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Quadratic pure-birth rates can explode

Statement

Let $E = {1, 2, \dots}$ , use one increment $ν = 1$ , and set the propensity $a (n) = n^{2}$ . Starting at $X_{0} = 1$ , the minimal reaction process has explosion time $τ_{\infty} = \sum_{n = 1}^{\infty} W_{n},$ where the independent holding times satisfy $W_{n} \sim Exp (n^{2})$ . This sum is finite almost surely, so finite per-state rates alone do not prevent explosion. The sufficient criterion in A proper Lyapunov bound prevents reaction-network explosion does not apply with $V (n) = n$ , since its local expression $Q V (n) = n^{2}$ is not bounded by $c n$ for any finite $c$ .

Facts & Assumptions

Given: Countable Choice, Dependent Choice, the reaction system above, and an independent stream of unit exponential variables.

[F1]

A reaction system permits countable $E \subseteq Z^{d}$ , a finite set of increments and finite total rate at each state, with active poststates in $E$ . (Finite-channel reaction propensity systems and their generator)

[F2]

A unit exponential law has density $e^{- t}$ for $t > 0$ and survival $e^{- t}$ ; rescaling by $n^{2}$ gives an exponential holding time with rate $n^{2}$ . (Exponential and Weibull survival specifications define probability laws)

[F3]

Expectation is integration against the probability measure. (Expectation of a nonnegative or integrable random variable)

[F4]

Increasing limits of nonnegative measurable functions pass through the integral. (Monotone convergence for the integral)

[F5]

On bounded intervals, the Lebesgue integral of a continuous function equals its Riemann integral, and the fundamental theorem evaluates that integral from an antiderivative. (A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ )

[F6]

The derivative of $e^{- t}$ is $- e^{- t}$ ; for $u \geq 0$ , the exponential series has nonnegative terms and dominates $u^{3} / 3! = u^{3} / 6$ . (Exponential and Weibull survival specifications define probability laws, The real exponential function and the number $e$ by a power series, A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum)

[F7]

A proper $V \geq 1$ with the local generator bound $Q V \leq c V$ is a sufficient hypothesis for countable-state nonexplosion. (A proper Lyapunov bound prevents reaction-network explosion)

[A1]

Countable Choice supplies the exponential law. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[A2]

Dependent Choice is used for the countable product probability space of holding-time inputs. (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, Assuming countable and dependent choice, countable products of arbitrary probability spaces)

Proof

technique · direct

1.1A1A2F1F2givenconstruct

At state $n$ , the only active channel has finite rate $n^{2}$ and moves to $n + 1 \in E$ , so [F1] defines a valid countable-state network. With independent unit exponentials $E_{n}$ , its $n$ th holding time is $W_{n} = E_{n} / n^{2}$ ; [F2] gives the stated law and the recursive path visits $1, 2, 3, \dots$ until its limiting time.

2.1F3F4F5F6step 1.1algebra

The density of $W_{n}$ is $n^{2} e^{- n^{2} t}$ for $t > 0$ . For $b > 0$ , [F5] and differentiation of the displayed antiderivative give $E [W_{n} 1_{{W_{n} \leq b}}] = \int_{0}^{b} t n^{2} e^{- n^{2} t} d t = \frac{1 - (n^{2} b + 1) e^{- n^{2} b}}{n^{2}} .$ As $b \to \infty$ , $(n^{2} b + 1) e^{- n^{2} b} \to 0$ : setting $u = n^{2} b$ , the exponential series gives $e^{u} \geq u^{3} / 6$ , so $(u + 1) e^{- u} \leq 6 (u + 1) / u^{3} \to 0$ . Applying [F4] to the increasing truncations yields $E W_{n} = 1 / n^{2}$ .

3.1F3F4step 2.1givenalgebra

For $n \geq 2$ , $1 / n^{2} \leq 1 / (n (n - 1)) = 1 / (n - 1) - 1 / n$ . Therefore $E \sum_{n = 1}^{m} W_{n} = \sum_{n = 1}^{m} 1 / n^{2} \leq 2$ . Since the partial sums increase, [F4] gives $E τ_{\infty} = E \lim_{m} \sum_{n = 1}^{m} W_{n} \leq 2$ . If $p = P (τ_{\infty} = \infty) > 0$ , then $E τ_{\infty} \geq k p$ for every integer $k$ , contradicting this finite bound. Thus $τ_{\infty} < \infty$ almost surely.

4.1F1F7step 1.1step 3.1algebra∎

For $V (n) = n$ , the finite local generator sum is $Q V (n) = n^{2} ((n + 1) - n) = n^{2}$ . If $Q V \leq c V$ for all $n$ , then $n \leq c$ for all positive integers, impossible; thus [F7] does not apply although step 3.1 proves explosion.

Depends on

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Sources

David F. Anderson, Introduction to Stochastic Processes with Applications to Biology (standard reference, not scraped)