We have just proved that T -meshes are either disjoint or one is contained in the other, so maximal T -meshes must be mutually disjoint. Let O
T
:= {x, t ∈ R
2 c
: t ≥ T }\N
T
. It is easy to see that O
T
⊂ R × T, ∞. Consider a point z = x, t ∈ R
2
with t T . If z 6∈ O
T
, then by Lemma 2.7 there exist ˆr
∈ ˆ W
r
z and ˆl ∈ ˆ
W
l
z with the property that there does not exist a z
′
= x
′
, t
′
with t
′
≥ T such that ˆr ∼
z
′
in
ˆl, hence by Lemma 3.3, z is not contained in any T-mesh. On the other hand, if z
∈ O
T
, then by Lemma 2.7 b and the nature of convergence of paths in the Brownian web see [SS08, Lemma 3.4 a], there exist ˆr
∈ ˆ W
r
and ˆl ∈
ˆ W
l
starting from points x
−
, t and x
+
, t, respectively, with x
−
x x
+
, such that ˆrs = ˆls for some s ∈ T, t. Now, setting
u := inf {s ∈ T, t : ˆrs = ˆls} and z
′
:= ˆru, u, by Lemma 3.3, M
z
′
ˆr, ˆl is a maximal T -mesh that contains z.
3.2 Reversibility
Recall from 1.15 the definition of the image set N
T
of the Brownian net started at time T . It follows from [SS08, Prop. 1.15] that the law of N
−∞
is symmetric with respect to time reversal. In the present section, we extend this property to T
−∞ by showing that locally on R × T, ∞, the law of N
T
is absolutely continuous with respect to its time-reversed counterpart. This is a useful property, since it allows us to conclude that certain properties that hold a.s. in the forward picture
also hold a.s. in the time-reversed picture. For example, meeting and separation points have a similar structure, related by time reversal. Note that this form of time-reversal is different from,
and should not be confused with, the dual Brownian net.
We write µ ≪ ν when a measure µ is absolutely continuous with respect to another measure ν, and
µ ∼ ν if µ and ν are equivalent, i.e., µ ≪ ν and ν ≪ µ.
Proposition 3.6. [Local reversibility] Let
−∞ S, T ∞ and let N
T
be the image set of the Brownian net started at time T . Define R
S
: R
2
→ R
2
by R
S
x, t := x, S − t. Let K ⊂ R
2
be a compact set such that K, R
S
K ⊂ R×T, ∞. Then
P[N
T
∩ R
S
K ∈ · ] ∼ P[R
S
N
T
∩ K ∈ · ]. 3.62
Proof By the reversibility of the backbone, it suffices to prove that
P[N
T
∩ K ∈ · ] ∼ P[N
−∞
∩ K ∈ · ]. 3.63
Choose some T s min {t : x, t ∈ K}. By [SS08, Prop. 1.12], the set
ˆ ξ
K s
:= { ˆ
πs : ˆ π ∈ ˆ
N K} 3.64
is a.s. a finite subset of R, say ˆ
ξ
K s
= {X
1
, . . . , X
M
} with X
1
· · · X
M
. 3.65
For U = T, −∞, let us write
ξ
U s
:= {πs : π ∈ N , σ
π
= U}. 3.66
By Lemma 2.7 and the fact that s is deterministic, for any z = x, t ∈ K, one has z ∈ N
U
if and only if there exist ˆr
∈ ˆ W
r
z and ˆl ∈ ˆ
W
l
z such that ˆr ˆl on [s, t and ξ
U s
∩ ˆrs,ˆls 6= ;. Thus, we can write
N
U
∩ K = [
i ∈I
U
N
i
, 3.67
838
where N
i
:= z = x, t ∈ K : ∃ˆr ∈ ˆ
W
r
z, ˆl ∈ ˆ
W
l
z s.t. ˆr ˆl on [s, t and ˆrs
≤ X
i
, ˆls ≥ X
i+1
3.68 and
I
U
:= i : 1 ≤ i ≤ M − 1, ξ
U s
∩ X
i
, X
i+1
6= ; .
3.69 It follows that
P[N
U
∩ K ∈ · ] = E h
P h [
i ∈I
U
N
i
∈ · {X
1
, . . . , X
M
} ii
, 3.70
where P
h [
i ∈I
U
N
i
∈ · {X
1
, . . . , X
M
} i
= X
I ⊂{1,...,M−1}
P h [
i ∈I
N
i
∈ · {X
1
, . . . , X
M
} i
P h
I
U
= I {X
1
, . . . , X
M
} i
. 3.71
Here the sum ranges over all subsets I of {1, . . . , M − 1}. The statement of the proposition now
follows from 3.70 and 3.71 by observing that P
h I
U
= I {X
1
, . . . , X
M
} i
a.s. 3.72
for all I ⊂ {1, . . . , M − 1} and U = T, −∞.
3.3 Classification according to incoming paths
