5Brownian motion
III Advanced Probability
5.3 Transience and recurrence
Theorem. Let (B
t
)
t≥0
be a Brownian motion in R
d
.
–
If
d
= 1, then (
B
t
)
t≥0
is point recurrent, i.e. for each
x, z ∈ R
, the set
{t ≥ 0 : B
t
= z} is unbounded P
x
-almost surely.
–
If
d
= 2, then (
B
t
)
t≥0
is neighbourhood recurrent, i.e. for each
x ∈ R
2
and
U ⊆ R
2
open, the set
{t ≥
0 :
B
t
∈ U}
is unbounded
P
x
-almost surely.
However, the process does not visit points, i.e. for all x, z ∈ R
d
, we have
P
X
(B
t
= z for some t > 0) = 0.
–
If
d ≥
3, then (
B
t
)
t≥0
is transient, i.e.
|B
t
| → ∞
as
t → ∞ P
x
-almost
surely.
Proof.
–
This is trivial, since
inf
t≥0
B
t
=
−∞
and
sup
t≥0
B
t
=
∞
almost surely,
and (B
t
)
t≥0
is continuous.
–
It is enough to prove for
x
= 0. Let 0
< ε < R < ∞
and
ϕ ∈ C
2
b
(
R
2
) such
that
ϕ
(
x
) =
log |x|
for
ε ≤ |x| ≤ R
. It is an easy exercise to check that this
is harmonic inside the annulus. By the theorem we didn’t prove, we know
M
t
= ϕ(B
t
) −
1
2
Z
t
0
∆ϕ(B
s
) ds
is a martingale. For
λ ≥
0, let
S
λ
=
inf{t ≥
0 :
|B
t
|
=
λ}
. If
ε ≤ |x| ≤ R
,
then
H
=
S
ε
∧ S
R
is
P
X
-almost surely finite. Then
M
H
is a bounded
martingale. By optional stopping, we have
E
x
(log |B
H
|) = log |x|.
But the LHS is
log εP(S
ε
< S
R
) + log RP(S
R
< S
ε
).
So we find that
P
x
(S
ε
< S
R
) =
log R − log |x|
log R − log ε
. (∗)
Note that if we let
R → ∞
, then
S
R
→ ∞
almost surely. Using (
∗
), this
implies
P
X
(
S
ε
< ∞
) = 1, and this does not depend on
x
. So we are done.
To prove that (
B
t
)
t≥0
does not visit points, let
ε →
0 in (
∗
) and then
R → ∞ for x 6= z = 0.
–
It is enough to consider the case
d
= 3. As before, let
ϕ ∈ C
2
b
(
R
3
) be such
that
ϕ(x) =
1
|x|
for ε ≤ x ≤ 2. Then ∆ϕ(x) = 0 for ε ≤ x ≤ R. As before, we get
P
x
(S
ε
< S
R
) =
|x|
−1
− |R|
−1
ε
−1
− R
−1
.
As R → ∞, we have
P
x
(S
ε
< ∞) =
ε
x
.
Now let
A
n
= {B
t
≥ n for all t ≥ B
T
n
3
}.
Then
P
0
(A
c
n
) =
1
n
2
.
So by Borel–Cantelli, we know only finitely of
A
c
n
occur almost surely. So
infinitely many of the A
n
hold. This guarantees our process → ∞.