Let the constant . Let be a jump process on whose embedded chain is a simple random walk, and whose transition rate at vertex is , where is the Euclidean distance from to the origin. For different values of , discuss whether explodes.

When , the jump process does not explode; when , the jump process explodes.

Stochastic Processes Solution – Problem 9: Let the constant

Question

Let the constant $\alpha\geq0$ . Let $X_{t}$ be a jump process on $\mathbb{Z}^{3}$ whose embedded chain is a simple random walk, and whose transition rate at vertex $v$ is $|v|^{\alpha}$ , where $|v|$ is the Euclidean distance from $v$ to the origin. For different values of $\alpha$ , discuss whether $X_{t}$ explodes.

Step-by-step solution

The lifetime of the jump process is defined as $\zeta = \lim_{n \to \infty} T_n$ , where $T_n = \tau_1 + \tau_2 + \cdots + \tau_n$ , and $\tau_k$ is the holding time of the $k$ -th jump of the embedded chain. If $\mathbb{P}(\zeta < \infty) > 0$ , the jump process is said to explode. The embedded chain is a simple random walk on $\mathbb{Z}^3$ , where at each step the walker moves to one of the 6 adjacent vertices with equal probability. The holding times $\tau_k$ are mutually independent and exponentially distributed with parameter $|X_{T_{k-1}}|^\alpha$ (where $X_{T_0} = 0$ is the starting point), i.e., the rate of $\tau_k$ is determined by the transition rate at the endpoint of the previous jump. By the Khasminskii explosion criterion: the jump process explodes if and only if $\sum_{v \in \mathbb{Z}^3} G(0, v)/|v|^\alpha = \infty$ , where $G(0, v) = \sum_{n=0}^\infty \mathbb{P}(X_{T_n} = v \mid X_{T_0} = 0)$ is the Green function of the embedded simple random walk. The simple random walk on $\mathbb{Z}^3$ is transient, i.e., $\lim_{n \to \infty} |X_{T_n}| = \infty$ (with probability 1), and satisfies: There exists a constant $C > 0$ such that as $|v| \to \infty$ , the Green function is asymptotically $G(0, v) \sim C/|v|$ . The key characteristic of transience: $\sum_{v \in \mathbb{Z}^3} G(0, v) = \infty$ , but $\sum_{v \in \mathbb{Z}^3} [G(0, v)]^2 < \infty$ .

Combining the asymptotic estimate of the Green function, the series $\sum_{v \in \mathbb{Z}^3} G(0, v)/|v|^\alpha$ can be approximated as: $\sum_{v \in \mathbb{Z}^3} G(0, v)/|v|^\alpha \sim C \sum_{v \in \mathbb{Z}^3} 1/|v|^{\alpha + 1}$ The convergence of the infinite series $\sum_{v \in \mathbb{Z}^3} 1/|v|^s$ ( $s > 0$ ) on $\mathbb{Z}^3$ is determined by the exponent $s$ : When $s > 3$ , the series converges; When $s \leq 3$ , the series diverges.

Case-by-case discussion for different values of $\alpha$ : (1) When $\alpha > 2$ , the jump process does not explode. In this case $\alpha + 1 > 3$ , so $\sum_{v \in \mathbb{Z}^3} 1/|v|^{\alpha + 1}$ converges, and hence $\sum_{v \in \mathbb{Z}^3} G(0, v)/|v|^\alpha < \infty$ . By the Khasminskii criterion, the series $\sum_{k=1}^\infty \tau_k$ (the lifetime $\zeta$ ) tends to $\infty$ with probability 1, i.e., $\mathbb{P}(\zeta = \infty) = 1$ , and the jump process does not explode.

(2) When $0 \leq \alpha \leq 2$ , the jump process explodes. In this case $\alpha + 1 \leq 3$ , so $\sum_{v \in \mathbb{Z}^3} 1/|v|^{\alpha + 1}$ diverges, and hence $\sum_{v \in \mathbb{Z}^3} G(0, v)/|v|^\alpha = \infty$ . By the Khasminskii criterion, there is positive probability that $\sum_{k=1}^\infty \tau_k < \infty$ , i.e., $\mathbb{P}(\zeta < \infty) > 0$ , and the jump process explodes.

Special case verification ( $\alpha = 0$ ): When $\alpha = 0$ , the transition rate is identically 1 ( $|v|^0 = 1$ ), and the holding times are i.i.d. exponential with parameter 1. Although $\tau_k$ has constant expectation 1, the transience of the simple random walk on $\mathbb{Z}^3$ implies $\sum_{v \in \mathbb{Z}^3} G(0, v) = \infty$ , satisfying the explosion criterion, so $\mathbb{P}(\zeta < \infty) > 0$ and the process still explodes.

Final answer

When $\alpha > 2$ , the jump process does not explode; when $0 \leq \alpha \leq 2$ , the jump process explodes.

Marking scheme

The following is the rubric for this problem and its official solution.

1. Key Steps (Checkpoints) (Total 7 pts)

Note: Grade exactly one path below; if the student uses a mixture of methods, take the highest-scoring path; do not add points across paths.

Path A: Green Function and Series Criterion (Official Solution Path)

Establish the explosion criterion [2 pts]: State that whether the jump process explodes depends on the convergence/divergence of the series $\sum_{v \in \mathbb{Z}^3} \frac{G(0, v)}{|v|^\alpha}$ (or the equivalent expected lifetime sum $\sum \mathbb{E}[\tau_n]$ ). Note: If only writing the formula without incorporating the specific rate $|v|^\alpha$ from the problem, award 1 pt.
Green function asymptotic estimate [2 pts]: Explicitly state the asymptotic property of the Green function for the three-dimensional simple random walk: $G(0, v) \sim \frac{C}{|v|}$ (or $O(|v|^{-1})$ ). This step is the core technical difficulty and must reflect the effect of dimension $d=3$ on the decay rate.
Series/integral convergence analysis [2 pts]: Substitute the Green function into the criterion to establish the series $\sum_{v} \frac{1}{|v|^{\alpha+1}}$ or the corresponding integral approximation $\int_{1}^{\infty} \frac{r^2}{r^{\alpha+1}} dr = \int_{1}^{\infty} \frac{1}{r^{\alpha-1}} dr$ . Correctly identify the critical value as $\alpha=2$ (i.e., $\alpha+1 > 3$ converges, $\alpha+1 \le 3$ diverges). Note: If the integral is correctly set up but an arithmetic error shifts the critical value, award 1 pt.
Conclusion [1 pt]: Strictly match the classification result of the official solution: when $\alpha > 2$ , series converges implies no explosion; when $0 \le \alpha \le 2$ , series diverges implies explosion.

Path B: Radial Process Comparison Method (Alternative Idea)

Dimension reduction modeling [2 pts]: Approximate the modulus $|X_n|$ of the three-dimensional random walk as a one-dimensional process with drift, or state that $|X_n| \sim \sqrt{n}$ (diffusive scaling).
Time/rate analysis [2 pts]: Construct the time sum series $\sum \frac{1}{|X_n|^\alpha} \sim \sum \frac{1}{n^{\alpha/2}}$ .
Convergence computation [2 pts]: Analyze the convergence of the series $\sum n^{-\alpha/2}$ , obtaining the critical condition $\alpha/2 > 1 \iff \alpha > 2$ .
Conclusion [1 pt]: Reach the same conclusion as the official solution ( $\alpha > 2$ no explosion, $\alpha \le 2$ explosion).

Total (max 7)

2. Zero-credit items

Only copying the transition rate $|v|^\alpha$ or definitions from the problem, with no subsequent derivation.
Incorrectly asserting that the three-dimensional simple random walk is recurrent, leading to $G(0,v) = \infty$ .
Only guessing the conclusion based on intuition (e.g., "the rate increases with distance, so it must explode") without mathematical derivation.
Citing theorems unrelated to this problem (e.g., the strong law of large numbers) without establishing a connection to the explosion time.

3. Deductions

Note: Deduct at most to 0 pts; no negative scores.

Green function dimension error (-2): Incorrectly citing the three-dimensional Green function as the two-dimensional form (e.g., $\ln|v|$ ) or one-dimensional/higher-dimensional form, causing completely wrong series order.
Series criterion logic error (-1): When analyzing $\sum 1/|v|^s$ , incorrectly remembering the convergence condition for $p$ -series (e.g., believing $s>1$ suffices for convergence in three-dimensional summation, without considering the volume element $r^2$ or the growth of the number of summation terms).
Ignoring constant terms/confusing random variables (-1): Seriously confusing random variables with constants during derivation, or failing to ensure symbol compatibility for boundary cases such as $\alpha=0$ (though separate discussion is not required, expressions like $\frac{1}{0}$ must not appear).
Missing conclusion (-1): Having computed convergence/divergence but not explicitly writing the final qualitative conclusion of "explodes" or "does not explode."

Stochastic Processes – Problem 9: Let the constant