Distribution of the Brownian motion on its way to hitting zero

For the one-dimensional Brownian motion $B=(B_t)_{t\ge 0}$, started at $x>0$, and the first hitting time $\tau=\inf\{t\ge 0:B_t=0\}$, we find the probability density of $B_{u\tau}$ for a $u\in(0,1)$, i.e. of the Brownian motion on its way to hitting zero.


Introduction
The following problem has been recently addressed in [5], [6].The authors considered a continuous time subcritical branching process Z = (Z t ) t≥0 , starting from the initial population of size Z 0 = x.As is well known, Z t gets extinct at the random time T = inf{t ≥ 0 : Z t = 0}, and T < ∞ with probability one.What can be said about Z T /2 , i.e. the population size on the half-way to its extinction?While the complete characterization of the law of Z uT with u = 1/2, or more generally u ∈ (0, 1), does not seem to be trackable, it turns out that under quite general conditions where the convergence is in distribution, C and c are constants, explicitly computable in terms of parameters of Z and η is a random variable with Gumbel distribution.
In this note we study the analogous problem for one-dimensional Brownian motion B = (B t ) t≥0 , started from x > 0. Hereafter we assume that B is defined on the canonical probability space (Ω, F , P x ) and let τ denote the first time it hits zero, i.e. τ = inf{t ≥ 0 : B t = 0}.Theorem 1.1.For x > 0 and u ∈ (0, 1), the distribution P x B uτ ≤ y is absolutely continuous with the density Remark 1.2.Notice that p(u, x; y) decays as ∝ 1/y 2 and hence its mean is infinite.Such behavior, of course, stems from the possibility of large excursions of B from the origin, before hitting zero.
Remark 1.3.The formula (1.2) implies that x −1 B uτ has the same law under P x as B uτ under P 1 , or using different notations, where B x stands for the Brownian motion, starting at x > 0, and τ (x) = inf{t ≥ 0 : uτ (1) has the density p(u, 1, y)).This scale invariance does not seem to be obvious at the outset and should be compared to (1.1), where the scaling depends on u and holds only in the limit.
In the following section we shall give an elementary proof of Theorem 1.1.In Section 3 our result is discussed in the context of Doob's h-transform conditioning.

Proof
Let δ > 0 and define 1 τ δ := δ⌊τ /δ⌋.Recall that τ has the probability density (see e.g.[2]): Let Ms,t := inf s≤r<t B r and φ(•) be a continuous bounded function, then 2 where q(x, t, y) is the probability density of P x M0,t > 0, B t ∈ dy with respect to the Lebesgue measure (see e.g.formula 1.2.8 page 126, [2]): 1 ⌊x⌋ stands for the integer part of x ∈ R and ⌈x⌉ := ⌊x⌋ + 1 2 I(•) denotes the indicator function By continuity of the densities (2.1) and (2.3), for any fixed x > 0 and u ∈ (0, 1), the function In Lemma 2.1 below we exhibit a function G(t, y), independent of δ, such that and hence, the dominated convergence and (2.2) imply On the other hand, lim δ→0 τ δ = τ , P x -a.s. and thus by continuity of B t , lim δ→0 B uτ δ = B uτ , P x -a.s. for any u ∈ (0, 1).Thus, by arbitrariness of φ, (2.5) implies that the distribution of B uτ has the density: A calculation now yields: and by a change of variables The statement of the Theorem 1.1 now follows from: Proof.Set t δ := ⌈t/δ⌉δ, so that t ≤ t δ ≤ t + δ, and Since the function z 2 e −Cz with C > 0 attains its maximum 4e −2 /C 2 on the interval [0, ∞) at z := 2/C, Similarly, for t > δ, Hence for δ ∈ (0, 1] we have the bound Since for u ∈ (0, 1) and x > 0, the quadratic function is lower bounded: the first function in the right hand side of (2.6) is integrable on R 2 + .Further, .
The latter function decays as ∝ 1/y 2 as y → ∞ and is bounded away from zero, uniformly in y ≥ 0, and thus is integrable on R + .Since the last term in the right hand side of (2.6) is nonnegative, by Fubini theorem it is an integrable function on R 2 + for all u ∈ (0, 1) and x > 0.

A connection to Doob's h-transform
In this section we show that the random variable B uτ has the same density as the so called scaled Brownian excursion at the corresponding time, averaged over its length.The latter process is defined by conditioning in the sense of Doob's h-transform, and it would be natural to identify this formal conditioning with the usual conditional probability.While in the analogous discrete time setting, such identification is evident, its precise justification in our case remains an open problem.
For a fixed time T > 0, let R = (R t ) t≤T be the 3-dimensional Bessel bridge R = (R t ) t≤T , starting at R 0 = x and ending at zero.Namely, R is the radial part3 of the 3-dimensional Brownian bridge V = (V t ) t≤T with V 0 = v and V T = 0: where v ∈ R 3 with v = x and W is a standard Brownian motion in R 3 .
The law of R coincides with the law of the scaled Brownian excursion process, which is defined as "the Brownian motion, started at x > 0 and conditioned to hit zero for the first time at time T ".Here the conditioning is understood in the sense of Doob's h-transform (see Ch. IV, §39, [7], and [1], [3] for the in depth treatment).