Dimension Determines the Origin Behavior
A Bessel process has a drift that depends on its dimension and becomes singular as the process approaches zero. That dimension decides whether the origin can be reached. The critical value is two: below it, a process started away from zero reaches zero in finite time; at two it can get arbitrarily close but does not hit; above two it avoids the origin and tends to move away. The zero-dimensional squared Bessel process has a particularly simple boundary convention: it is absorbed when it hits zero.
Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). For a progressively measurable integrand \(H\) in \(\mathcal{L}^2\), the Ito integral \(\int_0^t H_s\,dB_s\) is defined when \(\mathbb{E}\int_0^t H_s^2\,ds<\infty\). Bessel SDEs below are first considered before the process reaches zero; integrals can be localized to intervals bounded away from zero.
“Reaches zero” means that the hitting time \(\tau_0=\inf\{t\ge0:R_t=0\}\) is finite. This is stronger than the statement that the process visits arbitrarily small positive levels. The distinction matters: probabilities of hitting levels \(\varepsilon\) as \(\varepsilon\downarrow0\) do not by themselves prove that the limiting hitting time is finite.
Scale Probabilities on an Interval
For \(\delta>0\), the generator of the Bessel process, on twice continuously differentiable functions \(f\) on \((0,\infty)\), is \(\mathcal{L}f(r)=\tfrac12 f''(r)+\tfrac{\delta-1}{2r}f'(r)\). A scale function is a strictly increasing solution of \(\mathcal{L}s=0\). Choose \(s(r)=r^{2-\delta}/(2-\delta)\) if \(\delta\ne2\), and \(s(r)=\log r\) if \(\delta=2\). In both cases \(s'(r)=r^{1-\delta}\).
Proof. Direct differentiation gives \(s'(r)=r^{1-\delta}\) and \(s''(r)=(1-\delta)r^{-\delta}\), so \(\mathcal{L}s(r)=\tfrac12(1-\delta)r^{-\delta}+\tfrac12(\delta-1)r^{-\delta}=0\). Let \(\tau=\tau_\varepsilon\wedge\tau_A\). Ito's formula for \(s(R_{t\wedge\tau})\) has first derivative term \(s'(R_t)dB_t\), drift term \(\tfrac{\delta-1}{2R_t}s'(R_t)dt\), and correction term \(\tfrac12s''(R_t)dt\); the two drift terms sum to zero by the displayed calculation. On \([\varepsilon,A]\), \(s'\) is bounded, so the stopped stochastic integral is square-integrable on every finite time interval and has conditional mean-zero increments. Thus the stopped process is a martingale. The standard exit-time result for a nondegenerate one-dimensional diffusion on a bounded interval gives \(\tau<\infty\) almost surely. Since \(s\) is bounded on \([\varepsilon,A]\), bounded convergence and optional stopping yield \(s(x)=\mathbb{E}_x[s(R_\tau)]\). At exit, \(R_\tau\) is either \(\varepsilon\) or \(A\), and continuity prevents both events from occurring together. Writing \(p=\mathbb{P}_x(\tau_\varepsilon<\tau_A)\) gives \(s(x)=p\,s(\varepsilon)+(1-p)s(A)\); solving for \(p\) proves the formula. \(\square\)
What the Scale Calculation Establishes
The proposition gives exact probabilities for exiting an interval through its lower endpoint. It also shows why dimension two is critical. For \(0<\delta<2\), \(s(\varepsilon)\) has a finite limit as \(\varepsilon\downarrow0\). For \(\delta=2\), \(s(\varepsilon)=\log\varepsilon\) tends to \(-\infty\), and for \(\delta>2\), \(s(\varepsilon)\) tends to \(-\infty\) as well. When \(\delta\ge2\), the probability of reaching a level approaching zero before a fixed upper level tends to zero.
Scale probabilities alone do not establish the finite-time hitting assertion for \(0<\delta<2\). We therefore use the following standard hitting-time result explicitly. The result is a standard theorem about Bessel processes; its proof, which establishes the law of the first-passage time rather than just interval exit probabilities, is beyond this tutorial.
For \(\delta>0\), the scale calculation also clarifies the distinction between non-hitting and closeness to the origin. When \(\delta=2\), for fixed \(0<\varepsilon<x\), let \(A\) increase in the proposition: \(\mathbb{P}_x(\tau_\varepsilon<\tau_A)\) tends to one. Thus every fixed positive level \(\varepsilon<x\) is eventually reached almost surely. Taking a countable sequence of such levels tending to zero shows that the process gets arbitrarily close to zero, even though the hitting-time theorem says it never hits zero.
For \(\delta>2\), \(s(A)\to0\) as \(A\to\infty\). The probability of ever reaching \(\varepsilon\), obtained by letting \(A\to\infty\), is \((x/\varepsilon)^{2-\delta}\), which is less than one. In particular, the probability of hitting zero is at most this quantity for every \(\varepsilon\in(0,x)\); as \(\varepsilon\downarrow0\), that bound tends to zero. The standard transience property of Bessel processes of dimension greater than two further says \(R_t\to\infty\) almost surely. Combined with continuity, this implies \(\inf_{t\ge0}R_t>0\) almost surely for such a process started at \(x>0\).
Worked Boundary Comparisons
Worked Example: Dimension One Hits Zero
Take a dimension-one Bessel process started at \(x=1\), and ask for the probability of hitting zero before reaching \(A=4\).
For \(\delta=1\), the scale function is \(s(r)=r\). First use the interval formula with lower level \(\varepsilon\): $$ \mathbb{P}_1(\tau_\varepsilon<\tau_4) =\frac{s(4)-s(1)}{s(4)-s(\varepsilon)} =\frac{4-1}{4-\varepsilon}. $$ As \(\varepsilon\downarrow0\), this probability tends to \(3/4\). The scale limit describes reaching arbitrarily small levels before \(4\); the standard hitting-time theorem, not that limit alone, guarantees that zero is hit in finite time. The resulting probability of hitting zero before \(4\) is \(3/4\).
Check: The probability is between zero and one. Starting closer to the upper boundary would reduce it: for example, the formula at \(x=2\) gives \((4-2)/4=1/2\). This is consistent with the upper boundary being reached more readily from that starting point.
Worked Example: Dimension Two Approaches but Does Not Hit
Take \(\delta=2\), start at \(x=1\), and compare the lower level \(\varepsilon=\tfrac14\) with the upper level \(A=4\). The scale function is \(s(r)=\log r\), so $$ \mathbb{P}_1(\tau_{1/4}<\tau_4) =\frac{\log 4-\log 1}{\log 4-\log(1/4)} =\frac{\log 4}{\log 4+\log 4} =\frac12. $$ The probability of reaching a lower level before \(4\) is positive. If \(\varepsilon\downarrow0\) with \(x=1\) and \(A=4\) fixed, however, the numerator stays \(\log4\) and the denominator \(\log4-\log\varepsilon\) tends to infinity, so the probability tends to zero.
Check: Here \(\log(1/4)=-\log4\), so the denominator is \(\log4-(-\log4)=2\log4\), giving exactly \(1/2\). The process does not hit zero, by the hitting-time theorem. It nevertheless reaches each fixed positive level below \(1\) almost surely when allowed to run without a fixed upper boundary, as shown by letting \(A\to\infty\) in the scale formula.
Worked Example: Dimension Three Has a Positive Chance to Avoid a Level
Take \(\delta=3\), start at \(x=1\), and ask whether the process reaches \(\varepsilon=\tfrac12\) before it reaches \(A=2\). Since \(s(r)=r^{-1}/(-1)=-1/r\), the interval formula gives $$ \mathbb{P}_1(\tau_{1/2}<\tau_2) =\frac{s(2)-s(1)}{s(2)-s(1/2)} =\frac{-1/2-(-1)}{-1/2-(-2)} =\frac{1/2}{3/2} =\frac13. $$ So the probability of reaching \(2\) first is \(1-\tfrac13=\tfrac23\).
Check: The two exit probabilities sum to \(\tfrac13+\tfrac23=1\). Letting \(A\to\infty\), the probability of ever reaching \(\tfrac12\) is \((1/(1/2))^{2-3}=2^{-1}=\tfrac12\), which is less than one. This is consistent with avoiding the origin.
The Zero-Dimensional Squared Bessel Process
Dimension zero is best described through the squared process. If \(Y_0=y>0\), its equation before absorption is \(dY_t=2\sqrt{Y_t}\,dB_t\), with no drift term. Its first hitting time \(\tau_0\) is finite almost surely by the standard hitting-time result, and the stipulated boundary behavior sets \(Y_t=0\) for all \(t\ge\tau_0\). Consequently its radial process \(R_t=\sqrt{Y_t}\) also remains at zero after the first hit. This is different from the usual Bessel processes with \(0<\delta<2\), which return to positive values after reaching zero.
Worked Example: Zero-Dimensional Absorption Time
Let \(Y_0=4\) and let \(\tau_0\) be its first hitting time of zero. The standard zero-dimensional hitting-time identity gives \(\tau_0=4/(2G)=2/G\) in distribution, where \(G\) is exponential with rate \(1\). For \(t>0\), $$ \mathbb{P}(\tau_0\le t) =\mathbb{P}(2/G\le t) =\mathbb{P}(G\ge 2/t) =e^{-2/t}. $$ At \(t=2\), this probability is \(e^{-1}\). At \(t=1\), it is \(e^{-2}\). Both are positive, and the second is smaller because a shorter time gives less opportunity to hit zero.
Check: The distribution function \(e^{-2/t}\) tends to zero as \(t\downarrow0\), and to one as \(t\to\infty\). Thus the formula is consistent both with a positive initial value and with finite-time hitting almost surely. After the hit, absorption means \(\mathbb{P}(Y_t=0\mid \tau_0\le t)=1\).
Dimension Summary and a Common Pitfall
| Dimension | Origin behavior from a positive start | Behavior at zero |
|---|---|---|
| \(0<\delta<2\) | Hits zero in finite time almost surely | Standard Bessel process is instantaneously reflecting |
| \(\delta=2\) | Does not hit zero; reaches every fixed positive lower level almost surely | Not reached from a positive start |
| \(\delta>2\) | Does not hit zero; tends to infinity almost surely | Not reached from a positive start |
| \(\delta=0\), squared process | Hits zero in finite time almost surely | Absorbed at zero |
Common pitfall: confusing small-level visits with a finite-time hit. For example, a calculation showing that \(\mathbb{P}(\tau_{1/n}<\tau_A)\to1\) as \(n\to\infty\) does not, by itself, show \(\tau_0<\infty\). The times \(\tau_{1/n}\) could increase without bound. The scale calculation proves statements about exits from intervals; the finite-time assertion for dimensions below two requires the separate Bessel hitting-time theorem. For dimension two, that distinction is essential: the process gets arbitrarily close to zero but does not hit it.
The origin classification is useful whenever a square-root diffusion is transformed into a Bessel-type process. The transformed dimension predicts whether the lower boundary is inaccessible, accessible with immediate return, or absorbing in the zero-dimensional squared case. The classification must be matched to the process and its boundary convention: the same statement “zero is accessible” does not specify what happens after the first hit.
Check Your Understanding
Use the scale formula and the stated hitting-time result to distinguish interval-exit probabilities from claims about hitting zero.
- For dimension \(\delta=1\), initial value \(x=2\), and upper level \(A=6\), compute the probability of hitting zero before \(A\).
- For dimension \(\delta=2\), explain why a positive probability of hitting \(\varepsilon\) before \(A\) does not imply a positive probability of hitting zero.
- For dimension \(\delta=4\), compute the probability, starting at \(x=2\), of ever reaching \(\varepsilon=1\).
- State the boundary behavior after the first zero hit for a standard Bessel process with \(0<\delta<2\) and for a zero-dimensional squared Bessel process.
- For a zero-dimensional squared Bessel process started at \(Y_0=8\), write the distribution function of its first hitting time of zero.