Explosion Tests for Superlinear Drift Diffusions

Two Cubic Drifts, Two Different Outcomes

A drift that grows cubically can dominate Brownian fluctuations at large distances, but its sign matters. The drift \(x^3\) points away from the origin on both sides, whereas \(-x^3\) points toward it. Feller’s explosion test makes this distinction quantitative through the scale and speed densities. For the outward drift, a tail integral at each endpoint is finite; for the inward drift, the scale function diverges at both endpoints.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). For either equation, the diffusion coefficient is the constant \(1\), and the drift is locally Lipschitz. Thus, for each initial value \(X_0=x\), there is a pathwise unique strong solution up to its lifetime. Concretely, before explosion it satisfies \(X_t=x+\int_0^t b(X_s)\,ds+\int_0^t 1\,dB_s\). The integrand \(1\) is progressively measurable and belongs to \(\mathcal{L}^2\) on every finite interval, since \(\mathbb{E}\int_0^T1^2\,ds=T\). Local Lipschitz continuity does not ensure a global solution: define \(\tau_n=\inf\{t\ge0:|X_t|\ge n\}\) and \(\tau_\infty=\lim_{n\to\infty}\tau_n\), the lifetime at which explosion may occur.

We use the standard Feller explosion test from the earlier material on finite-time explosion: an endpoint with finite scale endpoint and finite corresponding integral is accessible as an exit endpoint, so there is positive probability of reaching it in finite time and hence of explosion there. If the scale endpoint is infinite, it cannot be reached in finite time. The test is an endpoint criterion; it does not say that every sample path explodes, nor does it give a deterministic explosion time.

Outward Cubic Drift: \(dX_t=X_t^3\,dt+dB_t\)

For \(b(x)=x^3\), integration gives \(\int_0^x2u^3\,du=x^4/2\). Therefore the scale density is \(p(x)=e^{-x^4/2}\), and the speed density is \(m(x)=2e^{x^4/2}\). Since \(p\) is even and decays rapidly, both scale endpoints are finite. The key point is to check that the scale tail multiplied by the speed density is integrable.

The coefficients and the scale density are symmetric under \(x\mapsto -x\). In particular, changing variables \(y=-v\) in the left-endpoint integral gives the same tail bound as on the right. Both \(I_-\) and \(I_+\) are finite. Feller’s test therefore permits finite-time explosion through either endpoint, with positive probability from a finite starting point. The symmetry of the coefficients does not mean that a given path explodes in both directions; the two endpoints describe alternative ways the lifetime can end.

Justification. The calculation above establishes finiteness of the right integral and, by symmetry, of the left integral. The scale density is positive and integrable on each tail by the same exponential estimate, so both scale limits are finite. The stated conclusion then follows from Feller’s explosion test. \(\square\)

Inward Cubic Drift: \(dX_t=-X_t^3\,dt+dB_t\)

For \(b(x)=-x^3\), the sign in the scale exponent reverses: \(p(x)=e^{x^4/2}\), and \(m(x)=2e^{-x^4/2}\). Because \(p(x)\ge1\), its integral over either unbounded half-line diverges. Thus \(s(+\infty)=+\infty\) and \(s(-\infty)=-\infty\). The scale test already rules out finite-time explosion at either endpoint. A separate stopped estimate confirms the conclusion without relying on endpoint classification alone.

Proof. Apply Itô’s formula, with \(f(x)=x^2\), \(f'(x)=2x\), and \(f''(x)=2\), to the solution stopped at \(\tau_n\). The drift is \(-x^3\) and the diffusion coefficient is \(1\), so the correction term is \(\tfrac12 f''(x)\,dt=dt\). Therefore

Before \(\tau_n\), \(|X_t|<n\), so the stopped stochastic integrand is bounded on \([0,T]\) and belongs to \(\mathcal{L}^2\). Its Itô integral is a true martingale: it is square-integrable by the Itô isometry, and its conditional expectation of each future increment given the current filtration is zero. Taking expectations, and using \(1-2X_t^4\le1\), yields \(\mathbb{E}[X_{T\wedge\tau_n}^2]\le x^2+T\). On \(\{\tau_n\le T\}\), continuity of the stopped path gives \(|X_{T\wedge\tau_n}|=n\). Hence

The stopping times \(\tau_n\) increase with \(n\), and \(\{\tau_\infty\le T\}\subseteq\{\tau_n\le T\}\) for every \(n\). Thus \(\mathbb{P}(\tau_\infty\le T)\le(x^2+T)/n^2\) for every \(n\), and letting \(n\to\infty\) gives \(\mathbb{P}(\tau_\infty\le T)=0\). This holds for each finite \(T\), proving nonexplosion. \(\square\)

Why the Scale Sign Matters

A common mistake is to omit the minus sign in the definition of the scale density. For \(b(x)=x^3\), the incorrect calculation \(p(x)=\exp(\int_0^x2u^3\,du)=e^{x^4/2}\) would imply infinite scale at both endpoints and would wrongly suggest nonexplosion. The correct calculation is \(p(x)=\exp(-\int_0^x2u^3\,du)=e^{-x^4/2}\). Reversing the drift to \(-x^3\) reverses this exponent. The scale calculation is therefore sensitive to the sign of the drift, not just to its growth rate.

These tests are useful because local existence and pathwise uniqueness do not settle whether a solution lasts for all time. Both cubic equations have locally Lipschitz coefficients, yet the outward drift admits explosion and the inward drift does not. The Feller calculation identifies the endpoint behavior; the Lyapunov estimate supplies a direct independent check for the nonexplosive case. Neither calculation asserts uniqueness in law as a substitute for pathwise uniqueness: here the local strong solution is pathwise unique up to its lifetime, and the stopped estimate rules out a finite lifetime for the inward equation.