ally in n” to mean, almost surely A
n
occurs for all but finitely many n in measure-
theoretic terms this corresponds to the assertion that the event S
n
T
m≥n
A
n
has null complement.
3.1 Case of first integral
Coupling of the first two iterated integrals is based on the Ben Arous et al. 1995 coupling construction for
B, R
B d t; we begin with a brief description of this in order to establish notation.
Figure 1: Plots of a two coupled Brownian motions A and B, b the difference
W = B − A = B0 − A0 + R
J − 1 d A. The coupling control J switches be- tween values
+1 “synchronous coupling” and −1 “reflection coupling”. In the figure, switches to fixed periods of
J = +1 are triggered by successive crossings of
±1 by W . Co-adapted couplings are built on two co-adapted Brownian motions
A and 386
B begun at different locations A0 and B0: we shall suppose they are related by a stochastic integral
B = B0 + R
J d A, where J is a piece-wise constant ±1-valued adapted random function. The coupling is defined by specifying J:
W =
B − A =
B0 − A0 + Z
J − 1 d A , 1
so that W is constant on intervals where J = 1 holding intervals, and evolves as
Brownian motion run at rate 4 on intervals where J = −1 intervals in which W
is run at full rate. The coupling is illustrated in Figure 1. So our coupling problem is reduced to a stochastic control problem: how
should one choose adapted J so as to control W and V = V 0 +
R W d t to
hit zero simultaneously? We start by noting that the trajectory
W, V breaks up into half-cycles ac- cording to successive alternate visits to the positive and negative rays of the axis
V = 0. We can assume V 0 = 0 without loss of generality; we can manipu- late
W and V to this end using an initial phase of controls We adopt a control strategy as follows: if the
n
th
half-cycle begins at W = ±a
n
for a
n
0 then we compute a level
b
n
depending on a
n
, with b
n
≤ a
n
≤ κb
n
for some fixed κ 1,
and run this half-cycle of W at full rate J = −1 until W hits ∓b
n
or the half- cycle ends. If
W hits ∓b
n
before the end of the half-cycle then we start a holding interval
J = 1 until V hits zero, so concluding the half-cycle. Set a
n+1
to be the absolute value of
W at the end of the half-cycle. We will call the holding interval the Fall of the half-cycle and will refer to the initial component as the Brownian
component or BrC. The construction is illustrated in Figure 2. With appropriate choices for the
a
n
and b
n
, it can be shown that this control forces
W, V almost surely to converge to 0, 0 in finite time. To see this, note the following. By the reflection principle applied to a Brownian motion
B begun at
0, P [BrC duration
≥ t
n
] ≤
P [2 |Bt
n
| ≤ a
n
+ b
n
] ≤
a
n
+ b
n
√ 2πt
n
. 2
Simple dynamical arguments allow us to control the duration of the Fall: Fall duration
≤ max
V during BrC b
n
≤ BrC duration × max W during BrC
b
n
, 3
387
Figure 2: Illustration of two half-cycles for the case b
n
= a
n
2, κ = 2, labelling Fall and BrC for first half-cycle.
which we can combine with the following for x
n
0: P [max W during BrC
≥ x
n
] =
P [2B + a
n
hits x
n
before − b
n
] =
a
n
+ b
n
a
n
+ b
n
+ x
n
. 4
We now use a Borel-Cantelli argument to deduce that Duration of half-cycle
n ≤
µ 1 +
x
n
b
n
¶ t
n
5 for all sufficiently large
n, so long as X
n
a
n
√ t
n
∞ , X
n
1 1 + x
n
2a
n
∞ . 6
Bear in mind, we have stipulated that b
n
≤ a
n
≤ κb
n
. Now this convergence is ensured by setting
√ t
n
= x
n
= a
n
n
1+α
for some α 0, in which case we obtain
Duration of half-cycle n
≤ µ
1 + κx
n
a
n
¶ t
n
≤ ¡
1 + κn
1+α
¢ a
2 n
n
2+2α
. 7
If we arrange for a
n
≤ κn
2+β
then the sum of this over n converges, since we
can choose α 2β3. Thus we have proved the following, which is a trivial
generalization of Ben Arous et al. 1995, Theorem 2.1: 388
Theorem 3.1 Suppose the evolution of
W, R
W d t is divided into half-cycles as described above: if the
n
th
half-cycle begins at W = ±a
n
, then it is run at full rate till
W hits ∓b
n
and then allowed to fall to the conclusion of the half-cycle. The fall phase is omitted if the half-cycle concludes before
W hits ∓b
n
. Our control consists of choosing the
b
n
; so long as a
n
κ ≤ b
n
≤ min{a
n
, 1n
2+β
} for all sufficiently large
n for some constants κ and β 0 , then W, R
W d t converges to
0, 0 in finite time.
Remark 3.2 By definition of
a
n
we know a
n
≤ b
n−1
≤ 1n − 1
2+β
, so it is feasible to choose
b
n
such that a
n
κ ≤ b
n
≤ min{a
n
, 1n
2+β
} for all large n.
Remark 3.3 Note that
a
n
is determined by the location of W at the end of half-
cycle n − 1.
Remark 3.4 We can assume the initial conditions W
= 1, V = 0 otherwise we
can run the diffusion at full rate till V hits zero, as can be shown to happen almost
surely, then re-scale accordingly. It then suffices to set b
n
= min{a
n
, 1n + 1
2+β
}. However this is not the only option; for example Ben Arous et al. 1995 use
b
n
= a
n
2. Note, in either case we find a
n+1
≤ b
n
≤ a
n
≤ κb
n
for κ = 2.
3.2 Controlling two iterated integrals
