
Having previously looked at Brownian bridges and excursions, I now turn to a third kind of process which can be constructed either as a conditioned Brownian motion or by extracting a segment from Brownian motion sample paths. Specifically, the Brownian meander, which is a Brownian motion conditioned to be positive over a unit time interval. Since this requires conditioning on a zero probability event, care must be taken. Instead, it is cleaner to start with an alternative definition by appropriately scaling a segment of a Brownian motion.
For a fixed positive times T, consider the last time σ before T at which a Brownian motion X is equal to zero,
(1) |
On interval [σ, T], the path of X will start from 0 and then be either strictly positive or strictly negative, and we may as well restrict to the positive case by taking absolute values. Scaling invariance says that c-1/2Xct is itself a standard Brownian motion for any positive constant c. So, scaling the path of X on [σ, 1] to the unit interval defines a process
(2) |
over 0 ≤ t ≤ 1; This starts from zero and is strictly positive at all other times.

Scaling invariance shows that the law of the process B does not depend on the choice of fixed time T The only remaining ambiguity is in the choice of the fixed time T.
Lemma 1 The distribution of B defined by (2) does not depend on the choice of the time T > 0.
Proof: Consider any other fixed positive time T̃, and use the construction above with T̃, σ̃, B̃ in place of T, σ, B respectively. We need to show that B̃ and B have the same distribution. Using the scaling factor S = T̃/T, then X′t = S-1/2XtS is a standard Brownian motion. Also, σ′= σ̃/S is the last time before T at which X′ is zero. So,
has the same distribution as B. ⬜
This leads to the definition used here for Brownian meanders.
Definition 2 A continuous process {Bt}t ∈ [0, 1] is a Brownian meander if and only it has the same distribution as (2) for a standard Brownian motion X and fixed time T > 0.
In fact, there are various alternative — but equivalent — ways in which Brownian excursions can be defined and constructed.
- As a scaled segment of a Brownian motion before a time T and after it last hits 0. This is definition 2.
- As a Brownian motion conditioned on being positive. See theorem 4 below.
- As a segment of a Brownian excursion. See lemma 5.
- As the path of a standard Brownian motion starting from its minimum, in either the forwards or backwards direction. See theorem 6.
- As a Markov process with specified transition probabilities. See theorem 9 below.
- As a solution to an SDE. See theorem 12 below.