1.1 Review of the Methods and Results

This paper develops a recursion formula for the conditional moments of the area under the absolute value of Brownian bridge given the local time at 0. The method of power series leads to a Hermite equation for the generating function of the coefficients which is solved in terms of the parabolic cylinder functions. By integrating out the local time variable, this leads to an integral expression for the joint moments of the areas under the positive and negative parts of the Brownian bridge.


Introduction 1.1 Review of the Methods and Results
There is considerable literature on the integral functionals of Brownian motion, going back to M. Kac [5].Recently, the results and methods have been unified by M. Perman and J.A. Wellner [9] who also give a good survey of the literature.The pur- pose of [9] was to obtain the law of the integral of the positive part of both Brownian motion and Brownian bridge.In short, they obtained the double Laplace transform of the laws of A + (t): f toB + (s)ds and Ao +" f U + (t)dt, where B(s) and U(t) are standard Brownian motion and Brownian bridge, respectively (Theorems 3.3 and 3.5 of [9]; actually they obtain the double Laplace transforms for an arbitrary linear combination of positive and negative parts).They also found (Corollary 5.1) a recur- sion formula for the moments.These results are obtained from excursion theory, by conditioning on the local time of B at an independent exponential random instant, and appealing to previous known results of Kac, Shepp, etc.Despite the considerable scope of these results, it seems to us worthwhile also to look at what can be done by conditioning on the local time t 0 of U at x--0.In prin- 100 FRANK B. KNIGHT ciple, all of the known results for the integrals of U follow from the corresponding con- ditional law, by integrating over the (known) joint distribution of local time at zero and the positive sojourn.This is because (a) the conditional law of the positive so- journ S + :rF f lI'''(U(t))dt of U, given the local time at 0, is known from P.
Lvy (see L,, Coroltary 1] that paper treats a problem analogous to the present but with the maximum replacing the area integral) and (b) given the positive sojourn S + and the local time 0 at 0, the local time processes of U with parameters x > 0 and x < 0 are independent and distributed as the local time processes of reflected Brown- inn bridges with spans S + and 1-S +, respectively (the corresponding assertions without conditioning on t 0 are.false:given only S +, the local time of U at x >_ 0 is not equivalent in law to the local time of a reflected b:idge of duration S + even if x 0).Accordingly, we are led to look for the law of f ol U(t) ldtlo x), 0 <_ x.
What we obtain below, however, is not an explicit expre'ssion for the law, but a recur- sion formula for the moments (as functions of x).The moments, in this case, deter- mine the law and conversely, but experience in similar cases (for example, that of Brownian excursion; see L. Takcs [14]) has shown that neither need follow easily from the other.Thus, finding the explicit conditional law seems to be still an open problem (as it is also for A0+ but to a lesser extent).
To describe our method, we consider the process defined by x W(x)" -(g(x), 1-/ g(u)du),O < x, (1.0) 0 conditional on t(0)-a > 0, where t(x)is the semimartingale (occupation)local time of IU(t) at x >_ 0. Thus, the second component is the residual lifetime of above x (we note the change of notation-t(0)'-2t 0 from above).Set E: [0, c) (R) (0, 1].It is not hard to realize that W(x)is a realization of a homogeneous Markov process on E, absorbed at (0, 0).This process, indeed, is the subject of a re- cent paper of J. Pitman [10] who characterizes it as the unique strong solution of certain S.D.E., and it appears earlier in the paper of C. Leuridan [8], who obtained the form of the extended infinitesimal generator by an h-path argument.We propose to call this process the "Pitman process".Our requirements for this process are rather different from those of [10].We wish to apply the method of Kac to the area functional y-(u)du dv- v(v)dv IV(u) ldu 0 0 0 given t(0)-a and fF(u)du-y, where U u is a Brownian bridge of span y_< 1.Thus, it is the integral of the second component of our process starting at (a, y).Con- sequently, we need to characterize this process W via its infinitesimal generator, as a two dimensional diffusion whose semigroup has the Feller property.Much of this may be obvious to a very knowledgeable reader, but it provides orientation and it seems to us that the methods may be more widely of use.In any case, the reader who can accept Corollary 1.3.5 (with A given by (1.2)) could go direction to Section 2.
We need the results of [10, Proposition 3, Theorem 4] only to the extent that there exists a diffusion process W (X,Y) (a strong Markov process with continuous paths) on E U (0,0) starting at (x,y) and absorbed at the state (0,0) at time To: inf { t > 0 "f toX(u)du y} < oc, of which the process (1.0) is a realization with x c, y-1, and the law of X(. for this process is weakly continuous in its dependence on (x, y).We also rely on the stochastic differential equation of [10] to determine the form of the generator of W, (see [8] for an alternative method).
Finally, we also need the scaling property [10, Proposition 3 (iii)].Let px, y denote the law of W starting at (x,y)E E. Then the equality of law PX'u{X(" e } Px/Y/'I{v/X(" /V/' e } (1.1) holds.
Our main assertion concerning W is as follows.
Proposition 1.1.1-ForN > O, x <_ N, let WN(t denote W(t A TN), 0 <_ t,where TN:-inf{t:X(t)-g}, and let E N denote [O,N](R)[O, 1] with the segment {(x,O), 0 <_ x <_ N} identified to the single point (0,0) and the quotient topology.Then W N has law that of a diffusion on the compact metrizable space E N absorbed at {x-N} [J {y-0}, whose semigroup has the Feller property on E N and is strongly continuous at t-O, and with infinitesimal generator extending the operator (02( x2--x 0 ) 2(Ev (12)   Af(x,y)" 2xx 2+ 4-y jox-f(x,y) for f eC c (interior compact support).
Remark 1.1"It seems non-trivial to ascertain the behavior of W starting at (x, 1) as xc (probably absorption at (0,0) occurs instantly).Hence the need for W y.
One might hope to appeal to the fundamental uniqueness theorem of Stroock and Varadhan (as stated, for example, in Rogers and Williams [13]), but there are insuper- able obstacles.To wit, the operator A is not strictly elliptic, the coefficients are un- bounded at y 0 and at x c, and A is undefined outside of E.
The proof of Proposition 1.1.1occupies Section 1.2 below.It uses a coupling argument, together with an extension of a strong comparison theorem of T. Yamada.
It seems of interest that this last, originally stated only for diffusions on R, extends without any difficulty to the Pitman process on R 2 (Lemma 1.2.1).Knowing that we have Feller processes to work with, while not indispensable, makes for a neuter treat- ment of Kac's method in Subsection 1.3.The form which we develop is doubtlessly familiar to many specialists, but we give a complete proof which should be adaptable to other analogous situations.In principle, the method applies to give Hu(x):--# f o Y(Xs)ds) whenever X is a Feller process absorbed on a boundary 0 at EXexp( T time T < cx, and Y(x) is sufficiently tractable.It then characterizes H,(x) as the unique bounded continuous solution of (A-#V)Hu-0 with H u 1 on 0, where A denotes the generator of X.In other words, H u is harmonic for the process X killed according to #V.
In Section 2, we specialize to the case when V(x,y)-y, and X is the Pitman process absorbed on {x-N or y-0}.We write H u 1/ n 1( #)nan(x'Y)' and try an expansion a n Ybn k(x,y)x k.Then a scaling argument leads to 1 y3n/2 bn, k (xy )kCn, k, where ca, k are constants, and the problem reduces to determining Gn(s)" = oVa, ks k.Some power series arguments lead (tentatively) for 0 _< n.The key to the solution for n > 1 lies in Lemma 2.3, where it emerges that Ks(s):-sGn(s solves the inhomogeneous Hermite equation (2.13) (this remains a surprise to us).. Since the forcing term (-1/2Gn_ (s))turns out inductively to be a finite linear combination of eigenfunctions (G o 1), this makes it possible to express the unique bounded solutions G n inductively in n, by a recursion formula for the coefficients (Theorem 2.4).This is our main result, but to establish it rigorously, by proving that the series for H o converges uniformly and absolutely on E and satisfies the uniqueness conditions of Kac's method, occupies the rest of Section 2. Since the series is not summed explicitly, we do not find H o in an invertible form, but it yields 1 the conditional moments, namely n!y3n/2Gn(xy-'), 1 <_ n.The recursion formula (2.17) for the coefficients is not particularly simple, but no doubt it can be program- med on a computer if high-order moments are desired.
In Section 3, we derive closed form expressions for the moments of the areas of the absolute value and the positive part of a Brownian bridge in terms of the coefficients in Section 2. These are not as simple as previously known recursion (see [9]), but they are simpler (perhaps) given the coefficients of Section 2. Anyway, they provide more checks on Section 2, and the method leads in Theorem 3.6 to integrals for the joint moments of the areas of the positive and negative parts of Brownian bridge.These can be done explicitly in the simplest cases, but the general case (which hints at orthogonality relations among the parabolic cylinder functions) is beyond our capa- bility.
1.2 Proof of Proposition 1.1.1 Let us show first that T N A T O tends to 0 uniformly in probability as (x,y) tends to the absorbing boundary {x N} U {y 0} of E g.There are really two separate pro- blems here: one as x increases to N and the other as y decreases to 0. For y > 5 > 0 as x---,N the coefficients of A near {x = N} are bounded, in such a way that one can read off from the meaning of A the uniform convergence in probability of T N to 0.
Unfortunately, to make this rigorous seems to require comparison methods as in Lemma 1.2.1 below (adapted from the one-dimensional case).Once the comparison is established, the convergence reduces to a triviality for one-dimensional diffusion with constant drift and need not concern us further.
The problem as y---0 is more interesting, and here it suffices to show that T O tends to 0 in probability as y0 / uniformly in x (for W, not for W N). For c > 0, 1 1 let E:-{(x,y) eE'xy 2}, and let R-inf{t>0"XtY}.Thus Ris the passage time to E, and it is a stopping time of W. We show first that T 0 A R 1 1 (dYy-7_ _XtY7 < _e for tends to 0 uniformly in x. Indeed, since k dt] < T 0 A Re, we have for the process starting at > e(T o A R). Thus T 0 A Re < 2e 19 uniformly in as asserted.Consequently, we see by the strong Markov property at time T 0 A Re, that it suffices to show that T 0 is uniformly small in probability for (x,y) E {y < e} as 0 +.
To this effect, we use the scaling (1.1) noting first that the process Y(. may be Moments of the Area 103 included on the left if we include yY(./V@)on the right.Indeed, Yt-Y-f toXsds, */X/, 1 which for P is equivalent to y-fov/X(/v/)d, whiCh equals y(1- fto/V/-Xsds -yY(t/v/-) as asserted From this, it is seen that the PX'U-law of T 0 equals the P -law of y2 To, and since xy 2< and y2< it is enough to show that lim px'I{T o > N}-0 uniformly for x < e small.Here we can use the fact discussed in [10], that the P'l-law of X is that of the local time of a standard Brownian excursion.As such, it does not return to the starting point 0 until 'time' To, i.e., T o is the excursion maximum value.Consequently, for small e > 0, 1 Then denoting the event in brackets pO, l{xtY 2 reaches e before Yt reaches 1/2} > . 1 by S e, and setting U(e)-{inft > 0: X Y2te } <_ oc, we have by the strong Markov >_ Pe'I{T 0 > Nv}P(Se) >_ -P {T O > Since this uniform in e (small), the assertion is now proved.
To derive the form (1.2) for the infinitesimal generator, it is enough to take n.= oc and consider the semigroup of W acting on the space %b(E) of bounded, Borel functions.Then from [10, p. 1], for (z, y) E E with y > 0, the PZ'U-law of W is that of the unique strong solution of dX u (4-X2u/Yu)du + 2V/XudBu; dY u X.du; (Xo, Yo) (x,y), (1.3) where the solution is unique up to the absorption time T o at (0,0). 2 0 formula for u < To, we have PX'U-a.s. for f Cc(EN) Since f z vanishes near y-0, we can take expectations to get u-l(E'Z(W,)-Z(x,y)) u u E '/ (2f(W,)X v + f(W,,)(4 X v/Yv)-fu(Wv)Xv)dv.
0 Let e-dist.(bdryE to supp f)> 0, where distance and boundary are Euclidean (without identifying the line y-0) and let C-{(x y) E:dist((x,y)suppf)< -} 3 Starting at (x,y) E-g, the process must first reach g before reaching suppl.
.6) u--O s < u Similarly, for (x,y)e C, (1.5) tends uniformly to Af(x,y) provided that (1.6) holds for every c > 0. Thus the assertion (1.2) for the (strong) infinitesimal generator follows if we show that (1.6) holds for all c > 0. Reducing e if necessary, the coeffi- cients of A are uniformly bounded on an -neighborhood Ca/3 of C, and the "local character" assertion (1.6) is familiar for diffusion, at least in one-dimensional.Unfor- tunately we lack a reference for dimension exceeding one so for the sake of complete- ness we sketch a proof by reduction of A to the one-dimensional case (fortunately A is 'almost' one-dimensional).Indeed, the a.s.identity Y(t)-b-f toX(u)du t< To, shows that it suffices to prove (1.6) with Xs-a in place of IWs-(a,b) l.To this end, choose constants 0 < c < d such that c < 4-x2/y < d holds on .., and let X (1) Cres, X (2) .e/-" be the solution of (1.3) starting at a but with c (rasp.d) replac- ing the coefficient 4-x2/y (using a single Brownian motion B throughout).
The last statement of Proposition 1.1.1pertaining to {x-0} is known for pO, u since then X has the law of an excursion local time (see [10]).For px, u it then follows from tP O'y and the strong Markov property at the passage time to x.It remains to discuss the Feller property of the semigroup.Since W was shown to be (,u) absorbed at (0,0) for P uniformly fast as y--,0+, it is clear that we must identify the segment {(x, 0), 0 < x < N} with (0, 0) in order to preserve continuity on the boundary of E N. It is well-known and easy to check that the absorbed process W N is again a diffusion (on EN).Let Tt N denote its semigroup on %b(EN).Then by the above remarks, for f EC(EN) and t>0, lim(x,u)(Zo, uo)Tf(x,y)f(xo, Yo) uniformly for (xo, Yo) absEN, where E N is compact with absEN:--{(N,y), 0 < y _< 1} U {(0,0)}.Actually, the segment {0, x < N, y 1} has yet to be discussed but it is obViously inaccessible except at t-0, and there is little difficulty now in seeing that limt__.0+ TtNf(x,y)f(x,y) uniformly on E N for f E C(EN) since limu__,0 + f(x, y) f(0, 0) uniformly on 0 _ x _ N (here we can resort again to P'{IX the comparison argument as in Lemma 1.2.1 to show that ]imt__,0 + x > c}--0 uniformly on E N-{y _ c}).In other words, we have strong continuity of Tt N at t= 0, and it remains only to show that, for f e C(EN) and t> 0, TtNf is continuous on EN absEN.
1 Now let (xn, yn)---,(x, y) E N -absEg, and for each n define two independent pro- cesses Wv and W N on the same product probability space, where yn) and WN(0) (x, y).Let where (Xv Yv)-Wv, etc.Since 0 < YN(O)-YN(t)< Nt, it is seen (for example, using Lemma 1.2.1) that limn_,oT n 0 in law, and of course, each T n is a stopping time for the usual product filtration hn(t).Then we have, if Ill < c, Exn'Unf(WN(t))-EX'Uf(WN(t))l < 2cP{T n >_ t} n T + E(E wy( n)f(t-Tn)-EWN(Tn)f(t-Tn)); Tn<t) l. (1.8) As noc, the first term on the right tends to 0. Setting 1 1 Zn(Tn): XV(Tn)(YV(Tn)) 2-XN(Tn)(YN(Tn)) 2, the difference in the second term becomes by (1.7) YN/v/yv(Tn)((t--Tn)/v/Yv(Tn) f (analogous)] (1.9) where (analogous) has the scale factor YN(Tn) in place of Yv(Tn).Now since Tn-0 in probability, it is clear that Yv(Tn)y and Y N(Tn)--,y in probability, and then by the remarks following (1.7), we see that the difference (1.9) tends to 0 in pro- bability, viz. each term converges in probability to EZn(Tn)'lf(x/XN/v/(t/x/ ), yYN/f:.(t//r))Vas n--oc, and since it is also bounded, (1.8) tends to 0 and the proof is complete.Remark 1.2:It is possible, but tedious, to show that the law of a Feller diffusion on E N absorbed at (x-N} 12 (y-0} and with strong generator satisfying (1.2)is thereby uniquely determined.For an indication of a proof, we observe that for 0 < e small, the coefficients of A satisfy a Lipschitz condition on E N ; [,N-] (R) [, 1), in such a way that they may be extended from N,o R 2 ac satisfy the conditions of [13, V, 22].Thus if (1.2) for the operator A determined by the extended coefficients is assumed on R2, there is a unique diffusion on R 2 which gives the unique solution to the "martingale problem."By optional stopping, this process absorbed on {x-} 12 {x-N-e} 12 {y-e} solves the martingale problem on N,e, and it is the unique such solution because any such can be extended to a solution on R 2 using the strong Markov property on the boundary.It remains only to let e-en--0, and to form the projective limit of these diffusions to obtain the law of WN uniquely.
1.3 A Form of M. Kac's Method for Functionals of an Absorbed Process We turn now to establishing a variant of Kac's method for obtaining the law of func- tionals of W N. For this we need to introduce a "killing" of W N according to the de- sired functional.But as an introduction to the problem we first make some observa- f e C(EN)VC2(Ev) and Af 0 on EV (the interior of EN).Suppose first that f e Cc2(v) (compact support).Then f is in the domain of the strong generator and by Dynkin's Formula we have Ex'YI(WN(t)) f(x,y)+ Ex'YfoAf(WN(s))dsf(x,y). Thus the martingale property follows by the Markov property of W N. Now supposing only f e 2(E), set EN, [,N ] (R) [, 1 ] and note that by stan- dard smoothing argument there is an fee c2(Ev) with f-f on E N It follows by optional stopping that, for (x,y) EN,, f(WN(t ATe) is a px'," Y-martingale where T" inf{t _> 0" WN(t {x or N-e} 12 {y e}}.Now f is uniformly bounded and, by continuity of paths we have limT-Tabs: ---0 inf{t > 0:WN(t e absEN} for (x,y) e EV.It follows that for (x,y) e Ev, f(WN(t/ Tabs) is a PX'Y-martingale.Since WN(t WN(t/ Tabs) and the result is trivial for (x,y) absEN, this finishes the argument except for px,1.But, of course, the Markov property for p,l shows that f(WN(t / )) is a martingale given {WN(S),S _< } for every > 0, and along with right-continuity of W N at t-0 this suffices trivially.
lemark 1.3.1"The converse assertion that if f is W N-harmonic then Af-0 on E is probably valid, but it is not needed for the purposes here.For applications it is the solutions of Af-0 which give the "answers."We note also the expression f(x,y)-E'Yf(W(Tabs)), which follows for WN-harmonic f by letting tcx under Now fix a Y(x,y) C + (EN) and let Eg,/x-E N 12 {/k}, where /k is adjoined to E N as an isolated point.For each pz, y, (x,y) EN, let e be an independent exponential random variable adjoined to the probability space of W N, and introduce: Moments of the Area 107 Definition 1.3.2:The process W N killed according to V is WN(t); < T( X)
Noting carefully that there is no "killing" (passage to A) on absEN, so that T(A) for pz, y if (x, y) e absE N, we have: Threm 1.3.3: With the initial probabilities px, y from WN, (x,y) eEy, WN, V(t becomes a Feller process on EN, A, strongly continuous at t-O, with con- tinuous paths except for (possibly) a single jump from E N to .The (strong) in- finitesimal generator is given by Avf:-A(f)-Vf for f C2c(E,A), (x,y) EN, and Avf(A O.The process is absorbed on absE g A. Prf: (Sketch) The killing formula used here goes back to G.A. Hunt, and is well-known to yield a strong Markov process from the Feller process W N. In proving the strong continuity and the Feller property, the main thing to use is that the killing occurs uniformly slowly on EN, i.e. limpX'y{T(A) < ) 0 uniformly on E N. 0 This not only suffices to derive the strong continuity at t 0 from that of W N, but it also preserves the main point of the coupling argument used to prove the Feller property, namely, that thecoupling time Tn( ) tends to 0 in lw when in its definition W N is replaced by W N, V.But a difficulty arises with the analog of (1.9) since, for general V, W N, y does not obey the scaling property (1.7).Instead, we have to introduce the killing operation on the paths (YN(Tn)Xn ((t-T,)/4Y(T)), etc.)in (1.9)starting at (X(T,),Y(T)), and analogously without superscript n.But there will be no change in the result if we use the same process W starting at (Z(T), 1)in both terms of the difference.In other words, we base the two futures after T n on a single process W(t) (using the probability kernel Zn(Tn)P Zn(Tn)' to define the conditional law of the future at T n given n(Tn), in the usual way for Markov processes).This implies that in introducing the killing 1 operations into the two terms we use the path with scale factor (Y(Tn)) for the 1 first, and (YN(Tn)) for the second, but the same path W(t), W(O)-(Zn(Tn),l), for each.Then convergence in probability of the scale Mctors to y implies that the (t-Tn) AT killing functionals converge in probability to f0 absv( fiXN/(s/ fi), yYN/,(s/))ds, i.e. their difference converges to 0. If we use (as we may)the same exponential random variable e to do the killing for both, it is clear that, with conditional probability near 1, either both are killed by time t-T n or neither, in such a way that the Feller property holds for the semigroup T' v of W N, V" Turning to the assertion about the infinitesimal generator of T' v, note first that f() 0 for f G C(E%, y)" We have t-l(TtN, Vf f) t-1 N -1 (T f f) E (f(WN(t)); WN, v(t) on E N, while the same expression is 0 at A. The first term on the right converges to A(f) uniformly on E N. Using (1.6)as before, we may assume IWN(t)--(x,y) < , with error o(t) uniformly on e as t-.0.Then f(WN(t)) may be replaced by f(.x,y) for small t, uniformly.onET, and we are left with f(x,y)t-lp (x'y) (fotA''absV(WN(s))ds > e).This'vanishes outside supp f, and P(X'U)(Tab s < t) o(t) uniformly on supp f.Thus we can extend the integral to t, and then p(X,U V(WN(s))ds > e 1-EX'Yexp V(WN(s))ds uniformly on E N.This completes the proof, the last assertion being obvious.
We come now to the key method (of Kac).
Theorem 1.3.4:Continuing the otation of Theorem 1.3.3,for # > 0 and (x,y) E E N set H. (x,y)-EX'Yexp(-#foabSV(WN(s))ds).Suppose there-exists anH G (EN) NC(EN) with A(H)-#VH=O on EN and H=l on absE N. Then H- Ht on E N. (x,y Proof: We have Hu(x,y P ){WN, uV reaches absE N before time T(A))- P(X'U){T(A) c}, when T(A) is defined for #V in place of V. Clearly, H u 1 on absEN, and if we set Hu(A --0 then H u is harmonic for the process WN, uV at least if it is continuous.Indeed we have Hu(x,y EX'UH (W N V(TabsAT(A))), and it follows by the Markov property of WN, uV that Hu(WN, uv(t)) is a px, y_ martingale for (x, y) E N.
On the other hand, if we set H(A) 0, then H H u on absE N t.J A, and we claim that H (being continuous by assumption is harmonic for W N, uV.As in the discuss- ion above for WN, this is taken to mean that H(WN, uv(t)) is a PX'U-martingale, (x,y) E N. The proof is much the same as above for W N (see (1.10)) only now the martingale has a (possible) jump.In short, using ENe = [e,i-e] (R) [e, 1 c] as before but H in place of f with H = H on EN, e, H C2c(EN) and He(A = 0, optional stopping of the martingale He(Wg, y(t)) for (x,y) G EN, e shows that H(WN, uv(tATe, uV)) is a PX'U-martingale, where Te, uy: =TeAT(A).Now e--,01imTe," V Tabs A T(A), and e01imH(WN, uy(t ATe, uV)) H(WN, uV(t)) except on the PX'U-nullset where Tabs --T(A).By dominated convergence of conditional ex- pectations, H(WN, uV(t))is PX'U-martingale if (x,y) EN The assertion is trivial for (x,y) EabsENUA and letting tcx) we have H(x,y)-EX'UH(WN, uV (Tab A T(A))).This is the same as with H u in place of H, so the proof is complete (except if y-1, but that case now follows from the continuity of H, which implies that H u extends by continuity to y-1).
What we need for Section 2 below is a form of Theorem 1.3.3applying to the case N-cx, or rather, to the limit as N---,.This means replacing W N by the process W of (1.0) ff.We first modify the definition of E slightly, by identifying the line {(0,x),0 _< x < oc} with (0,0), so that E u C E with the relative topology.We do not compactify E; however, we know from [10] that on E P(X'U){T 0 < oc} 1, follows as in the comparison Lemma 1.2.1 that Px'Y{X _ X2),.t _ To}-1 for d 2 (x,y) e E, where X 2) is a diffusion with generator 2x2 + 4d--on R +.For X 2) there are no "explosions" (oc is inaccessible) ([6, 4.5]) and it follows by comparison that as N--<x, px, y {X reaches N before To) tends to 0 uniformly on compact sets of E. It follows easily that the semigroup T of W preserves Cb(E (but of course, it is not strongly continuous at t 0), and its infinitesimal generator has the form Af for f e 2c(E), A as in (1.2), just as for TtN.
For Y e b + (E) (bounded, continuous on E) we define W y from W just as in Definition 1.3.2 for WN, V, where Tabs is replaced by T 0. The scaling (1.7) remains valid for T (only it is a little simpler here without absorption at N), and the coupling argument remains valid to show that the semigroup T of Wv preserves Cb(EI; Likewise, the argument after (1.9) for the generator of t N'V go'es through for T[.Thus we see that Theorem 1.3.3carries over to W y with only the changes noted: the generator is Ayf for f e c2(EA) and the process is absorbed on (0, 0) U A. We also have the analog of Theorem 1.3.4TaSfollows: Corollary 1.3.5:Set Hv(x,y)= EX'Vexp(-#fV(W(s))ds), (x,y)e E, with Y e Cb + (E), # >_ O. Suppose there exists an H E Cb(E gl C2(E0) with A(H)-

#VH-O one and H(O,O)-l. Then Ho-H onE.
Proof: There is nothing really new here, but it recapitulates the former proof.
We have Ho(x,y)-PZ'U{To<T(A)) where T(A) corresponds to #Y, and so gu(x,y) is Wuv-harmonic (if we set Hv(A)--0 apart from continuity considerations.On the other hand, H is continuous by assumption (and we set H(A) 0).Since g H v on (0, 0) U A and gv(z, y) E x' vg (Wvv(To A T(A))), it suffices to show that H(x,y)-EX'UH(Wvv(ToAT(A))), (x,y) e E. We note from the definition that if we absorb W#v on {x-N), for (x,y) E g we get a process with PX'V-law the same as WN,#V (actually, #V restricted to (x,y) EN).
The proof of Theorem 1.3.4shows that H(WN, vV(t)) is a PX'U-martingale.Now for (x,y) e E, W N, vV and W ov )coincide for g sufficiently large, (depending on the path), so it follows by bounded convergence of conditional expectations that H(Wuv(t)) is a martingale for px, v. Then letting tc we obtain the assertion.

Derivation of the Conditional Moments
We now specialize Corollary 1.3.5 to V(x,y)y, writing for brevity H for H T5 n, noted following ds) l(O)= x) where U is a Brownian bridge of term y, and (v) the loom time at Thus H. (x,y) is the Laplace transform whose inversion gives the law of of U, I.id s e,(0 ) (5ve,(v)dv e,(0) 1.We do not solve (A (f v() py)H,-0 per st, but instead we assume an expansion H,(x,y)-(f Y(v)dv) n, and then solve recursively for the terms.It is shown that the series converges for R and satisfies the conditions for H. Hence the expansion is justi- fied and the conditional moments are and they determine the (conditional) law of (f'glUu(s) lds eu(O) x).
In order not to prejudice the notation, let us write formally Ht(x, y) 1 + #)nan(x y), (2.0) and try to solve for the functions an(X,y).
(1.1) that we have First we note from the scaling property Hu(x,y H 3(xc-l,yc-2),O < c.
(2.2) lmark: We do not need to justify (2.2) rigorously, because it leads to the explicit solution, which is unique and verifiable.
Recalling (Corollary 1.3.5)that the equation satisfied by H u is )Hu (2.4) and we are lead to guess the existence of solutions in the form a0(x y) 1, an(x'Y)-Z bn, k (y)xk; 1 < n k=0 4 2 d when replaces t).
(by formal analogy of A with the heat operator t'

Y dx 2 d
We remark that an(O,y)(-b n 0(Y)) should be the n TM moment over n! of the 3n integral of the Brownian excursion of length y.By scaling this is cny 2 where M n cn:--.-! with M n denoting the n th moment for the integral of standard Brownian excursion.These moments figure prominently in Takcs [13], where Mk, k <_ 10, are tabulated (Table 4) and a recursion formula is given.Here they provide a check on our answers.When we work out the b n k(Y) by power series method, it turns out 3 that the series of even and odd terms commence with bn, o(y -cny and bn, l(y cn.-lyn (where c o -1), respectively, where at first the c n are arbitrary constants whos4e identity is known only from the (assumed) excursion connection.However later on, when we sum the series in terms of parabolic cylinder functions D_ n, it emerges that the values of c n are dictated uniquely by the behavior (limit 0) of the solution as y--0.Thus it turns out that the case of the excursion (x 0) follows as a consequence.
When we substitute the series (2.5) into (2.4), it emerges that there is a solution in (an k), Cn, O_Cn Cn,1 Cn-1.Indeed the form b,,k(y)--Cn, ky__ 2 so that and 4 granted a solution of the form (2.5), this form for bn, k follows by the scaling (2.2).
Thus we expect 3noo (1) k an(X,y y-T E Cn, k xy --(2.6)k=0 and the summation of the series reduces to identifying the generating functions Gn(s): E c,,k sk; Go(s 1. (2.7) k=0 Let us go through the case n-1 directly (although it is a consequence later of more general considerations) since (unlike n-2) we can derive the result by direct summation of the series, and it shows where the functions D_ n come from in this problem.
As for the uniqueness assertion, let fi'l denote the difference of two such solutions.
Then A 1 -0, and it follows as in Section 1 that l(W(t))is a bounded martingale for px,, (x,y) e E. Since Y(t)-f Tt(X(s))ds we see that as tTT o (and we have Y(t)< (T o-t)X(t) when X(t)-maxX(s), hence for a random sequence tn-*TO.Then X(tn)/Y(tn)-*c, and so l(W(tn))-*O.Hence ffl(W(t))-*0 as t-*To, and it follows that ffl(X, y) 0 as asserted.
In view of the extra k in the denominator of c,,k, we can see by induction on n that the series G,(s) converges absolutely for all real s.But we will find G j(s) M.
TO bounded for s>0 if and only if c= .--n, j<n, and then n!a,(x,y)-EX'(fo J 3. t-Y(v)dv) n.It also can be seen from (2.12) hat whenever cj > 0 for all j, we have ca, 2k > 0, ca, 2k + 1 < 0 for all n, k.
Taking generating functions of both sides of (2.12) leads to: 0 <_ n.
The general case is quite analogous, only complicated.The following is the main result of our paper.
Indeed, such H u satisfies the boundary condition H. (x,o)= 1.Note that the, limit 1 is even uniform in x as soon as n= 1(-/t) Gn(s s bounded for s > 0, because 3n of the powers y2, 1 _< n, in an(x,y).Moreover, it is not hard to see that 0 <_ Gn(s)O as sc for each n _> 1: we have remarked above that dk, n are nonnega- tive for all k, n while by the representation as transform of Orntein-Uhlenbeck pass- age probabilities to zero (see (2.14) ft.) it follows that 0 <_ (exP6)D_ j()0 as sx for 0_< j.
lemark: Since Gn(s has its maximum on R + at s-0, where Gn(O is (n!)-1 times the n th moment M n of the integral of the Brownian excursion, it would seem that convergence of the series could easily be proved by using the recursion formula of Takcs [14] for the excursion moments.But this argument is circular-we do not know that Gn(O has the required interpretation until our solution is rigorously esta- blished. Lemma 2.6: The series n n 1(-#) Gn(s) converges uniformly and absolutely in s >_ 0 for every # R and may be differentiated (twice) term-by-term for s > O.
Recapitulating, we have proved: Theorem 2.7: With Gn(s as in Theorem 2.4, we have for 0 <_ where Uy is a Brownian bridge of span y and y(O) is twice its local time at O.
3. Points of Contact with Perman and Wellner [9] As in [9], we set Ao + -fU +(t)dt, A fU-(t)dt, where U is a standard Brownian bridge.Thus we have f u(t)dt Ao + Aand f U(t) dt Ao + + A0-: A0, where the first is Gaussian with mean 0, hence its law is easy to find, while the law of the second has been obtained through a series of papers see [9,   Introduction].The moments #k: EAko follow from a recursion formula of L. Shepp (see [9, Theorem 5.1]), while #k:-E(Ao+)k follow from a recursion formula of Perman and Wellner [9, Corollary 5.1].By combining the above results, one can derive the joint moments #,,.n: E((Ao+)m(Ad-)n) for m + n _< 5.For example, 1 # E(Ao + A )2)'But for m + n > 5 the method does not suffice because Pl,1 ( 2 in the expansion of E(Ao + -bA d) there are already four distinct joint moments #m,n with m + n-6 (discounting 3 symmetries) while there are only three 'knowns' (including #g ).
All of these moments (including arbitrary (re, n) in #m,n) follow in principle by our method, by integrating out over the local time and bridge span variables (z,p).But if we require explicit expressions no involving integrals (i.e., that the integrations involved be done explicitly) a surprising fact emerges" we can do the integrations precisely for the bridge moments treated already in [9] neither more not less.And besides, our comprehensive recursion formula of Theorem 2.4 is doubtless more unwieldy to apply then the separate recursions noted above, which (apparently) do not follow easily from it.Thus our results, at present, seem to provide a strong vindication of the results of [9] for the moment problems treated here. 2 The density of el(0 (the local time of U at 0)is given by 4-1xexp(--), x 2 according to P. Lvy (see [7, p. 236]), and so #-n!f4-1xexp(--)G(x)dx.
Remark: Using the values of d j, n from Table 2.1 for n _< 4, we get values 19 in agreement with Table 1 of [9] (which continues to n-10).
To handle the moments #_ (or #-analogously) we note from [10] as in (1.0), that the pa, U_law of X is the same as that of the (spatial) local time process of [B[   at the time when it reaches a at 0, conditional on that time being y.Suppose that in this statement we replace [B[ by the reflected Brownian motion obtained from B by excising the negative excursion, i.e., B(z(t)) where f(t)I(o oo)(B(u))du t.Since this has the same law as B[, it also has the same local tme process (equality in law).But now the time when the local time at 0 reaches a is, in terms of B, the time spent positive at the instant T(a) when the local time of B at 0 reaches a.The local time of B(z(t)) at z_> 0 at time T(a)is the same as that of B (since the negative excursions do not contribute).Thus pa, is also the law of the local time of the process of positive excursions of B (holding the value 0 during the negative excursions) at T(c) given that the time spent positive equals y at the instant T(a).Since it is entirely defined from B(z(t)) is not hard to see that this remains true if the time spent negative at this instant, say z, is also given, in such a way that up to time y + z, B is in law a Brownian bridge of span y + z given that its local time at 0 is a and its time spent positive is y.But here y+z may be fixed (>_y), and in particular we may set y + z-1.Thus we have shown the following (see also [10, Corollary 18])" Lemma 3.2: The law Pa'u of (1.0) ff. is also the law of the local time process in parameter x >_ 0 of the standard Brownian bridge U conditional on its local time at 0 being a and its time spent positive being y.
Next we need the law of time spent positive by U conditional on local time a at 0. This is easily recognized as the time spent positive by B at time T(a) given T(a)-1.As noted by P. Lvy, this is the law of T() given Tl()+T2()- 1, where Gl(S -exp(])D_ 1(), where D_n(s denotes the parabolic cylinder function 102 FRANK B. KNIGHT property P'I{T o > N} k E'I{pX(U(e))'Y(U(e)){To > 1 E,I(p(e'I){y-ff(U(e))To > 1 el NV}" to the solutions (exp-)D_,x(x) and (exp-)D_.x(-x).For A > 0, the only 2 solutions bounded for x > 0 are c(exp-)D_.x(x). (For a discussion of the eigenfunc- FRANK B. KNIGHT tions of the Hermite equations, see[1]).