B Roynette

M Yor

Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX

Keywords: Limit theorems for additive functionals, Feynman-Kac functionals, long Brownian bridges. 2000 Mathematics Subject Classification 60

We obtain a local limit theorem for the laws of a class of Brownian additive functionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional : A - t := t 0 1 Xs<0 ds, t ≥ 0 . On the other hand, we describe Feynman-Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [RVY,I]).

0 Notations and introduction 0.1 Notations

• Ω, X t , F t t≥0 , F ∞ = ∨ t≥0 F t , P x (x ∈ R) denotes the canonical realisation of the onedimensional Wiener process. Ω = C [0, ∞[-→ R is the space of continuous functions on [0, ∞[, X t , t ≥ 0 the coordinates process on this space, F t , t ≥ 0 its natural filtration and P x , x ∈ R the family of Wiener measures on Ω, F ∞ , with P x (X 0 = x) = 1. When x = 0, we write simply P for P 0 .

• We denote by L x t , t ≥ 0, x ∈ R the jointly continuous family of the local times of X t , t ≥ 0 . We denote L t , t ≥ 0 for L 0 t , t ≥ 0 , the (continuous) local time process at level 0 and by τ l , l ≥ 0 its right-continuous inverse :

τ l := inf s > 0; L s > l l ≥ 0 (0.1)
• To q a positive Radon measure on R, q = 0, we associate the continuous additive functional :

A q t := R L x t q(dx) (0.2)
When q admits a density with respect to Lebesgue measure, we keep the former notation by still writing q for the density; we have :

A q t = t 0 q(X s )ds (0.3) from the occupation density formula for Brownian motion.

Throughout the following, we shall assume that q satisfies one of the three following hypotheses :

H1. (The integrable case) R 1 + |x| q(dx) < ∞.

H2. (The left unilateral case) 0 -∞ 1 + |x| q(dx) < ∞ and there exists α < 1 such that lim x→∞ x 2α q (a) (x) ≥ b > 0 where q (a) denotes the absolutely continuous part of q.

H3. (The right unilateral case) ∞ 0 1 + |x| q(dx) < ∞ and there exists α < 1 such that lim x→-∞ |x| 2α q (a) (x) ≥ b > 0.

Of course, if the pair (X t , A q t ), t ≥ 0 satisfies H2 (resp. H3), then the pair (-X t , A q t ), t ≥ 0 satisfies H3 (resp. H2).

Introduction

0.2.1

In [RVY, I], we obtained the following results : i) Under H1, H2 or H3, for any λ > 0 :

lim t→∞ √ t E x e -λ
2 A q t = ϕ λq (x). (0.4) (Later, we shall give other presentations of ϕ λq ).

ii) For any s ≥ 0 and Λ s ∈ F s :

lim t→∞ E 1 Λs e -λ 2 A q t E e -λ 2 A q t := Q (λq) Λ s . (0.5)
where formula (0.5) induces a probability Q (λq) on Ω, F ∞ (see [RVY,I] for more details about this penalisation result).

The first part of this work consists in :

• Using the result i) to obtain a limit theorem relative to the law of the additive functional A q t , t ≥ 0 . This is the content of Theorem 1.1.

• Obtaining a penalisation result, which is more general than (0.5) i.e, by replacing the exponential function x -→ e -λx 2 by a more general function. This is the content of Theorem 1.5.

0.2.2

In Section 2 of this work, we study in detail the situation where q = 1 ]-∞,0] i.e, A q t := A - t := t 0 1 Xs<0 ds. In particular, we prove a penalisation theorem for long Brownian bridges : this is the content of Theorem 2.1.

0.2.3

Section 3 of this work is devoted to the study of Feynman-Kac penalisation for long Brownian bridges, which generalizes what we have done in [RVY,I] for standard Brownian motion. This is the content of Theorem 3.1.

To summarize, this work extends, in the above directions, our preceding work [RVY,I].

1 A local limit theorem for the laws of some Brownian additive functionals and a penalisation result 1.1 A local limit theorem Theorem 1.1. Let q satisfy one of the hypotheses H1, H2 or H3, and let (A q t , t ≥ 0) be defined by 0.2) or (0.3) . Then, for every x ∈ R, there exists a positive, σ-finite measure ν x , carried by R + , such that : √ t P x (A q t ∈ dz) -→ t→∞ ν x (dz).

(1.1)

The convergence in (1.1) is understood in the following sense : for any function f : R + -→ R + Borel, and sub-exponential i.e. : there exist two positive constants C 1 and C 2 such that :

0 ≤ f (x) ≤ C 1 e -C 2 x . then √ t E x f (A q t ) -→ t→∞ R + f (z)ν x (dz).
The measure ν x is characterized by :

∞ 0 e -λ 2 y ν x (dy) = ϕ λq (x). (1.2)
1.2 Proof of Theorem 1.1

We first begin with some precisions, taken from [RVY, I], see also S. Kotani, [K] about ϕ λq , which was defined from (0.3) but admits at least another characterization, namely : ϕ λq is the unique solution of the Sturm-Liouville equation :

ϕ (dx) = λϕ(x)q(dx) (1.3)
This equation is taken in the sense of Schwartz distributions, and subject to the following boundary conditions :

Under H1. : ϕ (+∞) = -ϕ (-∞) = 2 π (1.4) Under H2. : ϕ (-∞) = - 2 π and ϕ(+∞) = 0 (1.5) Under H3. : ϕ (+∞) = 2 π and ϕ(-∞) = 0.
(1.6) Theorem 1.1. is now an immediate consequence of the next lemma.

Lemma 1.2. Under either of the hypotheses H1, H2, or H3, the function : λ -→ ϕ λq (x) (λ > 0) is, for any real x, completely monotone, i.e., it satisfies :

(-1) n ∂ n ∂λ n ϕ λq (x) ≥ 0. (1.7)
Consequently, there exists a positive, σ-finite measure ν x , carried by R + , such that :

ϕ λq (x) = ∞ 0 e -λ 2 z ν x (dz). (1.8)
We shall give two proofs for Lemma 1.2.

A first proof of Lemma 1.2

We define, for every f : R + -→ R and every real h = 0 :

D h f (λ) := f (λ + h) -f (λ) h • (1.9) For f (λ) := exp - λ 2 A q t , we get : (D h ) n (f )(λ) = e -λA q t 2 e -A q t h 2 -1 h n
and, hence for all h = 0 :

(-1) n (D h ) n (f )(λ) ≥ 0.
(1.10)

Consequently, taking the expectation of the LHS in (1.10), we obtain :

√ t(-1) n E x (D h ) n exp - • 2 A q t ≥ 0. (1.11)
Hence, from (0.4) :

√ t(-1) n E x (D h ) n exp - • 2 A q t -→ t→∞ (-1) n (D h ) n ϕ •q (x) .
Thus :

(-1) n (D h ) n ϕ •q (x) (λ) ≥ 0.
(1.12)

Letting h → 0 in (1.12), and using the fact that :

D h f -→ h→0 f , we get : (-1) n ∂ n ∂λ n ϕ λq (x) ≥ 0. (1.13) 1.4 A second proof of Lemma 1.2
We shall only give this second proof under the hypothesis H1 and for x = 0. In [RVY, I], Proposition 4.13, formula (4.43), we have obtained the following explicit formula for ϕ λq (0) :

ϕ λq (0) = 1 √ 2π ∞ 0 Q (0) l (exp -λ < Y, q ->) • Q (2) l (exp -λ < Y, q + >) +Q (2) l (exp -λ < Y, q ->)Q (0) l (exp -λ < Y, q + >) dl (1.14)
where, in this formula (1.14), the process (Y x , x ≥ 0) is, under

Q (0) l , resp. under Q (2) l
, a squared Bessel process with dimension 0, (resp. 2), starting from l, and we denote :

< Y, q + >= ∞ 0 Y x q(dx) ; < Y, q ->= 0 -∞ Y -x q(dx).
(1.15)

It is then clear from (1.14) that : λ -→ ϕ λq (0) is the Laplace transform of a positive measure, as an integral, with respect to the parameter l of the product of two Laplace transforms of positive measures (indexed by l).

We shall now give some examples for which the measure ν x may be computed explicitly. We recall that ν x is characterized by :

∞ 0 e -λ 2 z ν x (dz) = ϕ λq (x) where ϕ λq (x) is given by (1.3) ... (1.6).
1.5 Computation of ν x for q(dy) = δ 0 (dy)

In this case, the hypothesis H1 is verified and A q t = L t , is the local time at level 0

ϕ λq (x) = 2 π 2 λ + |x| cf [RVY, I], Ex. 4.8, p. 199-200 = ∞ 0 e -λ 2 z 2 π 1 z≥0 dz + 2 π |x|δ 0 (dz) .
(1.16) Thus :

ν x (dz) = 2 π 1 [0,∞] (z)dz + 2 π |x|δ 0 (dz).
(1.17)

1.6 Computation of ν x for q(dy) = δ a (dy) + δ b (dy) with (a < b)

In this case, the hypothesis H1 is satisfied and A q t = L a t + L b t where (L a t , t ≥ 0) resp. (L b t , t ≥ 0) denotes the local time at level a, resp. at level b. We know see [RVY,I],Ex. 4.8, that

ϕ λq (x) =                      2 π 1 λ + x -b if x > b 2 π 1 λ if x ∈ [a, b] 2 π 1 λ + a -x if x < a (1.18) = 2 π ∞ 0 e -λ 2 z 1 2 dz + (x -b)1 x>b δ 0 (dz) + (a -x)1 x<a δ 0 (dz) .
Hence :

ν x (dz) = 2 π 1 2 1 [0,∞[ (z)dz + (x -b) + δ 0 (dz) + (a -x) + δ 0 (dz) . (1.19)
1.7 Computation of ν x , for q(y) = e 2y

In this case, the hypothesis H2 is satisfied and A q t = t 0 e 2Xs ds.

To begin with, we show : To check (1.20), it suffices to see that :

ϕ λq (x) = 2 π K 0 ( √ λ e x ) (1.
ψ (x) = λe 2x ψ(x), ψ(x) -→ x→∞ 0, ψ (x) -→ x→-∞ - 2 π • (1.21)
Now (1.21) follows from see [Leb], p. 110 :

K 0 = -K 1 , -K 1 (z) = 1 z K 1 (z) + K 0 (z) and ψ(x) ∼ x→∞ 2 π π 2 √ λ e x e - √ λ e x -→ x→∞ 0 [Leb], p.123 ψ (x) = - 2 π √ λ e x K 1 ( √ λ e x ) ∼ x→-∞ - 2 π √ λ e x 1 2 2 √ λ e x -→ x→-∞ - 2 π • [Leb], p.111
This proves (1.20). But, we also have :

K 0 ( √ λ e x ) = 1 2 ∞ 0 e -t-λe 2x 4t dt t cf [Leb], p. 119 = 1 2 ∞ 0 e -λu 2 -e 2x 2u du u • Hence : ν x (dz) = 1 √ 2π e -e 2x 2z 1 [0,∞[ (z) dz z • (1.22)
1.8 Computation of ν x for q 0 (dx

) = 1 ]-∞,0] (x)dx
Here, it is the hypothesis H3 which is satisfied, and

A q 0 t = t 0 1 ]-∞,0] (X s )ds.
By scaling, one has, under P 0 : A q 0 t (law)

= t A q 0 1 , and it is well known that under P 0 , A q 0 1 follows the arc sine law, i.e., the beta 1 2 , 1 2 law. We shall recall the law of A q 0 t under P x for any x ∈ R, (see Subsection 2.1 below), which will allow to obtain the following result :

ν x (dz) = x + 2 π δ 0 (dz) + 1 π e - x 2 - 2z 1 [0,∞[ (z) dz √ z • (1.23)
For the moment, we shall prove (1.23) without using the explicit law of A q 0 t . For this purpose, we already observe that :

ϕ λq 0 (x) = 2 π e x √ λ 1 √ λ 1 x≤0 + x + 1 √ λ 1 x>0 .
(1.24) Indeed we have :

ϕ λq 0 (x) = λ 1 ]-∞,0] (x)ϕ λq (x), ϕ λq 0 (+∞) = 2 π , ϕ λq 0 (-∞) = 0.
Then, it remains to see that :

∞ 0 e -λ 2 z ν x (dz) = ϕ λq 0 (x) (1.25)
where ν x is defined via (1.23) and ϕ λq 0 (x) by (1.24). Now, for x > 0, one has :

∞ 0 e -λ 2 z ν x (dz) = x + 2 π + 1 π ∞ 0 e -λz 2 dz √ z = x + 2 π + 1 π 2 λ Γ(1/2) = x + 2 π + 2 π 1 √ λ = ϕ λq (x)
whereas for x < 0 :

∞ 0 e -λ 2 z ν x (dz) = 1 π ∞ 0 e -λ 2 z-x 2 2z dz √ z = 2 π K 1/2 |x| √ λ x 2 λ 1/4 see [Leb], p. 119 .
However, one has :

K 1/2 |x| √ λ = π 2|x| √ λ 1/2 e -|x| √ λ . Hence : ∞ 0 e -λ 2 z ν x (dz) = 2 π x 2 λ 1/4 π 2|x| √ λ 1/2 e -|x| √ λ = 2 π 1 √ λ e -|x| √ λ = ϕ λq (x).
1.9 Computation of ν x when q(y) = 1 [a,b] (y) (a < b)

The hypothesis H1 is satisfied and

A q t = t 0 1 [a,b] (X s ) ds.
We shall prove that :

ν (a,b) x (dz) =                        2 π (x -b)δ 0 (dz) + 1 π √ z 1 [0,∞[ (z)dz 1 + 2 ∞ n=1 e -n 2 (b-a) 2 2z if x > b 2 π (a -x)δ 0 (dz) + 1 π √ z 1 [0,∞[ (z)dz 1 + 2 ∞ n=1 e -n 2 (b-a) 2 2z if x < a 1 π √ z ∞ n=0 e - n(b-a)+b-x 2 2z + e - n(b-a)+(x-a) 2 2z 1 [0,∞[ (z)dz if x ∈ [a, b]
(1.26)

Here, the explicit form of ϕ (a,b) λq (x) is see [RVY,I],Ex. 4.7,p. 199 :

ϕ (a,b) λq (x) =                          2 π 1 √ λ tanh √ λ b-a 2 + x -b if x > b 2 π 1 √ λ tanh √ λ b-a 2 + a -x if x < a 2 π cosh √ λ x -a+b 2 ) √ λ sinh √ λ b-a 2 ) if x ∈ [a, b]
(1.27)

It now remains to prove that : via (1.27). But, (1.28) follows, after some elementary computations from the identities, for every real u and v > 0 :

∞ 0 e -λ 2 z ν (a,b) x (dz) = ϕ
cosh √ λu √ λ sinh √ λv = ∞ n=0 ∞ 0 dh e - √ λ h+(2n+1)v-u + e - √ λ h+(2n+1)v+u (1.29) = ∞ n=0 ∞ 0 dh ∞ 0 ds H h+(2n+1)v-u (s) + H h+(2n+1)v+u (s) e -λs (1.30) with H a (u) := a 2 √ πu 3 e -a 2 /4u = -1 √ πu ∂ ∂a (e -a 2 4u ) (a > 0).
Passing from (1.29) to (1.30) is obtained by using the elementary formula :

e - √ λa = ∞ 0 e -λu H a (u)du = ∞ 0 e -λu a 2 √ πu 3 e -a 2 4u du.
(1.31) (Note that (1.31) is nothing else but a translation of : E e -λ 2 2 Ta = exp(-λa), where T a denotes the hitting time of level a > 0 by Brownian motion starting from 0, and H a is the density of

T a = 1 √ λ e - √ λ(v-u) (1 + e -2 √ λu ) ∞ n=0 e -2n √ λv = 1 √ λ ∞ n=0 e - √ λ{v-u+2nv} + ∞ n=0 e - √ λ 2(u+nv)+(v-u) = 1 √ λ ∞ n=0 e - √ λ (2n+1)v-u + e - √ λ u+(2n+1)v = ∞ 0 e - √ λh ∞ n=0 e - √ λ (2n+1)v-u + e - √ λ u+(2n+1)v dh = ∞ n=0 ∞ 0 e - √ λ h+(2n+1)v-u + e - √ λ h+(2n+1)v+u dh. Remark 1.3 (i) If in formula (1.
26), we take : b = 0, and we let a tend to -∞, we obtain :

lim a→-∞ ν a,0 (dz) =        2 π x + δ 0 (dz) + 1 π √ z 1 [0,∞[ (z)dz if x > 0 1 π √ z e - x 2 - 2z 1 [0,∞[ (z)dz if x ≤ 0 (1.32)
We note that the RHS of (1.32) is nothing else but the measure ν x associated with q 0 (y) = 1 ]-∞,0] see (1.23) . This may be interpreted as "a continuity property" of ϕ a,b , as a -→ -∞.

(ii) In the same spirit, but taking up now the computation from subsection 1.9, where we choose for q the function :

q (c) (y) = 1 2c 1 [-c,+c] (y), we have : ∞ 0 e -λz 2 ν (c) x (dz) -→ c→0 ∞ 0 e -λz 2 ν x (dz) = 2 π 2 λ + |x| (1.33)
where ν x is the measure associated to q(dz) = δ 0 (dz) see (1.16) . In other terms, since :

1 2c t 0 1 [-c,c] (X s )ds -→ c→0 L t a.
s., we witness there also a "continuity property of ν (c)

x as c -→ 0". Let us show (1.33) for x = 0 ; from (1.27) :

∞ 0 e -λz 2 ν (c) 0 (dz) = 2 π cosh λ 2c c λ 2c sinh λ 2c • c -→ c→0 2 π • 2 λ
and for x = 0, and c small enough, we obtain from (1.27) that :

∞ 0 e -λz 2 ν (c) x (dz) = 2 π 1 λ 2c tanh λ 2c • c + |x -c| -→ c→0 2 π 2 λ + |x| . 1.10 Computation of ν x when q(y) = 1 [0,∞[ (y)y α , α > 0
The hypothesis H2 is satisfied, and we have :

A q t = t 0 1 (Xs>0)
X α s ds. We now show the existence of a constant C α > 0 such that :

ν 0 (dz) = C α z 1+α 2+α 1 [0,∞[ (z)dz. (1.34)
Indeed, thanks to the scaling property, we have :

E 0 e -λ 2 R t 0 1 Xs>0 X α s ds = E 0 e -λ 2 t 1+α/2 A q 1 = E 0 exp - 1 2 A q λ 2 2+α t
.

(1.35) Thus, multiplying (1.35) by √ t and letting t tend to +∞, we obtain :

ϕ λq (0) = 1 λ 1 2+α ϕ 1•q (0) = 1 λ 1 2+α c α = c α ∞ 0 e -λ 2 z dz z 1+α 2+α
.

The same computations, performed this time with x = 0, lead to :

ϕ λq (x) = 1 λ 1 2+α ϕ 1q xλ 1 2+α , i.e. ν x = 1 λ 1 2+α ν (λ) xλ 1 2+α
, where ν (λ) is the image of ν by the application z -→ λz.

Question 1.4

We know see [R,Y], chap. X that, if q is an integrable function, then :

1 √ t t 0 q(x + X s )ds law -→ t→∞ q(x + y)dy |N | = q(y)dy |N | (1.36)
where N is a standard Gaussian variable, and on the LHS of (1.36), (X s , s ≥ 0) is a Brownian motion starting from 0. Let g denote the density of the r.v. q|N | with q = q(y)dy that is :

g(z) = 1 q 2 π e -z 2 2(q) 2 1 [0,∞[ (z).
Let us now consider the supplementary hypothesis H, which seems reasonable enough in view of (1.36) that the density

g t (x, •) of the r.v. 1 √ t t 0 q(x + X s ) ds converges, as t -→ ∞,
uniformly on every compact, towards g. However, this would imply that, for every function h, which is continuous with compact support, one would have :

√ t E x h(A q t ) = √ t E x h A q t √ t • √ t = √ t ∞ 0 h(z √ t) g t (x, z) dz = ∞ 0 h(y)g t x, y √ t dy -→ t→∞ ∞ 0 h(y)g(0)dy.
But, from Theorem 1.1., we know that :

√ t E x h(A q t ) -→ t→∞ R + h(z)ν x (dz)
Thus, this would imply that the measure ν x (dz) would be equal to :

1 q 2 π 1 [0,∞[ dz (1.37)
so that, the measure ν x would not depend on x, and would be proportional to Lebesgue measure on R + . But clearly, this is not the case for either of the examples in subsections 1.4 to 1.9. Consequently, the hypothesis H is not satisfied for the corresponding q s. It would be of interest to know for which q s, if any, it is satisfied.

Penalisation by h(A q t )

Let q satisfy one of the previous hypotheses H1, H2 or H3, and denote, as before :

A q t = R L x t q(dx) = t 0 q(X s )ds if q admits a density . Let now h : R + -→ R + such that : √ t E x h(A q t ) -→ t→∞ ∞ 0 h(z)ν x (dz). (1.38)
Then, (1.38) is satisfied, from Theorem 1.1., as soon as h is sub-exponential (for example if h is continuous, with compact support). We shall now study the penalisation of Wiener measure by the functional h(A q t ), i.e. : we shall study the limit, as t -→ ∞, of :

E x 1 Λs h(A q t ) E x h(A q t ) (s ≥ 0, Λ s ∈ F s ) (1.39)
We have already made this study in two situations :

1) q(dy) = δ 0 (dy) then

A q t = L t cf [RVY, II] ; 2) A q t = R L y t q(dy) and h(u) = exp - λ 2 u cf [RVY, I] .
This time, Theorem 1.1. allows us to obtain :

Theorem 1.5. : Let q, A q and h as above. Then :

1) For every s ≥ 0, and every Λ s ∈ F s :

lim t→∞ E x 1 Λs h(A q t ) E x h(A q t )
exists.

(1.40)

2) This limit equals E x (1 Λs M h,q s ) := Q h,q (Λ s ), where

M h,q s := R + ν Xs (dz) h(z + A q s ) R + ν x (dz) h(z) • (1.41)
Furthermore,(M h,q s , s ≥ 0) is a positive martingale. In the case when h(u) := e -λ 2 u (u, λ ≥ 0), we then obtain :

M h,q s = ϕ λq (X s ) ϕ λq (x) exp - λ 2 A q s .
(1.42)

Proof of Theorem 1.5. We have :

E x 1 Λs h(A q t ) E x h(A q t ) = E x 1 Λs E b h(a + A q t-s ) E x h(A q t )
from the Markov property, where b = X s and a = A q s . Thus, from Theorem 1.1. :

E x h(A q t ) ∼ t→∞ 1 √ t ∞ 0 ν x (dz)h(z) and E b h(a + A q t-s ) ∼ t→∞ 1 √ t -s ∞ 0 ν b (dz)h(a + z).
Hence :

E x 1 Λs h(A q t ) E x h(A q t ) ∼ t→∞ √ t √ t -s E x 1 Λs R + ν Xs (dz)h(z + A q s ) R + h(z)ν x (dz) -→ t→∞ E x (1 Λs M h,q s )
In the preceding lines, we have been a little careless concerning the exchange of limit and expectation. Likewise, although it is easy to see that (M h,q s , s ≥ 0) is a local martingale, some care is needed in order to show that it is a true martingale. However, all this is correct as soon as h is sub-exponential. We leave details to the reader.

A detailed study for q

0 = 1 ]-∞,0] , A - t := t 0 1 (X s <0) ds
Throughout this section, we choose q 0 = 1 ]-∞,0] . Thus, the hypothesis H3 is now satisfied. We shall study this situation in detail, which we are able to do as we know see [Y] the law of A q 0 t = t 0 q 0 (X s )ds under P x , for every real x see (2.5) and (2.7) below . We shall, successively :

-compute explicitly the measure ν x starting from the knowledge of the law of A q 0 t and we shall recover the result of subsection 1.8 above ; -study the penalisation, not only of the process (X t , t ≥ 0) by h(A q 0 t ), but also the penalisation of the "long bridges" by this functional ; -describe precisely the behavior of the canonical process under the probability Q h,q 0 , where Q h,q 0 is defined via :

Q h,q 0 (Λ s ) = E(1 Λs M h,q 0 s ) (s ≥ 0, Λ s ∈ F s ). (2.1)
2.1 The law of A - t and the computation of ν x

To simplify notation, we denote :

A - t = t 0 1 (Xs<0) ds = t 0 q 0 (X s )ds. (2.2)
We recall the following result, which is found in [Y]. For any f : [0, 1] -→ R + , Borel, subexponential (see Theorem 1.1) and any y > 0 :

E 0 f 1 0 1 (Xs<y) ds = 1 0 du π u(1 -u) e -y 2 2u f (u) + f (1) 2 π y 0 e -α 2 2 dα (2.3)
whereas, for any y < 0, we use :

1 0 1 (Xs<y) ds law = 1 0 1 (Xs>-y) ds law = 1 - 1 0 1 (Xs<-y) ds (2.4)
and by the scaling property :

E x f t 0 1 (Xs<0) ds = E x f (A - t ) = E 0 f t 0 1 (Xs<-x) ds = E 0 f t 1 0 1 Xs<-x √ t ds .
Hence, from (2.3) and (2.4), if x ≤ 0 :

E x f (A - t ) = t 0 dv π v(t -v) e -x 2 2v f (v) + f (t) 2 π |x| √ t 0 e -α 2 2 dα (2.5) ∼ t→∞ 1 π √ t ∞ 0 dv √ v e -x 2 2v f (v) (2.6)
whereas, if x > 0 :

E x f (A - t ) = f (0) 2 π x √ t 0 e -α 2 2 dα + t 0 dv π v(t -v) e -x 2 2(t-v) f (v) (2.7) ∼ t→∞ f (0) 2 π x √ t 0 e -α 2 2 dα + 1 π √ t ∞ 0 dv √ v f (v).
(2.8) Thus, we obtain, for f : R + -→ R + , Borel, sub-exponential and then

∞ 0 dv √ v f (v) < ∞ √ t E x f (A - t -→ t→∞ ∞ 0 f (z)ν x (dz) with ν x (dz) = x + 2 π δ 0 (dz) + 1 π e - x 2 - 2z 1 [0,∞[ (z) dz √ z
which is precisely (1.23).

Penalisation by h(A - t ).

A study of "long bridges" and of the Q h -process

We recall that, from (2.5), the density of A - t under P 0 , which we denote by p A - t , equals :

p A - t (y) = 1 π 1 y(t -y) 1 [0,t] (y)
(: the arc sine law).

(2.9)

Throughout the following, h denotes a function from R + to R + such that :

∞ 0 dy √ y h(y) < ∞.
And we assume, without loss of generality, that :

∞ 0 dy √ y h(y) = 1.
(2.10)

Theorem 2.1 1) For every s ≥ 0 and every Λ s ∈ F s :

lim t→∞ E 0 (1 Λs |A - t = a) = Q (a) (Λ s ) (2.11) with Q (a) (Λ s ) := 2 π 1 a<s √ s -a E 0 (1 Λs X + s |A - s = a) + E 1 Λs a a -A - s 1 (A - s <a) e - (X - s ) 2 2(a-A - s )
(2.12)

recall that X + s = 0 ∨ X s , X - s = -(X s ∧ 0) and A - s = s 0 1 Xu<0 du .
2) For every function h which satisfies (2.10), for every s ≥ 0 and any Λ s ∈ F s :

lim t→∞ E 0 1 Λs h(A - t ) E 0 h(A - t ) = E(1 Λs M h s ) (2.13)
where (M h s , s ≥ 0) is the positive martingale given by :

M h s := √ 2π X + s h(A - s ) + ∞ 0 dy √ y e - (X - s ) 2 2y h(A - s + y) (2.14)
(note that M h 0 = 1). 3) Formula (2.13) induces a probability Q h on (Ω, F ∞ ), which admits the following disintegration :

Q h (Λ) = ∞ 0 Q (a) (Λ) h(a) √ a da (Λ ∈ F ∞ ) (2.15)
where Q (a) is given by (2.12).

4)

Under Q h , the canonical process X t , t ≥ 0 satisfies : i) A - ∞ is finite a.s., and admits as density

h(y) √ y 1 y>0 (2.16) ii) let g = inf{t ; A - t = A - ∞ } = sup{t ; X t ≤ 0}.
(2.17)

Then Q h (g < ∞) = 1
iii) the processes (X t , t ≤ g) and (X g+t , t ≥ 0) are independent ; iv) the process (X g+t , t ≥ 0) is a 3-dimensional Bessel process starting from 0.

Moreover, while proving Theorem 2.1, we shall give a precise description of the process (X t ; t ≤ g).

2.3 Proof of Theorem 2.1

2.3.1 Proof of point 1) in Theorem 2.1
For this purpose, we choose a function h, which is Borel, positive, and satisfies (2.10). We first write :

E 0 1 Λs h(A - t ) = t 0 E 0 (1 Λs |A - t = a)p A - t (a)h(a)da. (2.18)
Then, conditioning with respect to F s , we obtain :

E 0 1 Λs h(A - t ) = E 0 1 Λs E 0 h a + t-s 0 1 (Xu<-x) du (2.19)
with a = A - s and x = X s . Using now (2.5) and (2.6), we obtain :

E 0 1 Λs h(A - t ) = E 0 1 Λs 1 x<0 t-s 0 dv π v(t -s -v) e -x 2 2v h(a + v) + h(a + t -s)ψ |x| √ t -s +E 0 1 Λs 1 x>0 h(a)ψ x √ t -s + t-s 0 dv π v(t -s -v) e - x 2
2(t-s-v) h(a + v) (2.20) := (1) t + (2) t (2.21) where ψ x √ t := P |N | ≤ x √ t = 2 π x √ t 0 e -α 2 2 dα ∼ t→∞ 2 π x √ t •
We now study successively (1) t and (2) t . We rewrite (1) t in the form :

(1)

t = s 0 p A - s (a)da E 0 1 Λs 1 Xs<0 t-s 0 dv π v(t -s -v) e -X 2 s 2v h(a + v) + h(a + t -s) ψ |X s | √ t -s A - s = a = t-s 0 dv π v(t -s -v) v+s v da p A - s (a -v) E 0 1 Λs 1 Xs<0 e -X 2 s 2v |A - s = a -v h(a) + t t-s p A - s (a + s -t)E 0 1 Λs 1 Xs<0 ψ |X s | √ t -s A - s = a + s -t h(a)da. (2.22)
Similarly :

(

2) t = s 0 p A - s (a) E 0 1 Λs 1 Xs>0 ψ |X s | √ t -s A - s = a h(a)da (2.23) + s 0 da p A - s (a) t-s 0 dv π v(t -s -v) E 0 1 Λs 1 Xs>0 e - X 2 s 2(t-s-v) A - s = a)h(a + v) = s 0 p A - s (a) E 0 1 Λs 1 Xs>0 ψ |X s | √ t -s A - s = a h(a)da (2.24) + t-s 0 dv π v(t -s -v) v+s v p A - s (a -v) E 0 1 Λs 1 Xs>0 e - X 2 s 2(t-s-v) A - s = a -v h(a)da.
Then, comparing (2.18), (2.22), (2.24) and identifying the "coefficient of h(a)", it follows that :

E 0 1 Λs |A - t = a = ( 1) t + ( 2) t with : ( 1) t = 1 p A - t (a) t-s 0 dv π v(t -s -v) 1 v<a<v+s p A - s (a -v) E 0 1 Λs 1 Xs<0 e -X 2 s 2v A - s = a -v + 1 p A - t (a) 1 t-s<a<t p A - s (a + s -t) E 0 1 Λs 1 Xs<0 ψ |X s | √ t -s A - s = a + s -t -→ t→∞ s 0 √ a √ a -w p A - s (w) E 0 1 Λs 1 Xs<0 e - X 2 s 2(a-w) A - s = w dw = E 0 1 Λs 1 Xs<0 e - X 2 s 2(a-A - s ) √ a a -A - s 1 A - s <a (2.25) since p A - t (a) = 1 π a(t -a) 1 [0,t] (a).
Similarly, one has :

( 2

) t = p A - s (a) p A - t (a) 1 a<s E 0 1 Λs 1 Xs>0 ψ |X s | √ t -s A - s = a + 1 p A - t (a) t-s 0 1 v<a<v+s dv π v(t -s -v) p A - s (a -v) E 0 1 Λs 1 Xs>0 e - X 2 s 2(t-s-v) A - s = a -v -→ t→∞ 1 a<s √ s -a 2 π E 0 1 Λs X + s A - s = a + E 0 1 Λs 1 Xs>0 a a -A - s 1 A - s <a .
Hence, point 1 of Theorem 2.1 follows.

Proof of point 2 in Theorem 2.1

In fact, this point has already been shown while proving Theorem 1.5. With the help of the form (1.41) of M h and the explicit computation of ν x see formula (1.23) , we obtain :

M h s = ∞ 0 ν Xs (dy) h(A - s + y) ∞ 0 ν 0 (dy) h(y) = ∞ 0 h(A - s + y) X + s 2 π δ 0 (dy) + 1 π e - (X - s ) 2 2y dy √ y 1 π ∞ 0 h(y) dy √ y = √ 2π X + s h(A - s ) + ∞ 0 dy √ y e - (X - s ) 2 2y h(A - s + y).
(2.26) Now, clearly, this point 1 of Theorem 2.1 which we just proved implies also point 2 of the same Theorem 2.1. Indeed, we have :

E 0 1 Λs h(A - t ) E 0 h(A - t ) = t 0 E 0 1 Λs |A - t = a h(a) p A - t (a)da t 0 h(a) p A - t (a)da

•

From the above point 1, and with the help of the explicit form of p A - t (a) as given by (2.9) the above quantity converges, as t → ∞, towards :

∞ 0 da √ a Q (a) (Λ s )h(a) ∞ 0 h(a)da √ a = ∞ 0 h(a) √ a Q (a) (Λ s )da (2.27)
since we assumed :

∞ 0 h(a)da √ a = 1.
It now remains to compute :

∞ 0 h(a) √ a Q (a) (Λ s )da = 2 π s 0 1 a(s -a) E 0 1 Λs X + s |A - s = a h(a)da (2.28) + ∞ 0 h(a) √ a E 0 1 Λs a a -A - s 1 A - s <a e - (X - s ) 2 2(a-A - s ) da from (2.12) = √ 2π s 0 p A - s (a) E 0 1 Λs X + s |A - s = a h(a)da + ∞ 0 dy √ y E 0 1 Λs e - (X - s ) 2 2y h(A - s + y after the change of variable a -A - s = y = √ 2π E 0 1 Λs X + s h(A - s ) + ∞ 0 E 0 1 Λs e - (X - s ) 2 2y h(A - s + y) dy √ y = E 0 (1 Λs M h s ) from (2.14) .
(2.29)

We now remark that point 3 in Theorem 2.1 states precisely formula (2.29) we just established.

Proofs of points 4i

) and 4ii) in Theorem 2.1 a) From formula (2.15) and from Doob's optional sampling theorem, we deduce :

Q h (A - ∞ > a) = E[M h σa ], with σ a := inf{t ; A - t > a} (2.30) But : M h σa = √ 2π h(a) X + σa + ∞ 0 dy √ y e - (X - σa ) 2 2y h(a + y) = ∞ 0 dy √ y e - (X - σa ) 2 2y h(a + y) since X + σa = 0.
We recall that the process (X - σa ; a ≥ 0) is distributed as the reflecting Brownian motion |X a |, a ≥ 0 , where (X a , a > 0) is a standard Brownian motion starting from 0 see, e.g., [K.S],Th. 3.1,p. 419 . Hence, we obtain :

E[M h σa ] = ∞ 0 dy √ y h(a + y) E(e - X 2 a 2y ) = ∞ 0 dy √ y h(a + y) y y + a = ∞ 0 dy √ y + a h(a + y) = ∞ a dy √ y h(y).
b) We now remark that it is easy to recover the law of A - ∞ under Q h from points 1 and 2 in Theorem 2.1. We may already prove that, under Q (a) , one has A - ∞ = a a.s. Indeed, this follows from :

if b > a, Q (a) (A - s > b) = 2 π 1 a<s √ s -a E 0 X + s 1 A - s >b |A - s = a + E 0 a a -A - s 1 b<A - s <a e - (X - s ) 2 2(a-A - s ) = 0.
Hence, passing to the limit as

s → ∞, if b > a : Q a (A - ∞ > b) = 0 On the other hand, it is clear that E 0 1 A - s ≤a |A - t = a) = 1 (t > s)
, hence, passing to the limit as t → ∞, and then, letting s → ∞ we obtain : Finally,from (2.15), we get :

Q (a) (A - ∞ ≤ a) = 1.
Q h (A - ∞ ≤ b) = ∞ 0 h(a) a Q (a) (A - ∞ ≤ b)da = b 0 h(a) √ a da.

Computation of Azéma's supermartingale Z

t := Q h (g > t|F t )
Our proof of points 4iii) and 4iv) in Theorem 2.1 is based on the theory of enlargements of filtration (cf [J] or [JY]). In order to apply this theory, we need to calculate Azéma's supermartingale Q h (g > t|F t ). We start with this computation.

Let g = inf{t ≥ 0 ; A - t = A - ∞ } = sup{t ≥ 0 ; X t ≤ 0}. (2.31) Lemma 2.2
The following explicit formula holds :

Z t := Q (h) g > t|F t ) = 1 (Xt<0) + 1 (Xt>0) ∞ 0 dv √ v h(A - t + v) M h t • (2.32)
Proof of Lemma 2.2 We note that, for Λ t ∈ F t :

Q h (1 g>t 1 Λt ) = Q h (1 Λt 1 Xt<0 ) + Q h (1 Λt 1 Xt>0 1 dt<∞ )
where d t denotes the first return time to 0 after time t)

= Q h (1 Λt 1 Xt<0 ) + E(1 Λt 1 X t>0 M h dt ) .
We have :

M h dt = √ 2π h(A - dt )X + dt + ∞ 0 dv √ v h(A - dt + v) = ∞ 0 dv √ v h(A - t + v) from (2.14)
since X dt = 0 and A - dt = A - t on the set X t > 0 . Hence :

Q h (1 g>t 1 Λt ) = Q h 1 Λt 1 Xt<0 + 1 Xt>0 ∞ 0 dw √ w h(A - t + w) M h t .
This proves (2.32) and Lemma 2.2.

Proof of

Q h (g < ∞) = 1
We deduce from (2.32) that :

Q[g < t] = 1 -Q[g > t] = 1 -E 1 Xt<0 M h t + 1 Xt>0 ∞ 0 dv √ v h(A - t + v) = √ 2π E X + t h(A - t ) from (2.26) and since (M h t , t ≥ 0) is a martingale s.t. E(M h t ) = 1 = √ 2π 2 E t 0 h(A - s ) d L s from Itô-Tanaka formula -→ t→∞ π 2 E ∞ 0 h(A - s ) d L s = π 2 E ∞ 0 h(a) dL σa
where (σ a , a ≥ 0) denotes the right continuous inverse of (

A - t , t ≥ 0) = 2 π 2 E ∞ 0 h(a) dL a since (X - σa , a ≥ 0) is distributed as |X a |, a ≥ 0 ; cf point a of Subsection 2.3.3 = 2 π 2 ∞ 0 h(a) E(dL a ) = ∞ 0 h(a) √ a da = 1 since E(L a ) = √ a • 2 π • 2.3.6 Description of the canonical process (X t , t ≥ 0) under Q h
For this purpose, we shall use the technique of enlargement of filtrations. Thus, let (G t , t ≥ 0) denote the smallest filtration which makes g a (G t t ≥ 0) stopping time, and which contains (F t , t ≥ 0). The application of Girsanov's Theorem and (2.14) imply the existence of a (F t , Q h ) Brownian motion (β t , t ≥ 0) such that, under Q h :

X t = β t + t 0 1 M h s √ 2π h(A - s ) 1 Xs>0 - ∞ 0 dw w 3/2 e -(X - s ) 2 2w h(A - s + w) X - s ds. (2.33)
We now apply the enlargement formulae cf [J], [JY], [MY] . We first observe that :

dZ t = - ∞ 0 dw √ w h(A - t + w) (M h t ) 2 √ 2π h(A - t )1 Xt>0 dX t + dV t (2.34)
where (V t , t ≥ 0) has bounded variations and therefore :

d < Z, X > t = - ∞ 0 dw √ w h(A - t + w) (M h t ) 2 √ 2π h(A - t ) 1 (Xt>0) dt. (2.35)
Thus, there exists a (G t , t ≥ 0), Q h Brownian motion ( β t , t ≥ 0) such that :

dX t = d β t + 1 M h t √ 2π h(A - t ) 1 Xt>0 - ∞ 0 dw w 3/2 e -(X - t ) 2 2w h(A - t + w) X - t dt +1 t<g - ∞ 0 dw √ w h(A - t + w) (M h t ) 2 √ 2π h(A - t ) 1 Xt>0 • M h t ∞ 0 dw √ w h(A - t + w) dt -1 t>g - ∞ 0 dw √ w h(A - t + w) (M h t ) 2 √ 2π h(A - t ) 1 Xt>0 • M h t √ 2π h(A - t ) X + t dt.
This yields, after some simplifications :

X t = β t - t∧g 0 1 M h s ∞ 0 dw w 3/2 e -(X - s ) 2 2w h(A - s + w) X - s ds + t t∧g ds X s (2.36)
since, after g, X - t = 0, hence X + t = X t . Points 4 iii) and iv) now follow immediately from (2.36).

Remark 2.3 When h(x) = e -λx 2 (λ > 0, x ≥ 0), the equation (2.36) simplifies as :

X t = β t - t∧g 0 X - s λ 1 2 λ 1 4 √ 2π X + s -X - s ds + t t∧g ds X s • (2.37)
This formula (2.37) follows from :

∞ 0 dw w 3/2 e -(X - s ) 2 2w -λ 2 w dw = (X - s ) 2 λ -1/4 2 K -1/2 ( √ λX - s ) ∞ 0 dw w 1/2 e -(X - s ) 2 2w -λ 2 w dw = (X - s ) 2 λ +1/4 2 K 1/2 ( √ λX - s )
and from :

K -1 2 (z) = K 1 2 (z) = π 2z
1 2 e -z [Leb], p. 112 and p. 119 .

Markovian limit process

Theorem 2.1 shows that the process (X t , t ≥ 0) is not Markovian under Q h , whereas the 2-dimensional process (X t , A - t ), t ≥ 0 is Markovian. Indeed, g is not a (F t , t ≥ 0) stopping time and the dynamics of (X t ) is not the same before and after g. On the other hand, we know see [RVY, I]

that if h(x) := e -λ 2 x (λ, x ≥ 0), then the Q h -process is Markovian. It is the diffusion with infinitesimal generator L h : L h f (x) = 1 2 f (x) + ϕ ϕ (x)f (x), f ∈ C 2 b where ϕ denotes the unique solution of ϕ = λϕ, ϕ(-∞) = 0 ; ϕ (+∞) = 2 π • In this case,
the solution of this equation see (1.24) takes the explicit form :

ϕ λ (x) = 2 π e x √ λ 1 √ λ 1 x≤0 + x + 1 √ λ 1 x>0 . (2.38)
Under Q h , we obtain :

X t = B t + t 0 du X + u + 1 √ λ compare with (2.37) (2.39) where (B t , t ≥ 0) is a (F t , t ≥ 0), Q h Brownian motion. The martingale (M h s , s ≥ 0) is equal to : M h s = ϕ λ (X s ) exp - λ 2 s 0 1 ]-∞,0] (X u ) du .
(2.40)

This example motivated us to raise the question : which are the functions h such that the Q h process is Markovian ? The answer is given by the following :

Proposition 2.4 : Let h be regular, bounded, satisfying equation (2.10) i.e, ∞ 0 dy √ y h(y) = 1 and such that the process (X t , t ≥ 0) is Markov under Q h . Then, there exists λ ≥ 0 such that

h(x) = λ 2π e -λ 2 x (x ≥ 0).

Proof of Proposition 2.4

To answer this question, we come back to equation (2.33). The problem is to find under which conditions the drift term :

√ 2π h(A - t ) 1 Xt>0 - ∞ 0 dw w 3/2 e -(X - t ) 2 2w h(A - t + w) X - t √ 2π h(A - t ) X + t + ∞ 0 dw w 1/2 e -(X - t ) 2 2w h(A - t + w)
(2.41)

does not depend on A - t . Considering this expression when X t < 0, the problem amounts to study the functions h for which :

x ∞ 0 dw w 3/2 e -x 2 2w h(a + w) ∞ 0 dw w 1/2 e -x 2 2w h(a + w) := k(x) (2.42)
does not depend on a. (2.42) may be written :

∂ ∂x log θ(x, a) = -k(x)
where we have

denoted : θ(x, a) := ∞ 0 dw √ w e -x 2 2w h(a + w).
Hence, by integration we obtain the existence of two functions ϕ 1 and ϕ 2 such that :

∞ 0 x √ 2πw 3 e -x 2 2w h(a + w) dw = ϕ 1 (a)ϕ 2 (x).
(2.43)

Letting x -→ 0 in (2.43), we obtain h(a) = ϕ 1 (a)ϕ 2 (0). Note that the LHS in (2.43) writes E h(a + T x ) , where (T x , x ≥ 0) is the 1 2 -stable subordinator of Brownian first hitting times. Hence we have :

E h(a + T x ) = P x (h)(a) = E ϕ 1 (a + T x )ϕ 2 (0) = ϕ 1 (a)ϕ 2 (0) (2.44)
where (P x , x ≥ 0) denotes the semi-group associated with the subordinator (T x , x ≥ 0), whose infinitesimal generator is ∂ 2 ∂x 2 1 2 . In other terms, from (2.43), we get :

P x ϕ 1 (a) = ϕ 2 (x) ϕ 2 (0) ϕ 1 (a).
(2.45) ϕ 1 is an eigenfunction of P x , and consequently an eigenfunction of ∂ 2 ∂x 2 • ϕ 1 being positive and bounded : ϕ 1 (a) = c e -λ 2 a (a, λ ≥ 0) and h(a) = c e -λ 2 a ϕ 2 (0) = c e -λ 2 a .

3 A local limit theorem for a class of additive functionals of the "long Brownian bridges"

3.1 Statement of Theorem 3.1

In this section, our aim is to obtain results similar to those in Section 1, but, now, Brownian motion (X s , s ≥ 0) is being replaced by the Brownian bridge with length t, with t → ∞. q denotes a function from R to R + , which is Borel, and such that :

0 < ∞ -∞ (1 + x 2 ) q(x) dx < ∞. (3.1)
We let :

A q t := t 0 q(X s ) ds.

(3.2)

Theorem 3.1 1) For every x and y ∈ R, and µ > 0 :

E x exp - µ 2 t 0 q(X s ) ds X t = y ∼ t→∞ π 2 ϕ µq (x)ϕ µq (y) t (3.3)
where ϕ µq denotes the unique solution of :

ϕ = (µq) • ϕ, lim x→+∞ ϕ (x) = -lim x→-∞ ϕ (x) = 2 π (3.4) 2) lim t→∞ t P x A q t ∈ dz|X t = y = ν x * ν y (dz) (3.5)
where ν x and ν y have been defined in Theorem 1.1. The convergence in (3.5) has the same meaning as in Theorem 1.1.

Proof of Theorem 3.1

Without loss of generality, we shall assume that µ = 1.

An auxiliary lemma

Lemma 3.2 There exists a constant C > 0, depending only on q, such that :

E x exp - 1 2 t 0 q(X s ) ds X t = y ≤ C e (x-y) 2 2t (1 + |x| 1 + |y| 1 + t • (3.6) 3.2.2 Proof of lemma 3.2 1)
As an intermediary result, we already show that :

E x exp - 1 2 t 0 q(X s ) ds X t = y ≤ C e (x-y) 2 2t 1 + |x| √ 1 + t (3.7)
for a constant C which does not depend on x, y, t.

To prove (3.7), we condition with respect to X t/2 , and we get :

E x exp - 1 2 t 0 q(X s ) ds X t = y e -(x-y) 2 2t = 1 √ πt ∞ -∞ E x exp - 1 2 A q t/2 X t/2 = c E c exp - 1 2 A q t/2 X t 2 = y • e -(x-c) 2 t - (y-c) 2 t dc. (3.8)
In (3.8), we majorize E c exp -1 2 A q t/2 |X t/2 = y by 1, and we get : [RVY, I]. Thus, we have obtained (3.7).

E x exp - 1 2 A q t |X t = y e -(x-y) 2 2t ≤ 1 √ πt • ∞ -∞ E x exp - 1 2 A q t/2 |X t/2 = c e - (x-c) 2 2•t/2 dc ≤ E x exp - 1 2 A q t/2 ≤ C 1 + |x| √ 1 + t from Lemma 4.3 in
2) Then, plugging the estimate (3.7) in (3.8), we obtain :

E x exp - 1 2 A q t X t = y ≤ e (x-y) 2 2t C 1 + |y| √ 1 + t ∞ -∞ E x exp - 1 2 A q t/2 |X t/2 = c e - (x-c) 2 2•t/2 2πt/2 dc ≤ C 1 + |x| 1 + |y| 1 + t • e (x-y) 2 2t
since :

E c exp - 1 2 A q t/2 |X t/2 = y = E y exp - 1 2 A q t/2 |X t/2 = c and e (x-y) 2 2t • C 1 + |y| √ 1 + t E x exp - 1 2 A q t/2 |X t/2 = c ≤ e (x-y) 2 2t C 1 + |y| 1 + |x| 1 + t
by applying once again Lemma 4.3 in [RVY, I].

Another auxiliary lemma

Lemma 3.3 Let Z(t, x, y) := E x exp -1 2 A q t |X t = y . We also denote by U (t, x, y) the solution of :

   ∂U ∂t (t, x, y) - 1 2 ∂ 2 U ∂x 2 (t, x, y) + 1 2 U (t, x, y) q(x) = 0 U (0, •, y) = δ y (3.9) Then : Z(t, x, y) = √ 2πt e (x-y) 2 2t
U (t, x, y). In particular, it follows from Lemma 3.2, that :

U (t, x, y) ≤ C 1 + |x| 1 + |y| (1 + t) 3/2 (t ≥ 1).
(3.10)

Proof of Lemma 3.3

We know that, for every regular function f :

Z f (t, x) := E x exp - 1 2 A q t f (X t )
is solution of :

∂Z f ∂t - 1 2 ∂ 2 Z f ∂x 2 + 1 2 Z f • q = 0, Z f (0, x) = f (x). (3.11)
It suffices, in order to obtain Lemma 3.3, to write :

Z(t, x, y) = lim ε↓0 E x exp - 1 2 A q t f ε (X t ) E x f ε (X t )
where f ε is a family of functions which converges weakly towards δ y , and to use (3.11).

Use of the Laplace transform

We define, for every λ > 0 :

A(λ, x, y) = ∞ 0 e -λt U (t, x, y) dt.
(3.12)

Since Z(t, x, y) is a decreasing function of t, we deduce the following equivalences from the Tauberian theorem :

i) Z(t, x, y) ∼ t→∞ π 2 ϕ q (x)ϕ q (y) t ii) U (t, x, y) ∼ t→∞ √ π 2 √ 2 ϕ q (x)ϕ q (y) t 3/2 iii) ∂ ∂λ A(λ, x, y) = - ∂ ∂λ A(λ, x, y) ∼ λ→0 1 2 √ 2 1 √ λ ϕ q (x)ϕ q (y). (3.13)
We shall now show (3.13). We already deduce from Lemmas 3.2 and 3.3 that : ∂ 2 ∂x 2 ψ = ψ q, with adequate limit conditions in x = ±∞.

lim λ→0 A(λ, x, y) = ∞ 0 U (t, x, y) dt < ∞ (3.14) A(λ, x, y) ≤ C 1 + |x| 1 + |y| (3.15) ∂ ∂λ A(λ, x, y) ≤ C √ λ 1 + |x| 1 + |y| . ( 3
(3.17)

Convergence of the Laplace transform : first step

We get, from (3.9) :

U (t, x, y) = 1 √ 2πt e -(x-y) 2 2t - 1 2 t 0 ds ∞ -∞ e - (x-z) 2 2(t-s) 2π(t -s) U (s, z, y) q(z)dz. (3.18)
Thus, after taking the Laplace transform in the variable t of the two sides of (3.18), we obtain :

A(λ, x, y) = g λ (x, y) - 1 2 ∞ -∞ g λ (x, z) A(λ, z, y) q(z)dz (3.19)
where g λ denotes the density of the resolvent kernel of Brownian motion :

g λ (x, z) = 1 √ 2λ e -|x-z| √ 2λ .
We write (3.19) in the form :

A(λ, x, y) = G λ δ y - 1 2 A(λ, •, y) q(•) (x) (3.20)
with for any Radon measure µ(dz) :

G λ µ (x) := ∞ -∞ g λ (x, z) µ(dz) (3.21)
and we use the resolvent equation :

∂ 2 ∂x 2 G λ µ = -2µ + 2λG λ µ, to obtain : ∂ 2 ∂x 2 A(λ, x, y) = 2λ A(λ, x, y) -2δ y -A(λ, x, y) q(x) . (3.22)
As a consequence, differentiating with respect to λ, then multiplying by √ λ, we obtain :

∂ 2 ∂x 2 √ λ ∂A ∂λ (λ, x, y) - √ λ ∂A ∂λ (λ, x, y) q(x) = 2 √ λ A(λ, x, y) + 2λ
∂ 2 ∂x 2 A(λ, x, y) -A(λ, x, y) q(x) ≤ C √ λ 1 + |x| 1 + |y| (λ → 0) (3.24)
(3.24) is the first step to prove that A(λ, x, y) converges, as λ → 0, to a solution of the Sturm-Liouville equation (3.17).

3.2.7

Convergence of the Laplace transform : limit conditions in x = ±∞

We now examine the limit conditions in x = ±∞.

We come back to equation (3.19) which we differentiate with respect to λ, then we multiply by λ :

√ λ A(λ, x, y) = - 1 2 A(λ, x, y) - 1 2 ∞ -∞ e -|x-z| √ 2λ |x -z| δ y (dz) -A(λ, z, y) q(z)dz - 1 √ 2 ∞ -∞
e -|x-z| √ 2λ A(λ, x, z) q(z)dz. 3.2.8 End of the proof of point 1 of Theorem 3.1

Thus, from the equivalence between i), ii) and iii) which we recalled in (3.13), we get :

E x exp - 1 2 t 0 q(X s )ds X t = y ∼ t→∞ ψ(x, y) t (3.30)
where ψ is solution to :

∂ 2 ψ ∂x 2 (x, y) = ψ(x, y) q(x), lim Thus, from the definition of ϕ q see(3.4) , we get :

ψ(x, y) = C(y) π 2 ϕ q (x). Now, since Z(t, x, y) is symmetric in x and y : ψ(x, y) = Kϕ q (x)ϕ q (y).

(3.32)

It remains to determine the value of K. For this purpose, we write :

ϕ q (x) = E x exp - 1 2 t 0
ds q(X s ) ϕ q (X t ) since ϕ q (X t ) exp -1 2 A q t , t ≥ 0 is a martingale = ∞ -∞ E x exp -1 2 A q t X t = y e -(x-y) 2 2t √ 2πt ϕ q (y)dy

∼ t→∞ K ∞ -∞
ϕ q (x)ϕ q (y) t ϕ q (y) e -(x-y) 2 2t √ 2πt dy

= K ϕ q (x) t E x ϕ 2 q (X t ) ∼ t→∞ K ϕ q (x) t 2 π t since ϕ q (z) ∼ |z|→∞ 2 π |z|. Hence K 2 π = 1, that is : K = π 2 •
Thus, we have obtained point 1 of Theorem 3.1.

3.2.9 Point 2 of Theorem 3.1 may be proven with the help of (3.3), exactly as Theorem 1.1.

Remark 3.2 Under our hypothesis H1 on q, there is the equivalence : Z(t, x, y) ≡ E x exp -1 2 t 0 q(X s )ds X t = y ∼ t→∞ π 2t ϕ q (x)ϕ q (y).

(3.33)

Intuitively, we may think of the bridge of duration t going from x to y as "resembling", as t → ∞, to the concatenation of two brownian motions each being defined on a time interval 0, t 2 , with the first one starting from x and the second one, after time reversal, starting from y, these two parts being independent. If this were true, then :

Z(t, x, y) = E x exp - 1 2 A q t |X t = y = E x exp - 1 2 A q t/2 • E y exp - 1 2 A q t/2 ∼ t→∞ ϕ q (x) t/2 • ϕ q (y) t/2 = 4 π π 2
ϕ q (x)ϕ q (y) t • Thus, comparing with (3.33) the factor 4 π which we just obtained measures, in some sense, the default of independence of these two Brownian components.

Remark 3.3 Theorem 3.1 allows to "penalize long Brownian Bridges". More precisely, for every s ≥ 0 and Λ s ∈ F s :

E x 1 Λs exp - 1 2 A q t X t = y E x exp - 1 2 A q t X t = y -→ t→∞ E x (1 Λs M ϕ s ) (3.34)
with M ϕ s := ϕ q (X s ) ϕ q (x) exp -1 2 A q s , et (M ϕ s , s ≥ 0) is a positive martingale. In other terms, comparing with Theorem 5.1 in [RVY, I], the penalisation is the same for "long bridges" as for Brownian motion itself. Once more see [RVY, III] , we obtain that a long bridge of duration t, as t → ∞, behaves as a standard Brownian motion.

3.2.10 Finally, we show (3.34)

E x 1 Λs exp - 1 2 A q t |X t = y E x exp - 1 2 A q t |X t = y = E x 1 Λs exp - 1 2 A q s E Xs,s exp - 1 2 t s q(X u )du |X t = y E x exp - 1 2 A q t |X t = y ∼ t→∞ E x 1 Λs exp - 1 2 A q s •
ϕ q (X s )ϕ q (y) t -s ϕ q (x)ϕ q (y)

t -→ t→∞ E x (1 Λs M ϕ s ).

  20) where K 0 denotes the Bessel-Mc Donald function with index 0 see [Leb], p

  x, y) satisfies the Sturm-Liouville equation (for any fixed y) :

  hand, differentiating (3.25) with respect to x, we obtain :


Keywords: antiretroviral treatment, men, health care experiences, gender, South Africa

"No one can ask me 'Why do you take that stuff?"': Men's experiences of antiretroviral treatment in South Africa

Molly Fitzgerald, Martine Collumbien, Victoria Hosegood

Introduction

As public HIV treatment programmes providing antiretroviral therapy (ART) continue to expand and develop in Africa, it is important to understand how gender shapes ART access, adherence, health, treatment experience, as well as, the consequences of treatment [START_REF] Hirsch | Gender, sexuality, and antiretroviral therapy: using social science to enhance outcomes and inform secondary prevention strategies[END_REF][START_REF] Hirsch | Social aspects of antiretroviral therapy scaleup: introduction and overview[END_REF]. To date, few studies in developing countries have explored the experiences of men accessing treatment [START_REF] Seeley | HIV/AIDS in fishing communities: challenges to delivering antiretroviral therapy to vulnerable groups[END_REF]. Such a knowledge gap is regrettable given the level of participation by men in ART programmes. In South Africa, of the more than 400,000 people enrolled in public sector programmes, 45% of people receiving ART are men (Department of Health, 2007). Furthermore, there is an increasing, multi-disciplinary evidence base in developed countries documenting the influence of gender on men's health, health behaviours, and health care [START_REF] Courtenay | Constructions of masculinity and their influence on men's well-being: A theory of gender and health[END_REF][START_REF] O'brien | It's caveman stuff, but that is to a certain extent how guys still operate': men's accounts of masculinity and help seeking[END_REF][START_REF] Robertson | Understanding Men and Health. Masculinities, Identity and Well-being[END_REF]. Given many ART programmes are relatively new and in a phase of expansion and development, closely observed studies of individual experiences of treatment in Africa are also timely.

In this paper we examine health, health behaviours and health care experiences of HIV positive men enrolled in a rapidly expanding rural district ART programme in KwaZulu-Natal, South Africa. The study area has high rates of HIV prevalence in the general population; in 2003/4 the peak prevalence in men was 44% (95% CI 38-49%) in 30-34 year olds [START_REF] Welz | Continued very high prevalence of HIV infection in rural KwaZulu-Natal, South Africa: a population-based longitudinal study[END_REF]. No declines in HIV incidence have been observed, overall HIV incidence for the period 2003-2007 was 3.4 per 100 person-years (95% CI 3.1-3.7) [START_REF] Barnighausen | High HIV incidence in a community with high HIV prevalence in rural South Africa: findings from a prospective population-based study[END_REF].

The study uses data collected during in-depth interviews with men, as well as their family members and programme staff. We present men's case studies to provide insight into gendered dimensions to men's health and health care experiences. These include treatment disclosure, attitudes to coping and support, and men's relationship with their families and health care providers. We consider the way in which gender intersects with men's health experiences around ART. [START_REF] Fitzgerald | Stigma and the Social Context of Anti-retroviral Treatment in KwaZulu-Natal, South Africa[END_REF]. In 2005/6, in-depth interviews were conducted with 30 people (8 men, 22 women) initiating treatment in a public ART programme based at a district hospital in rural KwaZulu-Natal. Programme participants attending education sessions were invited to enrol in the research project. The sex ratio of respondents recruited was similar to that in the ART programme. A second interview was conducted after four months with 26 people (6 men, 20 women). By which time, two participants had died, one was hospitalised, and one had left the study area. Additional interviews were conducted with nine family members who acted as treatment supporters and five health service providers from the ART programme. The study used ethnographic methods, including participant observation and a daily log of informal discussions. A research team consisting of Fitzgerald (female, age 36 years) and a research assistant (female, age 29 years) was based in the hospital and lived in the study community.

F

The in-depth interviews with programme participants began by exploring key life events and treatment history. Topic guides were used to facilitate a chronological discussion of events from the time of the individual learning about his or her HIV status to accessing ART treatment. Some of the central topics included stigma, disclosure of HIV status, social support, experiences related to treatment, including health outcomes, adherence, and relationships with staff and other participants. Responses in the first interview were explored in greater detail in the second interview with a particular emphasis on understanding the changing experience over the course of diagnosis and treatment. Most interviews were conducted in Zulu. All interviews were taped, transcribed and translated into English. A detailed analysis was conducted by the lead author [START_REF] Fitzgerald | Stigma and the Social Context of Anti-retroviral Treatment in KwaZulu-Natal, South Africa[END_REF]. In this paper, we select narratives from, or about, male ART participants. All names used in this paper are pseudonyms.

Study area

The district health service serves approximately 168,500 people and covers an area of 1,417 square kilometres. The area is predominantly rural but also includes a township and densely populated peri-urban areas. The population is almost exclusively Zulu-speaking. Few households are engaged in subsistence agriculture, and most are dependent on waged income and state grants [START_REF] Hosegood | Dispensing with marriage: Marital and partnership trends in rural KwaZulu-Natal, South Africa 2000-2006[END_REF]. The unemployment rate, as a percentage of the economically active, is high (39% of people aged 15-65 years in 2001) [START_REF] Case | ACDIS Monograph: Socioeconomic Factors. Mtubatuba: Africa Centre for Health and Population Studies[END_REF]. HIV prevalence estimated through population-based surveys in part of hospital catchment area is very high. In 2005, 38% of antenatal women were HIV infected (38%) [START_REF] Rice | Population and antenatal-based HIV prevalence estimates in a high contracepting female population in rural South Africa[END_REF].

In 2003/4, HIV prevalence in the general population was 22% [START_REF] Welz | Continued very high prevalence of HIV infection in rural KwaZulu-Natal, South Africa: a population-based longitudinal study[END_REF].

The roll-out by the public health service of ART in the district started in September 2004. The project was conducted during a phase of rapid expansion in the district ART programme. At the start of the project in 2005, the programme provided treatment to 142 adults with all initiation and monitoring occurring in the district hospital. In 2006 when the project ended, 5,000 people were being enrolled and monitored at the hospital and local clinics. Voluntary counselling and testing (VTC) was available in all clinics throughout the period.

Results

Responses to questions about HIV treatment

Male participants were acutely aware of the risk that taking ART or attending the hospital might result in family and public gossip about their HIV status. Men's responses ranged from emphatically stating that nobody would ask them outright, to being prepared to deny that they were HIV positive or refusing to answer, should they be accused. Some men took authoritative and defensive positions suggesting that their position in the family, household and community should be protection against being challenged about their HIV status or becoming the subject of gossip.

Mandla was a 31 year old man who had been extremely ill before and after starting ART. Previously a migrant worker, he lost his job due to ill health and returned to live with his wife, children, parents and adult siblings. His family had cared for him during his repeated bouts of illness but despite their help and support, Mandla was adamant that he had not and would not discuss his HIV status with them. Being too weak to walk unaided, Mandla's father accompanied his son for his routine visits to the ART clinic. But despite this, father and son had not discussed the kind of treatment he received. Mandla at first claimed that his father did not know the reason he went to the hospital. Later he reconsidered and suggested that his family probably were aware that he was HIV positive and on treatment despite him never having told them. He described openly taking his ART pills, confident that nobody in his family would dare to ask him directly about them. 'As I said I'm not hiding my pills…I'm not shy about that. But no one can ask me why do you take that stuff, you see.' Mandla appeared to feel that his status as a husband and adult son was enough to secure the support of his family without full disclosure, and should protect him against gossip and discrimination. However, Mandla's interactions with his family around illness and treatment were far more complicated. While he sought to create an impression during interviews of being in control, he swung between expressions of confidence and feelings of failure in his family role and desperation about his situation.

Men's desire to demonstrate confidence in being able to deal with HIV stigma and discrimination infused their language and behaviour during the interviews. Interviews, particularly the first, with men were often characterized by laughter and joking, with men seemingly casual and taking everything lightly. However, men's moods during interviews were more volatile than women's. The bravura with which men started the interview could quickly dissipate when they became less confident and clowning gave way to anxiety and pensiveness. Furthermore, the casual stance was at odds with the content of their responses and the extent to which they had given serious thought to issues related to their health and the implications for themselves and their families. The repeated interviews with men provided important time and rapport needed for men to moderate the style and level of reflection when responding to questions. In the first interview, men tended to focus very much on their health and employmentrelated experience. Although remaining largely pragmatic and with little emotional expression, in subsequent interviews, men became more expansive about social aspects such as concerns about their families or partners. Male participants held strong opinions about the behaviour of health service providers, particularly staff in the treatment programme. It was of the utmost importance to men that staff behaved professionally and that they could be trusted to maintain confidentiality. These high expectations created a potentially fragile relationship between male participants and staff, one in which men might readily stop attending if they perceived a breach in trust or an unsupportive reaction. The identity of health providers as being a 'professional' was felt by some participants as the reason why only health providers should be 'allowed' to talk about HIV and AIDS.

Shifting roles and identities

Men's confidence about their future was undermined by how they, and others, perceived their roles within their families. Most men were unemployed, many having lost their jobs because of illness. In an area of high unemployment, not having paid work is not necessarily stigmatised, however, male participants were unable to be seen to be even actively looking for work given continued illness and the need to regularly attend the hospital. Thus, having failed to fulfil their primary roles as household breadwinners or fathers who provided for their children, many young male men felt that they did not have an essential role in their families.

The story of Dingane and Mandla illustrates the extent to which HIV stigma is layered upon preexisting norms related masculinity and fatherhood. Dingane, a 26 year old man had moved back to live with his parents when he became very ill, lost his job, and his girlfriend died. Dingane had not disclosed his HIV status to anyone in his family. His father, an alcoholic, used abusive and stigmatizing language towards his son. His parents' disappointment and accusations related primarily to their son's 'failure' to start and provide for a family of his own. This was combined with blaming and moral judgements about the cause of Dingane's illness whilst not directly referring to HIV: '…the prostitutes have finished you.' his father was reported to have told him. Mandla, a 31 year old married man whose father had also made disparaging remarks about his inability to provide for his wife, children and parents, appeared to have internalised these negative sentiments: 'Previously I handled [driving in building posts] myself. But now am not able. Maybe I try to do something...and then, I fail. Yes [whispering]...like building a house, I used to build for myself....I was a breadwinner and all -a lot of things.'

Strongly gendered expectations that fathers should only be proud of strong, masculine and successful sons placed an enormous strain on relationships between fathers and their sick adult sons. Such a profound influence was not observed between fathers and their HIV positive daughters. For men, the low self-esteem resulting from failing as a son, father, husband and 7 breadwinner, also appeared to reinforce a particular resistance to disclosing HIV status or treatment to other men. The need to protect oneself against disapproval or being '…made fun of' by other men, particularly those who drank heavily, was a deep concern for men. Fears about the consequences of disclosing to other men had in some cases hindered men's ability to secure care and support from their families.

Men's feelings of failure with regards to their families differed by age. When in confident spirits, younger men were often hopeful that they could succeed in being household heads, partners and fathers. However, such buoyancy was fragile and younger men would readily switch to feelings of failure in building and supporting families. Older men appeared to have a more resigned attitude to their circumstances. Some older men described themselves as having been 'beaten' by life and others were sanguine about their ability to fulfil family and community expectations regardless of their HIV status.

Alcohol and ART

Several men described how, knowing that the ART programme discouraged alcohol use, they had waited to start treatment until managing to stop or reduce alcohol use. While all the men in this study had eventually started treatment, some described having been unsure in the past about whether ART was in fact an option for them at all given that they drank. Those men who initiated changes in their alcohol use themselves, did so without seeking advice or support from health providers. Men differed in their attitudes to long-term treatment and alcohol use. While some men matter-of-factly reported that they did not drink anymore because of treatment, other men were more conflicted. Alcohol was described as an important way that men cope with the kind of anxiety and stress brought about by being HIV-positive. Nonetheless these justifications were accompanied by considerable guilt about drinking while taking ART.

Men also frequently referred to alcohol when discussing HIV disclosure; emphasizing they neither could confide in, nor rely on support from, family members or friends who used alcohol. For Dingane, his fathers' history of alcohol abuse had fostered a sense that he could not rely on his father for support. Furthermore, in trying to cope with HIV, Dingane's father was now an actual threat given the risk of uninhibited talk, rudeness, and violence: 'How can I put it? My father is a drunk. Maybe he will talk about it anywhere he is. Maybe he is in taverns with neighbours [.

..] I have to look at him and see how he reacts at home [...]'

Discussion

In 2007, Jennifer Hirsch wrote 'The intersection of gender and ART represents a critical area for further research to identify the structures that create disparities in access, in order to consider how they might be transformed.' This study is one such starting point with its documentation of men's experiences of ART in a rural district of South Africa.

The study suggests that men's experiences need to be seen in the context of their life course and age. Important stressors for younger men in the programme were feelings of being thwarted in their relationships, ability to form their own household, provide financially for their families, and thus, be unable to meet their own and others' normative expectations. Older men established in their social roles were more secure and better able to disclose and negotiate family support.

Other authors have suggested that older HIV positive people face specific challenges for example, social age-related differentials in HIV disease progression and treatment effects and health provider attitudes around sexuality and perceptions of risk [START_REF] Auerbach | HIV/AIDS and aging: interventions for older adults[END_REF][START_REF] Knodel | AIDS and older persons: an international perspective[END_REF][START_REF] Levy | HIV/AIDS interventions for midlife and older adults: current status and challenges[END_REF], and exclusion from HIV prevention and education programmes [START_REF] Cooperman | Current sexual activity and risky sexual behavior in older men with or at risk for HIV infection[END_REF][START_REF] Lieberman | HIV in older Americans: an epidemiologic perspective[END_REF]. We observed the greatest variation in treatment experiences by life stage rather than simply age, i.e. whether men had a living and geographically accessible wife, partner or parents, in particular mothers.

The study identified alcohol as an important aspect of men's lives before and during treatment.

Men described the impact that alcohol use had on their treatment experience. Several men reported a self-imposed delay in starting treatment until they had been able to stop or reduce alcohol use. It is possible that for some HIV positive men, alcohol dependence allows selfexclusion from treatment programmes, fostering denial about their HIV status. Respondents also thought using alcohol was a particularly masculine way of coping with the stresses of HIV, illness and treatment. Alcohol as a method of coping has been previously reported in studies of recently diagnosed HIV positive men and women in South Africa [START_REF] Olley | Post-traumatic stress disorder among recently diagnosed patients with HIV/AIDS in South Africa[END_REF]. In rural KwaZulu-Natal, strong gender norms exist about alcohol use behaviour and desirability bias may influence drinking behaviour and reporting. In interviews we conducted with 22 women participating in the programme [START_REF] Fitzgerald | Stigma and the Social Context of Anti-retroviral Treatment in KwaZulu-Natal, South Africa[END_REF], none mentioned alcohol in connection to their own treatment. In our study we are unable to assess what role that alcohol use and abuse in men's experiences of depression, psychological distress or ability to adhere to treatment. Alcohol use has been shown to be negatively associated with treatment adherence in South Africa [START_REF] Dahab | That is why I stopped the ART": Patients' & providers' perspectives on barriers to and enablers of HIV treatment adherence in a South African workplace programme[END_REF], but unexplored as a barrier to treatment uptake.

The findings suggest that building understanding about alcohol and substance abuse by participants themselves, and their family and household members, might be useful complement for primary health care and programme providers and counsellors. Increasing awareness by health promoters of issues related to alcohol during the pre-treatment period may assist in timely initiation among alcohol-dependent HIV positive men. Based on the results of a clinical intervention trial to reduce alcohol use by HIV positive people, [START_REF] Kalichman | HIV/AIDS risk reduction counselling for alchol using sexually transmitted infections clinic patients in Cape Town, South Africa[END_REF] suggest intensive alcohol risk reduction and maintenance interventions may be required. The finding that alcohol may be used as a coping strategy among HIV positive men on treatment has potentially important implications for HIV prevention too. The inter-relationships between alcohol use, HIV/STD transmission risk and intimate partner violence has been well documented in South Africa [START_REF] Dahab | That is why I stopped the ART": Patients' & providers' perspectives on barriers to and enablers of HIV treatment adherence in a South African workplace programme[END_REF][START_REF] Dunkle | Transactional sex with casual and main partners among young South African men in the rural Eastern Cape: prevalence, predictors, and associations with gender-based violence[END_REF][START_REF] Kalichman | HIV/AIDS risk reduction counselling for alchol using sexually transmitted infections clinic patients in Cape Town, South Africa[END_REF][START_REF] Olley | Post-traumatic stress disorder among recently diagnosed patients with HIV/AIDS in South Africa[END_REF] and other African countries [START_REF] Wolff | Women who fall by the roadside: gender, sexual risk and alcohol in rural Uganda[END_REF].

To what extent are our findings generalisable? Two aspects need to be noted. First, the participants we interviewed were newly enrolled in the ART programme. As a consequence, their narratives are strongly interwoven with their experiences of recent or ongoing illness. The intensity with which men's described their loss of identity and role within families might have been less acute had we interviewed participants longer established in the programme who were responding well to treatment [START_REF] Nguyen | Adherence as therapeutic citizenship: Impact of the history of access to antiretroviral drugs on adherence to treatment[END_REF] , 2007). Second, the study was conducted at the start of the public ART programme. Since then, it has expanded considerably and received considerable inputs in terms of staff and training. However, it is unlikely that the gendered influences on HIV positive men's health and treatment experiences have undergone any similarly substantive change since 2005/6.

Local treatment programmes require the support of the public health system. In the South African national guidelines on treatment or adherence (Department of Health, 2004), no specific reference is made to issues related to men and treatment. Furthermore, while the guidelines encourage providers to facilitate support for programme participants by engaging with families and communities, no further guidance is offered about how this may vary with the age or gender of participants. Research and programmatic efforts related to gender and ART need to be broader than merely identifying barriers to men's access and adherence to treatment. As ART programmes become established and develop, it is also important to examine how HIV positive men understand and experience health and treatment.

"

  No one can ask me 'Why do you take that stuff?'": Men's experiences of antiretroviral treatment in South Africa

  S'bu, a 43 year old participant felt that it would be wrong to talk about his own or another persons' HIV status: 'You have to keep that secret. You don't have to tell him straight… No, it's not right because he will ask "How do you know? You are not a doctor!'' On the other hand, when male participants felt confident that the health care professionals would assure their privacy, they sought, or reported wanting to seek, their advice and support. Elaborating about his anxiety over disclosure to his fiancé, 26 year old Jabulani explained 'No I didn't inform her [nervously laughing]. I went to the clinic hoping that would give me advice about informing her. They were too busy'

http://mc.manuscriptcentral.com/ac-phm-vcy Health Sciences

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Ciprian Dariescu

Marina-Aura Dariescu

The Axially Symmetric p-Modes of Maxwell Radiation in Milne Spacetime

Keywords: Maxwell equations, Milne Universe, Dipole Radiation PACS: 02.40.-k, 11.27.+d, 98.80.-k, 03.50.De

The aim of this work is to give a fully covariant treatment of the (l = 1, m = 0)-electromagnetic excitations into the linearly expanding (k = -1)-Universe. For the corresponding radiating dipole, the closed form solutions, in the axially symmetric Lorentz-Coulomb gauge, are derived in terms of the Legendre associated functions. In order to get completeness in the physical interpretation, we have turned to hypergeometric functions and compute, in the far-field region, the observables of the entirely radiating field and the effective power.

Introduction

Many years after E. A. Milne published his works, [1], the Universe bearing his name has become a target of investigations within modern cosmology, especially after it has been seen that its predictions are consistent with data on apparent distances and red shift relationship for supernovae, [2]. It seams that our observation platform, as we look out upon the galaxies, is the Minkowski spacetime, while the cosmological events kinematics in this frame are subjected to the rules that give rise to the equations of Milne Cosmology, [3]. The geometrodynamical analysis of the extremal spacetime structures derived as exact solutions of the Einstein's field equations, for a generic FRW background with a quartically self-interacting scalar field as matter-content, has revealed that, while the central system fixed point is gravitationally inconsistent only with the spatially-closed FRW geometries, each of the other two is consistent only with the (k = -1)-family of FRW manifolds. So, once the vacuum is set in one of these two ground states, the spontaneous Z 2 -symmetry breaking creates an anti-de Sitter Universe, which gets filled with massive particles representing the physical field excitations, when slightly perturbed around the settled ground state. Thus, setting k = -1 as compulsory, the central fixed point corresponds to a Milne phase, which, being unstable against the coherent field fluctuations, does primarily turn into an anti-de Sitter one of small curvature value, [4]. In the area of investigation on the transitions between a contracting and expanding phases of the Universe there have been used different theoretical models, as for example the 11-dimensional M-theory on the R 9 × M background, where M is the two dimensional Milne universe, [5], or the one bubble inflationary scenario in an open universe, as an alternative to the standard inflation, [6]. Since one can define a natural vacuum state in the Milne spacetime that corresponds to the usual Minkowski vacuum, it has been concluded that we can gain an insight into the quantum state of a field inside an expanding bubble in terms of the particle spectrum observed by the comoving observers in the Milne universe at late times.

The present paper is generalizing the SO(3, 1)-gauge invariant theory of the dipole radiation in Minkowski Universe [7], where, using the components of the vector potential, solutions to the Maxwell equations, we have written down, for the pure dipole-like radiation, the observables, the Umov-Poynting vector components and the well-known Larmor formula for the effective ra-diated power. The general results can be related to the ones from [8], where the authors are deriving a solution of Maxwell's equations in an anisotropic Universe, for which one direction is linearly expanding and the others are static, and a dilaton field is present. Our interest in considering a Milne Universe is sustained by somewhat recent papers of A. Dev where, based on a careful analysis of the latest gravity lensing and high-z supernova data, it has been concluded that the existence of a present (quasi)Milne stage has not been completely ruled out, [9]. Apart from the internal beauty of the subject, the general problematic has deep implications in explaining the origin of the cosmological dipole, generally interpreted as a Doppler effect arising from the motion of the Earth relative to the CMBR frame. Recently, the idea that it actually results from ultra-large scale isocurvature perturbations has been examined in the context of open cosmologies, leading to the conclusion that the open universe must be extremely close to a flat manifold [10].

The Axially Symmetric Lorentz-Coulomb

Gauge for the Milne Metric

The linearly expending, k = -1, hyperbolic image of the Minkowski spacetime does actually describe the so-called Milne Universe,

ds 2 = t 2 (dχ) 2 + sinh 2 χ dΩ 2 -(dt) 2 , dΩ 2 = (dθ) 2 + sin 2 θ(dϕ) 2 , ( 1
)
where θ and ϕ stand for the usual angular spherical coordinates, and the local coordinates χ and t are properly related to the radial coordinate r and the Minkowski time τ by the Milne relations

r = t sinh χ , τ = t cosh χ , ( 2
)
so that the initial Minkowski metric in spherical coordinates

ds 2 (0) = (t cosh χdχ + sinh χdt) 2 + t 2 sinh 2 χdΩ 2 -(cosh χdt + t sinh χdχ) 2
(3) does actually become the celebrated Milne-Universe spacetime interval.

As the title goes, in the followings, we would like to give a full treatment on the equivalent vertical dipole radiation as it is percieved by an observer from (within) the Milne world, i.e. the complete set of solutions to the Maxwell equations

F ab ; b = 0 , where F ab = A b;a -A a;b , ( 4
)
in the axially symmetric Coulomb-Lorentz gauge-fixing condition,

A a ;a = 0 , with A 4 = 0 = A 3 and

∂A A ∂ϕ = 0 , ( 5
)
where the index A = 1, 2. The semi-colons stand for the Levi-Civita covariant derivatives with respect to the metric (1) within the Milne orthogonal frame whose dual bases is

ω 1 = t dχ , ω 2 = t sinh χdθ , ω 3 = t sinh χ sin θdϕ , ω 4 = dt . ( 6
)
One can straightly work out the Levi-Civita connection coefficients, Γ abc = η ad Γ d

• bc , employing the first Cartan structure equation

dω a = Γ a • [bc] ω b ∧ ω c , 1 ≤ b < c ≤ 4 , (7)
where Γ (ab)c = 0 comes from the metricity condition g ab;c = 0, with g ab = η ab . It yields therefore the algebraic essential coefficients

Γ 212 = Γ 313 = coth χ t , Γ 323 = cot θ t sinh χ , Γ 141 = Γ 242 = Γ 343 = 1 t , ( 8
)
all of them being anti-symmetric in the first two indices. Now, because of the primarly gauge-fixing conditions, A 3 = 0 = A 4 , the U (1)-connection reads A = A B ω B , with B = 1, 2, leading, within the light of the axial symmetry A B|3 = 0, to the U (1)-curvature

F = dA = A B|C ω C ∧ ω B -A B|4 ω B ∧ ω 4 + A B dω B , ( 9
)
which actually states the essential components of the Maxwell tensor,

F 12 = A 2|1 -A 1|2 + coth χ t A 2 , F A4 = -A A|4 + 1 t A A , A = 1, 2 . ( 10
)
As for the Lorentz condition,

A B|B + A B Γ a •Ba = 0 , ( 11
)
this does concretely read, using (8),

1 sin θ ∂ ∂θ (sin θA 2 ) = -sinh χ ∂A 1 ∂χ + 2 coth χA 1 . ( 12
)
Now, let us launch the quest for the dipole-like solutions of the Maxwell equations ( 4) in the Milne Universe, parametrized by the metric (1). As it can be easily checked, because of the chosen gauge, the third Maxwell equation, F 3b ; b = 0, goes straightly trivial, yielding zero everywhere. With respect to the fourth equation, things are a bit more involved, yet the final result is the same. In what it concerns the second Maxwell equation, F 2b ;b = 0, this is always fulfilled by the component

A 2 = - sinh χ sin θ ∂A 1 ∂χ + 2 coth χA 1 sin θdθ , ( 13
)
gotten from [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF], once the component A 1 evolves as a solution to its own propagation equation, F 1b ; b = 0. Thence, this is the one we have to deal with for it contains, by its consequences, all the information we need about the structure of the radiating field. Evolving the covariant derivatives, plugging in the Maxwell tensor components, (10), and the connection coefficients (8), it can be written as

A 2|12 + cot θ t sinh χ A 2|1 + coth χ t A 2|2 + cot θ coth χ t 2 sinh χ A 2 = A 1|22 + cot θ t sinh χ A 1|2 -A 1|44 - 3 t A 1|4 - 1 t 2 A 1 , ( 14
)
where the left-hand-side should be entirely converted to an expression in A 1 alone. This can be done applying a trick with respect the non-commuting derivatives A 2|12 and the gauge-fixing condition. It finally yields the equation

A 1|11 + 4 coth χ t A 1|1 + A 1|22 + cot θ t sinh χ A 1|2 -A 1|44 - 3 t A 1|4 + 2 t (coth χ) |1 + 4 t 2 coth 2 χ - 1 t 2 A 1 = 0 , ( 15
)
which becomes, switching to the derivatives

{∂ A (•)} = ∂(•) ∂χ , ∂(•) ∂θ and ∂(•) ∂t , 1 t 2 ∂ 2 A 1 ∂χ 2 + 4 coth χ ∂A 1 ∂χ + 1 sinh 2 χ ∂ 2 A 1 ∂θ 2 + cot θ ∂A 1 ∂θ + 2A 1 - ∂ 2 A 1 ∂t 2 + 3 t ∂A 1 ∂t - 3 t 2 A 1 = 0 . ( 16
)
In principle, once we have the components A 1 and A 2 , the structure of the electromagnetic field is characterized by the classical observables

E 1 = - ∂A 1 ∂t + 1 t A 1 , E 2 = - ∂A 2 ∂t + 1 t A 2 , B 3 = 1 t ∂A 2 ∂χ - 1 sinh χ ∂A 1 ∂θ + coth χ A 2 , ( 17
)
standing for the components of the electric field intensity and the magnetic induction, respectively. In practice, however, things are not so simple because of the complexity induced by the special functions involved with the solutions.

Considering

A 1 = F (χ)Y (θ)T (t) , ( 18
)
in the propagation equation, one can easily separate the function Y : [0, π] → C, as being the Legendre polynomial of degree , namely Y (θ) = P (cos θ).

Since the case of multiple excitations = 2, 3, . . . ∈ N is too much involved (mathematically) to make it readable in some simple terms, we shall stick with the pure dipole radiation case, = 1. Thus, for the F and T parts, we come to the two eigenvalue problems [START_REF] Williams | Topics in Quantum Mechanics[END_REF],

(a)

d 2 F dχ 2 + 4 coth χ dF dχ + µF = 0 (b) t 2 d 2 T dt 2 + 3t dT dt -ΩT = 0 . ( 19
)
The algebraic constraint between the constant scalars, µ + Ω = 3 is coming effectively from the isotropic uniform expansion of the Milne Universe and has no obvious analogue into Minkowski spacetime.

Let us start with the equation (19.b) and solve the problem considering the eigenvalue equation on (t ∈ R + ),

Dψ λ = λ ψ λ , λ ∈ R (20)
for the Hermitian operator

D = t 2 d 2 dt 2 + 2t d dt . ( 21
)
We notice that, performing the eigenfunction substitution

ψ λ (t) = t a 1/2 T λ (t) , a ∈ R + , ( 22
)
we get precisely the (19.b) equation for T , with Ω = λ -3/4. Thence, both Ω and µ are not only real-valued constants, but we can also specify their structure with respect to a single essential wave-number-like, real-valued, parameter k, which turns the continuous spectrum of fundamental solutions to (20),

φ k (t) = (2πa) -1/2 ψ λ(k) (t) , (∀) k ∈ R , ( 23
)
into a complete set of orthonormal eigenfunctions; both of these properties being expressed in terms of δ-functions. That is coming around as it follows. With (21), the eigenvalue equation (20) reads

t 2 d 2 ψ k dt 2 + 2t dψ λ dt = λψ λ , ( 24
)
leading to the set of independent fundamental solutions. Choosing the sign for the so-called positive frequency solutions,

e -iωt ∼ exp -ik ln t a ,
and setting

λ = - 1 4 -k 2 , ( 25
)
the corresponding generic solution is

ψ λ(k) (t) = t a -1/2 exp -ik ln t a ≡ ψ k (t) , ( 26
)
filling the property

(ψ k , ψ k ) = 2πa δ(k -k) .
Thus, the whole set of eigenfunctions,

φ k (t) = (2πt) -1/2 exp -ik ln t a , ( 27
)
is not only orthonormal but also complete. Thence, from Ω = λ -3/4 and µ = 3 -Ω, one gets, because of (25), the k-dependent structures

Ω = -(1 + k 2 ) , µ = 4 + k 2 , ( 28
)
which have to be considered in working out the mode-solutions of the basic equations (19). Since (19.b) does already have the Euler canonical form of the second order differential equation, its integration is straightforward, leading to the positive-frequency-like solution

T k (t) = a t exp -ik ln t a ( 29
)
which is perfectly consistent with what we have obtained from the two formulae (22, 26).

Concerning the amplitude function F (χ), solution of (19.a), we have to work on it a little bit more. First, the variable substitution z = cosh χ is needed in order to cast the equation into the more familiar form (z 2 -1)

d 2 F dz 2 + 5z dF dz + µF = 0 . ( 30
)
Secondly, there is a catch suggested by its similarity to the canonical equation of the Legendre associated function,

(z 2 -1) d 2 S dz 2 + 2z dS dz -n(n + 1) + m 2 z 2 -1 S = 0 , ( 31
)
where

S(z) = {P m n (z) , Q m n (z)}.
Once the following substitution is applied,

S = (z 2 -1) ν F , with ν ∈ R , ( 32
)
(31) becomes, in terms of F , (z 2 -1)

d 2 F dz 2 + (4ν + 2) dF dz + (2ν) 2 + 2ν -n(n + 1) + (2ν) 2 -m 2 z 2 -1 F = 0,
(33) and turns into the form (30) for the completely determined parameters

ν = 3 4 , m = 3 2 , n = - 1 2 ± ik , ( 34
)
where the latter comes from the second degree equation

n 2 + n + [µ -2ν(2ν + 1)] = 0.
Putting everything back, we have shown that the Milne radial functions F (χ) are given by the expressions

F k (χ) = sinh -3/2 χ P 3/2 -1 2 ±ik (cosh χ) , Q 3/2 -1 2 ±ik (cosh χ) , ( 35
)
where the signs in front of ik do respectively account for the out-going and in-coming modes. Thence, for completeness reasons, by including a constant factor N , of physical dimension T esla • meter, the Milne radial component of the four potential A, does explicitly read

A 1 (χ, θ, t) = N cos θ sinh 3/2 χ P 3/2 -1 2 ±ik (cosh χ) , Q 3/2 -1 2 ±ik (cosh χ) × a t exp -ik ln t a . ( 36
)
Concerning the declination component, A 2 , inserting into (13) the Laplace decomposition (18) and the expression Y = cos θ, it yields

A 2 (χ, θ, t) = G(χ)T (t) sin θ , ( 37
)
where the corresponding radial function G is defined by

G(χ) = - sinh χ 2 dF dχ + 2 coth χ F . ( 38
)
In terms of the variable z and using (32), namely

F = (z 2 -1) -3/4 S 3/2 -1 2 ±ik (z) , ( 39
)
the G-function is given by the expression

= - 1 2 (z 2 -1) -3/4 (z 2 -1) d dz S 3/2 -1/2±ik (z) + z 2 S -1/2±ik (z) , ( 40
)
which, by employing the properties of the Legendre associated functions [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF], turns into

G(z) = (z 2 -1) -3/4 4 (1 ± ik)S 3/2 -3/2±ik (z) + (1 ∓ ik)S 3/2 1/2±ik (z) , ( 41
)
so that the Milne declination component does explicitly read

A 2 (χ, θ, t) = N sin θ 4 sinh 3/2 χ (1 ± ik) P 3/2 -3 2 ±ik (cosh χ) , Q 3/2 -3 2 ±ik (cosh χ) + (1 ∓ ik) P 3/2 1 2 ±ik (cosh χ) , Q 3/2 1 2 ±ik (cosh χ) a t exp -ik ln t a . ( 42
)
3 The Observables

Although good looking as closed form solutions, the two expressions, (36, 42), are not physically transparent, particularly with respect to their longrange behavior, where exp(-2χ) 1. To get this matter clear, we have to come back to (30) and to work out directly its mode-solutions, in terms of easier handling special functions, such as the hypergeometric ones. In this respect, the following variable substitution is needed:

z = 1 -2x ⇔ x = - 1 2 (z -1) ⇔ x = -sinh 2 χ 2 .
It turns the amplitude function equation (30) into the form

x(1 -x) d 2 F dx 2 + 5 2 -5x dF dx -µF = 0 , where µ = 4 + k 2 , ( 43
)
which is the same with the hypergeometric functions' equation

x(1 -x) d 2 F dx 2 + [γ -(α + β + 1) x] dF dx -αβF = 0 , ( 44
)
possessing the two linearly independent solutions

F R (x) = F [α, β, γ; x] , F S (x) = x 1-γ F [β -γ 1, α -γ + 1, 2 -γ; x] . ( 45
)
The notation F [a, b, c; x] stands for the corresponding hypergeometric func-2 F 1 [a, b, c; x] and the subscripts R, S indicate the respective behavior at the origin, x = 0. R stands for non-singular, i.e. regular, while S signals out the presence of a singularity at x = 0 and therefore, it stands for singular. Thence, by simply identifying the coefficients in the two equations ( 43) and (44), it yields

α = 2 -ik , β = 2 + ik , γ = 5 2 (46)
so that, according to (45), the generating solutions of the differential equation ( 43), for the Milne radial function F , are given by the expressions

F R (x) = F 2 -ik, 2 + ik, 5 2 ; x , F S (x) = x -3/2 F 1 2 -ik, 1 2 + ik, - 1 2 ; x . ( 47
)
The only problem with them is that, because of x = -sinh 2 (χ/2) and χ ∈ [0, ∞), the variable x goes beyond the "traditional" domain of convergence, |x| ≤ 1, of the hypergeometric series 2 F 1 and therefore, we have to use in (47) the property

F [a, b, c; x] = (1 -x) -a F a, c -b, c ; x x -1 , ( 48
)
to clear up this matter. Accordingly, taking the steps backward to the radial modes basic equation (19.a), its fundamental the solution(s) read

(a) F R (χ) = cosh 2ik (χ/2) cosh 4 (χ/2) F 2 -ik, 1 2 -ik, 5 2 ; tanh 2 χ 2 , (b) F S (χ) = i cosh 2ik (χ/2) sinh 3 (χ/2) cosh(χ/2) F 1 2 -ik, -(1 + ik), - 1 2 ; tanh 2 χ 2 . ( 49
)
The first good thing about them regards the very short-range behavior of the respectively generated Maxwell field. Indeed, as χ → 0, the first amplitude function, given by (49.a), describes a vertically directed homogeneous electric field; extremely similar to the one inside a plane capacitor, while the Milne radial F S (χ) does actually account for the Maxwell field structure of an origin-centered, vertically directed, electric dipole of momentum p =

E (cl) 1 = -8 (kN ) a 2 r 3 cos θ , E (cl) 2 = -4 (kN ) a 2 r 3 sin θ , (50)
and

E (dip) 1 = - ∂V ∂r = p 2πr 3 cos θ ≡ E (cl) 1 , E (dip) 2 = - 1 r ∂V ∂r = p 4πr 3 sin θ ≡ E (cl) 2 , ( 51
)
where the constant amplitude factor N is defined by

kN = - p 16πa 2 . ( 52
)
Thence, in the case = 1, the amplitude function (49.b) is the basic ingredient we need in order to describe the electric dipole radiation in the Milne Universe. Envisaging the relation (38), the two essential components of the 4-potential A do respectively read

(a) A 1 = N F S (χ)T k (t) cos θ , (b) A 2 = N G S (χ)T k (t) sin θ , (53)
where the time-dependent part, for positive-frequency modes is given by (29). It yields therefore, from ( 17),

(a) E 1 = i N ka t 2 F S (χ) exp -ik ln t a cos θ , (b) E 2 = i N ka t 2 G S (χ) exp -ik ln t a sin θ , (c) B 3 = N a t 2 dG S dχ + coth χG S + 1 sinh χ F S exp -ik ln t a sin θ . ( 54
)
Now, it comes the second good thing about the radial amplitude function given by (49.b). Since, at large positive values of χ, accounting for the radiation zone, it yields tanh(χ/2) → 1 and therefore, [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF],

F 2 -ik, -1 -ik, - 1 2 ; 1 = Γ -1 2 Γ(-1 + ik) Γ 1 2 + ik where Γ(2ik) = 2 2ik 2 √ π Γ(ik) Γ 1 2 + ik , Γ(ik) = (-1 + ik) Γ(-1 + ik) , one gets F 1 2 -ik, -1 -ik, - 1 2 ; 1 =
(1 -ik) 2 2ik ,
so that, in the far-field region, where

i sinh 3 χ 2 • cosh 2ik χ 2 cosh χ 2 ≈ 16i • 2 -2ik • e -2χ • e ikχ ,
the F -radial function reads

F S (χ) = 16(k + i) e -2χ e ikχ . ( 55
)
In light of (38), -within the same approximation -the G-radial function becomes

G S (χ) = -4ik(k + i)e -χ e ikχ , (56)
casting the 4-potential essential components (53), into the form

(a) A 1 = 16 N a t (k + i) e -2χ exp ik χ -ln t a cos θ , (b) A 2 = -4i N ka t (k + i) e -χ exp ik χ -ln t a sin θ , (57)
so that, the entirely radiating Maxwell field of an electric dipole in the Milne Universe, turns out to be described by the classical observables

(a) E 1 = 16i N ka t 2 (k + i) e -2χ exp ik -ln t a cos θ , (b) E 2 = 4 N k a t 2 (k + i) e -χ exp ik χ -ln t a sin θ , (c) B 3 = 4 N k 2 a t 2 (k + i) e -χ exp ik χ -ln t a sin θ , ( 58
)
where, in the far-field region, exp(-2χ) 1, it survive only

E 2 = 4 N a ka t 2 k + i e χ exp ik χ -ln t a sin θ = B 3 , ( 59
)
expressed in Heaviside units. The imaginary unit, i, accompanying the dimensionless wave-number parameter, k, has no obvious analogue in the plane Minkowski spacetime. Nevertheless, the structure (59) of the radiating field should not take us by surprise if we have in mind the cosmological principle related to the linear expansion (of the considered Universe). The combination χ -ln(t/a), appearing in the phase is nothing else but "minus" the retarded null-coordinate, u = ln(t/a) -χ, for out-going wave fronts, exactly similar to the retarded null-coordinate

U = τ -r = t cosh χ -t sinh χ = te -χ ⇒ U = ae u
in Minkowski spacetime. In addition, from the red-shift formula,

- ∂Φ ∂t = ω(t) = a t ω 0 , (60)
where φ = k (χ -ln(t/a)), it yields the physical meaning of the wave parameter, k = aω 0 .

The Effective Radiated Power

Globally, we have come to the problem of the effective radiated power, at the Milne cosmic time, t, defined as

P(t) = (4π) Π 1 (χ, θ, t)ω 2 ∧ ω 3 , ( 61
)
where, in the far field region,

ω 2 = 1 2 te χ dθ , ω 3 = 1 2 te χ sin θdϕ , ( 62
)
and Π 1 (χ, θ, t) stands for the essential radial component of the Poynting vector

Π 1 = 2 |E 2 | 2 = 8 |N 2 a 2 t 4 e 2χ k 4 (k 2 + 1) sin 2 θ . ( 63
)
Thence, the expression (61) gets explicitly

P(t) = 16π 3 |N | 2 a t 2 k 4 (k 2 + 1) (64)
and the effective radial component of the Poynting vector can be written as,

Π = p 2 0 ω 4 0 128π 2 k 2 + 1 k 2 a t 2 sin 2 θ r 2 , ( 65
)
where we have inserted the heuristic norm factor

N = - p 0 16πa 2 k , ( 66
)
ω 0 = k/a and p 0 stands for the Minkowskian dipole p 0 = q 0 d, d = 2h.

In the radiation zone, these results can be expressed in terms of a better parameter than the maximum charge q 0 , namely the excitation current amplitude, I 0 , that feeds-in the radiating system. It comes from the definition law

I(t) = d dt q(t) ,
where I(t) is the instantaneous feeding current and q(t) is the "accumulated" charge at one of the electric dipole, given by the Milne time-variation law

q(t) = q 0 exp -ik ln t a . ( 67
)
Thence, it yields

I(t) = -i kq 0 a a t exp -ik ln t a = I 0 a t exp -ik ln t a ( 68
)
and therefore the excitation current amplitude reads

I 0 = -i kq 0 a , (69)
which also implies q 0 = i(a/k)I 0 . It is precisely the latter relation one needs in order to get the concrete meaning of the heuristic factor

N = - 2q 0 h 16πka 2 , N = - i 8π I 0 h k 2 a . ( 70
)
Thus, whole structure of the pure dipole radiation, in the Milne Universe, turns out to be described by the relations

(a) E 2 ( r, t) = -i I 0 h 4πa (k + i) a t exp ik χ -ln t a sin θ r = B 3 ( r, t) , (b) Π( r, t) = I 2 0 h 2 32π 2 k 2 + 1 a 2 a t 2 sin 2 θ r 2 , (c) P(t) = I 2 0 h 2 12π k 2 + 1 a 2 a t 2 , (71)
with the physically important meaning of the wave number-like parameter, k = aω 0 and the physical radius given by r = t sinh χ ⇒ te χ ≈ 2r, at large (positive) values of χ.

Conclusions

The present paper deals with the SO(3, 1)-gauge invariant treatment of the (l = 1, m = 0)-electromagnetic excitations in the Milne Universe. Using the complete set of solutions to the Maxwell equations in the axially symmetric Lorentz-Coulomb gauge, we get the radial and the declination components of the four potential, in terms of the Legendre associated functions. In order to put the results in a more physically transparent form, we turn to hypergeometric functions and compute, in the radiation zone, the classical observables. In the far-field region, the structure of the phase appearing in the explicit expressions of the surviving electromagnetic components, E 2 = B 3 , allows us to give a natural explanation to the red-shift formula. In what it concerns the effective radial component of the Poynting vector given by (65), one can make the following comment: since a is the cosmological length scale, being at least few billion (light) years, the values of k = ω 0 a are extremely high, even in the ULF-portion of the electromagnetic spectrum, so that (k 2 + 1)/k 2 ≈ 1 and therefore in the (local) Minkowskian limit, where t = a + τ and |τ | the expression (65) turns into the form

Π p 2 0 ω 4 0 128π 2 sin 2 θ r 2 , ( 72
)
which leads to the effectively radiated power (in Heaviside units)

P = p 2 0 ω 4 0 48π . ( 73
)
Seemingly, there is a difference with respect to the usual Larmour law, given by a steady (1/4)-factor which affects both the well-known results for the Minkowski spacetime:

Π 0 = p 2 0 ω 4 0 32π 2
sin 2 θ r 2 ; P 0 =

p 2 0 ω 4 0 12π . ( 74
)
Actually, that does only come from the naive identification of the effective electric dipole p from the Milne Universe with the one, p 0 , from Minkowski.

In fact, because of the Milne horizon at t = 0 (in a manner of speaking) just one of the charges of the initial Minkowski dipole makes it into the Milne patch (χ, t) > 0, which we have considered, while the other one gets trapped, behind the horizon, into the lower patch. Thence, for all the observers living within the upper Milne Universe, the concretely oscillating dipole has the momentum p = q 0 h = p 0 /2 ⇔ p 0 = 2p. Once the heuristic norm factor is rescaled according to the above relation, i.e. being entirely expressed in terms of p, the Umov-Poynting vector and the radiated power get 4-times bigger and the complete agreement with Larmour's expressions (74) is restored. Also, one may notice the clear derivation of the 1/(distance) 2 -law of the radiating field intensity with respect to the physical (cosmological) distance d ph (t) = a(t) • r = t • r. Last but not least, it can be proven, from (71.c), that the radiation resistance of the dipole (antenna), namely

R = 80π 2 h λ 2 ,
where λ = 2πc/ω 0 , is Milne cosmologically invariant.

Acknowledgement

The warm hospitality of the Gravitation and Cosmology Group from the Federal University of Paraiba is deeply acknowledged. Special thanks go to Floyd Williams and Carlos Romero for inspiring discussions. The authors are most grateful to the anonymous referee for insightful suggestions in improving the original form of our paper.