Stochastic Processes Solution – Problem 48: Prove that if is a three-dimensional standard Brownian motion, then almost surely

Question

Prove that if $(B_{t})_{t\geq0}$ is a three-dimensional standard Brownian motion, then almost surely $\lim_{t\to\infty}|B_{t}|=\infty$ .

Step-by-step solution

Step 1. Three-dimensional standard Brownian motion $B_t=(B_t^{(1)},B_t^{(2)},B_t^{(3)})$ has independent one-dimensional standard Brownian motion components, continuous paths, and modulus $|B_t|=\sqrt{(B_t^{(1)})^2+(B_t^{(2)})^2+(B_t^{(3)})^2}$ . The key harmonic function is $f(x)=\frac{1}{|x|}$ on $\mathbb{R}^3\setminus\{0\}$ , satisfying $\Delta f(x)=0$ . By Ito's formula, $f(B_t)$ is a local martingale when $B_t\neq 0$ .

Step 2. For $0<r<R$ , define the exit time $T_{r,R}=\inf\{t\geq0 \mid |B_t|\leq r \text{ or } |B_t|\geq R\}$ . The process $f(B_{t\wedge T_{r,R}})$ is a bounded martingale since $\frac{1}{R}\le f\le\frac{1}{r}$ on the annulus.

Step 3. By the optional stopping theorem, $E[f(B_{T_{r,R}})]=f(B_0)$ . After adjusting for the singularity at the origin using the first exit time $T_r$ from the ball of radius $r$ , we obtain the boundary probability equation. Setting $P_1$ as the probability of returning to the inner sphere and $P_2$ as reaching the outer sphere: $P_1\cdot\frac{1}{r}+P_2\cdot\frac{1}{R}=\frac{1}{r}$ , which gives $P_1=0$ and $P_2=1$ .

Step 4. Letting $R\to\infty$ , the probability of returning to the ball of radius $r$ after leaving it is zero. This means three-dimensional Brownian motion is transient.

Step 5. For any $M>0$ , take $r=M$ . By transience, there exists a stopping time $T_r<\infty$ a.s. such that for all $t>T_r$ , $|B_t|>M$ a.s. Since $M$ is arbitrary, $\lim_{t\to\infty}|B_t|=\infty$ a.s.

Final answer

Three-dimensional standard Brownian motion $(B_t)_{t\geq0}$ almost surely satisfies $\boxed{\lim_{t\to\infty}|B_t|=\infty \quad \text{a.s.}}$ QED.

Marking scheme

The following is the rubric for this problem. Please score according to the guidelines below.

1. Checkpoints (max 7 pts)

Apply exactly one path for scoring (if the student mixes methods, take the highest-scoring single path).

Chain A: Martingale and Harmonic Function Method (Official Solution)

Key function and martingale property [2 pts]
State that $f(x) = 1/|x|$ is harmonic on $\mathbb{R}^3\setminus\{0\}$ (i.e., $\Delta f = 0$ ), or
Use Ito's formula to show $f(B_t)$ is a local martingale.
*Note: If only writing the function without mentioning harmonicity or the martingale property, 0 pts.*
Construct stopping time and boundedness argument [2 pts]
Introduce the annular region $\{x : r < |x| < R\}$ and the first exit time $T = \inf\{t : |B_t| \le r \text{ or } |B_t| \ge R\}$ . (1 pt)
State that the process is bounded before this stopping time, hence the optional stopping theorem applies. (1 pt)
Establish probability equation [2 pts]
Apply the optional stopping theorem to set up the equation relating boundary probabilities, e.g., $P(\text{hit } r) \cdot \frac{1}{r} + P(\text{hit } R) \cdot \frac{1}{R} = \frac{1}{|x_{start}|}$ .
Take limit and final conclusion [1 pt]
Let $R \to \infty$ to deduce the escape probability is 1 (or the return probability is $<1$ combined with the Markov property), concluding $\lim_{t\to\infty} |B_t| = \infty$ .

Chain B: Green's Function / Analytic Method (Alternative)

Integral / Green's function computation [3 pts]
Correctly write and compute the time integral of the 3D Brownian motion transition density (heat kernel), or directly cite the Green's function $G(x,y) \sim C/|x-y|$ .
Key point: show $\int_1^\infty t^{-3/2} dt < \infty$ (convergence).
Transience derivation [2 pts]
Argue that the finite integral implies: for any bounded set $K$ , the total occupation time in $K$ is finite ( $\int_0^\infty 1_K(B_t) dt < \infty$ a.s.).
Final conclusion [2 pts]
From "finite occupation time in any bounded set" logically deduce $\lim_{t\to\infty} |B_t| = \infty$ .

2. Zero-credit items

Merely copying definitions of Brownian motion or standard properties (normal increments, independence) without substantive derivation.
Merely citing the conclusion "Brownian motion is transient for $d \ge 3$ " without proof.
Incorrectly using the 2D harmonic function $f(x) = \log|x|$ or the 1D function $f(x)=x$ .
Claiming $\lim_{t\to\infty} B_t$ (vector limit) exists or equals infinity (conceptual error; it is the modulus that diverges).

3. Deductions

Note: Deductions cannot reduce the total below 0. Apply the most severe single deduction.

Ignoring singularity/domain issues (-1 pt): Directly setting initial point $B_0=0$ when applying $1/|x|$ without any treatment (e.g., starting from $\epsilon$ or using a stopping time to avoid the origin), causing a division-by-zero logical gap.
Martingale rigor deficiency (-1 pt): Claiming $1/|B_t|$ is a martingale on the entire interval $[0, \infty)$ (it is only a local martingale; it must be truncated/stopped to form a bounded martingale).
Logical confusion (-2 pts): Confusing "transience" ( $\lim |B_t|=\infty$ ) with "recurrence" (infinitely many returns to a region), e.g., computing return probability as 1 yet concluding the modulus diverges.

Stochastic Processes – Problem 48: Prove that if is a three-dimensional standard Brownian motion, then almost surely