞),

(4.5)  P_{y}^{↑}(F) = n_{y}^{↑}(F |S_{y} = +∞),

whenever x > y > A. The coordinate process is a diffusion under any of these laws: P_{y}^{x↓} is the law of X started at x, conditioned to converge to A, and then killed at T_{y}; P_{y}^{↑} is the law of X started at y and conditioned to never return to y. These conditionings are accomplished by means of the appropriate h-transforms. In particular, the associated infinitesimal generators are given by

(4.6)  G_{y}^{x↓}f(z) = r(z)^{−1}G(fr)(z),  z > y;
(4.7)  G_{y}^{↑}f(z) = r_{y}(z)^{−1}G(fr_{y})(z),  z > y,

where r_{y}(z) = 1 − r(z)/r(y).

We can now state the general version of Williams’ theorem. Recall that γ = H_{ζ−} and ρ = inf{t > 0 : X_{t} = γ}.

(4.8) Theorem. (a) The joint law of (γ, ρ, ζ) is given by

(4.9)  P^{x}(f(γ)e^{−αρ−βζ}) = ∫_{A}^{x}[g_{2}^{α+β}(x)/g_{2}^{α+β}(y)]f(y)P_{y}^{↑}(e^{−βζ})\frac{−dr(y)}{r(y)}.

(b) For F, G ∈ bF^{∘} and ψ bounded and Borel on E,

(4.10)  P^{x}(F^{∘}k_{ρ}ψ(γ)G^{∘}θ_{ρ}) = P^{x}(P_{γ}^{x↓}(F)ψ(γ)P_{γ}^{↑}(G)).

(4.11) Remark. The intuitive content of (4.10) is that the processes (X_{t} : 0 ≤ t < ρ) and (X_{ρ+t} : 0 ≤ t < ζ − ρ) are conditionally independent under P^{x}, given γ; and that the conditional distributions, given that γ = y, are P_{y}^{x↓} and P_{y}^{↑} respectively.

Proof of (4.8). Define J(y, ω) = 1_{{S_{y}=+∞}}(ω) and observe that ρ(ω) = u if and only if u ∈ G(ω) and J(X_{u}(ω), e^{u}(ω)) = 1. Thus, using (2.11),

(4.12) \begin{aligned}P^{x}(F^{∘}k_{ρ}ψ(γ)G^{∘}θ_{ρ}) & = P^{x} ∑_{u∈G} F^{∘}k_{u}ψ(X_{u})G(e^{u})J(X_{u}, e^{u}) \\ & = ∫_{A}^{x} P^{x}(F^{∘}T_{y}; T_{y} < +∞)ψ(y)n_{y}^{↑}(G · J(y, ·)) ds(y) \\ & = ∫_{A}^{x} P^{x↓}(F)ψ(y)P_{y}^{↑}(G)P^{x}(T_{y} < +∞)n_{y}^{↑}(S_{y} = +∞) ds(y). \end{aligned}

Taking F = G = 1 in (4.12) we see that

(4.13)  P^{x}(γ ∈ dy) = P^{x}(T_{y} < +∞)n_{y}^{↑}(S_{y} = +∞) ds(y).

Now (4.13) substituted into the last line of (4.12) yields (4.10). To obtain (4.9) use (4.10) with F = e^{−(α+β)ζ} and G = e^{−αζ}, noting that F^{∘}k_{ρ} = e^{−(α+β)ρ} and ζ = ρ+ζ^{∘}θ_{ρ} (P^{x}-a.s.) since ρ < ζ, P^{x}-a.s. Thus

\begin{aligned}P^{x}(f(γ)e^{−αρ}e^{−βζ}) & = P^{x}(f(γ)[e^{−(α+β)ζ}]^{∘}k_{ρ} [e^{−βζ}]^{∘}θ_{ρ}) \\ & = P^{x}(P_{γ}^{x↓}(e^{−(α+β)ζ})f(γ)P_{γ}^{↑}(e^{−βζ})). \end{aligned}

Formula (4.9) now follows since

\begin{aligned}P_{y}^{x↓}(e^{−(α+β)ζ}) & = P^{x}(e^{−(α+β)T_{y}})/P^{x}(T_{y} < +∞) \\ & = [g_{2}^{α+β}(x)/g_{2}^{α+β}(y)]/P^{x}(T_{y} < +∞), \end{aligned}

and since n_{y}^{↑}(S_{y} = +∞) = −r^{+}(y)/r(y) (Lemma (4.3)).

(4.14) Corollary. P^{x}(ρ ∈ dt, γ ∈ dy) = P^{x}(T_{y} ∈ dt)\frac{−dr(y)}{r(y)}.

5. A local decomposition

Fix t > 0 and define

ρ_{t} = inf{u > 0 : X_{u} = H_{t}} ∧ t.

Arguing as in [17] one can show that, almost surely on {t < ζ}, ρ_{t} is the unique u ∈]0, t[ such that X_{u} = H_{t}. Our purpose in this section is to describe the conditional distribution of {X_{u} : 0 ≤ u ≤ t} under P^{b}, given that H_{t} = y, ρ_{t} = u, and X_{t} = x. This conditional distribution has been computed by Imhof [6] for the Brownian motion (and closely related processes). The joint law of (H_{t}, ρ_{t}, X_{t}), again in the case of Brownian motion, has been found by Shepp [16]. See also [2, 9, 15] for related results.

We begin by computing the joint law of (H_{t}, ρ_{t}, X_{t}). Recall from [8; §4.11] that the first passage distribution P^{x}(T_{y} ∈ dv) has a density f(v; x, y) on ]0, +∞[ relative to Lebesgue measure. Note that if we set F_{t,y}(x) = P^{x}(t < T_{y} < +∞), then (see [8; p. 154])

(5.1)  f(t; x, y) = −\frac{∂}{∂t}F_{t,y}(x) = GF_{t,y}(x),  x > y ∈ E, t > 0.

Applying Q^{y}(z, dx) to both sides of (5.1) and integrating over x ∈]y, +∞[∩E (making use of the relation Q_{s}^{y}G = GQ_{s}^{y} on ]y, +∞[), we obtain

f(t + s; z, y) = ∫_{]y,+∞[} Q_{s}^{y}(z, dx)f(t; x, y).

In other words, (t, x) ↦ f(t; x, y) is an exit law for the semigroup (Q_{s}^{y}).

Next, recall from [8; §4.11] that the semigroup (Q_{t}^{y}) has a density q^{y}(t; x, z) (for x ∧ z > y) relative to the speed measure m(dz); we have q^{y} > 0 on ]0, +∞[×(]y, B[)^{2} and q^{y}(t; x, z) = q^{y}(t; z, x). The entrance law for n_{y}^{↑} can now be expressed as

(5.2)  n_{y}^{↑}(X_{t} ∈ dx) = q_{y}^{↑}(t; x) m(dx),

where

(5.3)  q_{y}^{↑}(t; x) = ∫_{]y,B]} n_{y}^{↑}(X_{t−u} ∈ dz)q^{y}(u; z, x).

Substituting (5.2) into (5.3) and using the symmetry of q^{y}, we see that

(5.4)  q_{y}^{↑}(t + u; x) = ∫_{]y,B]} Q_{u}^{y}(x, dz)q_{y}^{↑}(t; z).

But (5.4) means that (t, x) ↦ q_{y}^{↑}(t; x) is also an exit law for (Q_{t}^{y}). Finally, using (3.6), if α > 0 and h is positive, measurable, and vanishes off ]y, B], we may compute

∫_{0}^{∞} e^{−αt} ∫_{E} q_{y}^{↑}(t; x)h(x) m(dx) dt = W^{α}h(y)
= ∫_{]y,B]}[g_{2}^{α}(x)/g_{2}^{α}(y)]h(x) m(dx)
= ∫_{]y,B]} P^{x}(e^{−αT_{y}})h(x) m(dx)
= ∫_{0}^{∞} e^{−αt} ∫_{E} f(t; x, y)h(x) m(dx).

By Laplace inversion,

(5.5)  q_{y}^{↑}(t; x) = f(t; x, y)

for dt ⊗ dm-a.e. (t, x) in ]0, +∞[×]y, B]. Since both sides of (5.5) are exit laws (and so excessive functions in time-space), it follows that (5.5) holds identically for t > 0, y ∈ E^{∘} and E ∋ x > y. See §3 of [5], and especially (3.17) therein.

(5.6) Proposition. For b ∈ E, x ∈ E, u ∈]0, t[, and y ∈]A, b ∧ x[,

(5.7)  P^{b}(H_{t} ∈ dy, ρ_{t} ∈ du, X_{t} ∈ dx) = f(u; b, y)f(t − u; x, y) ds(y) du m(dx).

Proof. Let g, h, and ϕ be bounded positive Borel functions on IR, vanishing off E, E, and ]0, t[, respectively. Put J(v, y, ω) = 1_{{S_{y}>v}}(ω). Using (2.11) we have, since u = ρ_{t}(ω) if and only if u ∈ G(ω) and J(t − u, X_{u}(ω), e^{u}(ω)) = 1,

P^{b}(g(H_{t})ϕ(ρ_{t})h(X_{t})) = P^{b} ∑_{u∈G} g(X_{u})ϕ(u)h(X_{t−u}^{∘}θ_{u})J(t − u, X_{u}, e^{u})

= P^{b} ∑_{u∈G} g(X_{u})ϕ(u)h(e_{t−u}^{u})J(t − u, X_{u}, e^{u})

= ∫_{A}^{b} ∫_{Ω} g(y)ϕ(T_{y}(ω))n_{y}^{↑}(h(X_{t−T_{y}(ω)})P^{b}(dω) ds(y)

= ∫_{A}^{b} ∫_{]0,t[} g(y)ϕ(u)n_{y}^{↑}(h(X_{t−u})f(u; b, y) du ds(y).

The proposition now follows from (5.2) and (5.5).

Our local decomposition of X will be expressed in terms of certain “bridges” of X. First, let K̂^{y,ℓ,x} denote the h-transform of P_{y}^{↑} by means of the time-space harmonic function

h_{ℓ,x}(t, z) = q^{y}(ℓ − t; z, x) [\frac{r(y) − r(x)}{r(y) − r(z)}] 1_{]0,ℓ[}(t),

where ℓ > 0 and x > y. Straightforward computations show that the absolute probabilities and transition probabilities under K̂^{y,ℓ,x} are given by

K̂^{y,ℓ,x}(X_{t} ∈ dz) = \frac{q^{y}(ℓ − t; z, x)f(t; y, z)}{f(ℓ; y, x)} m(dz),

and

K̂^{y,ℓ,x}(X_{t+v} ∈ dw|X_{t} = z) = \frac{q^{y}(v; z, w)q^{y}(ℓ − t − v; w, x)}{q^{y}(ℓ − t; z, x)} m(dw).

Moreover (cf. [15])

(5.8)  K̂^{y,ℓ,x}(ζ = ℓ, X_{ζ−} = x) = 1,
∫_{]y,B]} K̂^{y,ℓ,x}(F) P_{y}^{↑}(X_{ℓ} ∈ dx) = P_{y}^{↑}(F^{∘}k_{ℓ}).

Thus, {K̂^{y,ℓ,x} : x ∈]y, B]} is a regular version of the conditional probabilities F ↦ P_{y}^{↑}(F^{∘}k_{ℓ}|X_{ℓ} = x).

Now let K^{x,ℓ,y} denote the image of K̂^{y,ℓ,x} under the time-reversal mapping, taking ω to the path γ_{ℓ}ω defined by

(γ_{ℓ}ω)(t) = \{ \begin{matrix}ω(ℓ − t), & 0 < t < ℓ \\ ω(ℓ−), & t = 0 \\ Δ, & t ≥ ℓ. \end{matrix}

Like K̂^{y,ℓ,x}, K^{x,ℓ,y} is the law of a non-homogeneous Markov diffusion; from (5.8) we see that

K^{x,ℓ,y}(X_{0} = x, ζ = ℓ, X_{ζ−} = y) = 1.

Moreover, computation of finite dimensional distributions shows that the transition probabilities for K^{x,ℓ,y} are given by

(5.9)  K^{x,ℓ,y}(X_{t+v} ∈ dw|X_{t} = z) = \frac{q^{y}(v; z, w)f(ℓ − t − v; w, y)}{f(ℓ − t; z, y)}.

It follows that {K^{x,ℓ,y} : ℓ > 0} is a regular version of the conditional probabilities

P_{y}^{x↓}(·|ζ = ℓ).

(5.10) Theorem. Let b ∈ E. Then under P^{b} the path fragments (X_{t} : 0 ≤ t < ρ_{t}) and (X_{ρ_{t}+u} : 0 ≤ u < t − ρ_{t}) are conditionally independent given (H_{t}, ρ_{t}, X_{t}) on {X_{t} ∈ E} = {t < ζ}. Moreover, given that H_{t} = y, ρ_{t} = u, and X_{t} = x (0 < u < t, y > x), the above processes have conditional laws K^{b,u,y} and K̂^{y,t−u,x} respectively.

The proof of (5.10) is similar to that of (4.8) and is left to the interested reader as an exercise.

6. A result of W. Vervaat In this last section we use the decomposition of §5 to give a new proof of a result of Vervaat [17] which concerns a path transformation carrying Brownian bridge into Brownian excursion.

In this section we take the basic process (X_{t}, P^{x}) to be standard Brownian motion on IR. Let P_{0} denote the law of Brownian bridge; namely,

P_{0}(F) = P^{0}(F |X_{1} = 0), F ∈ F_{1}.

Under P_{0} the coordinate process is centered Gaussian with continuous paths, X_{0} = 0, and covariance P_{0}(X_{u}X_{t}) = u(1 − t) for 0 ≤ u ≤ t ≤ 1.

Next, Let P_{+} denote the law of scaled Brownian excursion. Under P_{+} the coordinate process (X_{t} : 0 ≤ t ≤ 1) is a non-homogeneous Markov diffusion with absolute probabilities

(6.1)(a)  P_{+}(X_{t} ∈ dx) = \frac{2x^{2}}{\sqrt{2πt^{3}(1 − t)^{3}}} e^{−x^{2}/2t(1−t)}

and transition probabilities

(6.1)(b) P_{+}(X_{t+v} ∈ dy|X_{t} = x) = p(v; y − x) (\frac{1 − t}{1 − t − v} )^{3/2} \frac{y exp(−y^{2}/2(1 − t − v))}{x exp(−x^{2}/2(1 − t))},

where p(v; x) = (2πv)^{−1/2}e^{−x^{2}/2v} is the Gauss kernel, and 0 < t < t + v < 1, 0 < x, y. Also, P_{+}(ζ = 1) = P_{+}(X_{t} > 0, ∀t ∈]0, 1[) = P_{+}(X_{0} = X_{1−} = 0) = 1.

Computation of finite dimensional distributions now shows that the following identities hold:

\begin{aligned}k_{u}(P_{+}(·|x_{u} = y)) & = K̂^{0,u,y}, \\ θ_{u}(P_{+}(·|X_{u} = y)) & = K^{y,1−u,0}, \end{aligned}

where K̂^{0,u,y} and K^{y,1−u,0} are as defined in the last section, the basic process being standard Brownian motion.

Now let Ω_{0} = \{ω ∈ Ω : ω(0) = ω(1−) = 0, ζ(ω) = 1\} and \overline{Ω} = Ω_{0}×]0, 1[. Define a map Φ: \overline{Ω} → Ω_{0} by

Φ(ω, u)(t) = Φ_{u}(ω)(t) = \{ \begin{matrix}ω(u + t) − ω(u), & 0 ≤ t < 1 − u, \\ ω(u + t − 1) − ω(u), & 1 − u ≤ t < 1. \end{matrix}

In the following we regard P_{+} and P_{0} as measures on Ω_{0}. Define \overline{P} on \overline{Ω} by \overline{P} = P_{+} ⊗ λ, where λ is Lebesgue measure on ]0, 1[. Set U(ω, u) = u and V = 1 − U on \overline{Ω}. (6.2) Proposition. The joint law of (Φ, V, X_{U}) under \overline{P} is the same as the joint law of (ω, ρ_{1}, −H_{1}) under P_{0}. Proof. For paths ω and ω′, and t ∈]0, 1[ let ω/t/ω′ denote the spliced path

(ω/t/ω′)(u) = \{ \begin{matrix}ω(u), & 0 ≤ u < t, \\ ω′(u − t), & t ≤ u < 1, \end{matrix}

and let τ_{y}ω(t) = ω(t) − y. Let p_{+}(u, y) = P_{+}(X_{u} ∈ dy)/dy. Note that if ω(u) = ω′(0), then

Φ(ω/u/ω′, u) = (τ_{y}ω′/1 − u/τ_{y}ω),

where 0 < u < 1 and y = ω(u). Thus,

\overline{P} (F^{∘}Φ ψ(V, X_{U})) = ∫_{0}^{1} P_{+}(F^{∘}Φ_{u} ψ(1 − u, X_{u})) du
= ∫_{0}^{1} ∫_{0}^{∞} ∫_{Ω} ∫_{Ω} F (Φ_{u}(ω/u/ω′))ψ(1 − u, y)K̂^{0,u,y}(dω)K^{y,1−u,0}(dω′)p_{+}(u, y) dy du

= ∫_{0}^{1} ∫_{0}^{∞} ∫_{Ω} ∫_{Ω} F (τ_{y}ω′/1 − u/τ_{y}ω)ψ(1 − u, y)K̂^{0,u,y}(dω)K^{y,1−u,0}(dω′)p_{+}(u, y) dy du

= ∫_{0}^{1} ∫_{0}^{∞} ∫_{Ω} ∫_{Ω} F (ω′/1 − u/ω)ψ(1 − u, y)K^{0,1−u,−y}(dω′)K^{−y,u,0}(dω)p_{+}(u, y) dy du

= P_{0}(F · ψ(ρ_{1}, −H_{1})).

(6.3) Corollary. (Vervaat): Define a transformation Ψ : Ω_{0} → Ω_{0} by

(Ψω)(t) = \{ \begin{matrix}ω(ρ_{1}(ω) + t) − H_{1}(ω), & 0 ≤ t < 1 − ρ_{1}(ω), \\ ω(ρ_{1}(ω) + t + 1) − H_{1}(ω), & 1 − ρ_{1}(ω) ≤ t < 1. \end{matrix}

Then Ψ(P_{0}) = P_{+}. That is, the P_{0}-law of (X_{t}^{∘}Ψ : 0 ≤ t < 1) is P_{+}.

Proof. It is easy to check that Ψ^{∘}Φ(ω, u) = ω for all (ω, u) ∈ \overline{Ω}. Using Proposition (6.2),

P_{0}(F^{∘}Ψ) = \overline{P} (F^{∘}Ψ^{∘}Φ)
= \overline{P} (F^{∘}π_{1})
= P_{+}(F),

where π_{1} : (ω, u) → ω.

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