Non-explosion by Stratonovich noise for ODEs

We show that the addition of a suitable Stratonovich noise prevents explosion for ODEs with drifts of super-linear growth.

Condition 1.1. Fix m > 1 and η > (m − 1)/2. The drift b : R d → R d is locally Lipschitz and verifies, for some C ≥ 0, The diffusion coefficient σ : R d → R d×d is C 1 with locally Lipschitz derivative and satisfies, for some R > 0, We have used the notation I d for the d × d-dimensional matrix, B R for the open ball of center 0 and radius R. Under this conditions, the SDE admits a unique local strong solution.
Here is our main result: Theorem 1.2. Under assumptions 1.1, the SDE (1.1) in R d , d ≥ 2, admits a global-in-time (unique, strong) solution.
The particular form of the noise is due to the following remark: under the transformation y = φ(x) := |x| −η−1 x, (1.2) the noise σ(x) • dW becomes simply an additive noise. This remark reveals the idea behind the non-explosion: by the transformation φ, explosion becomes passage through 0, and such passage can be prevented, for d ≥ 2, by an additive noise. We will give two proofs of this theorem: one exploits directly this idea of transforming explosion in passage through 0, the other uses the Lyapunov function method. The second proof may be useful in infinite dimensions and yields also the existence of a stationary solution. The possibility to use the noise to restore existence or uniqueness or stability has been widely explored. Many works are concerned with uniqueness and regularity for ODEs with irregular drifts, perturbed by additive noise, and transport equations, perturbed by linear transport noise, see e.g. [ Here instead every direction is possibly explosive and, to prevent explosion, a multiplicative noise is used. This research direction has been also studied in a number of works, see [MMR02, WH09, LS12] among many others, mostly exploiting the Lyapunov function method. The closest paper to ours seems [AMR08], which shows non-explosion by noise for a large class of drifts and diffusion coefficients, using power-type Lyapunov functions. In particular, writing our SDE (1.1) in Itô form, outside the ball B R , we can recognize that our SDE falls in the class of [AMR08] for d ≥ 3. The novelty here, with respect to [AMR08], is the idea behing the first proof, based on the Stratonovich noise and the transformation in (1.2), and the use of a logarithmic Lyapunov function to deal with the case d = 2. The Stratonovich noise arises as limit of smooth noises and is also widely used in SPDEs. In the SPDE context, we point out the papers [GG19, GS19] (among others) which also take non-linear noises to show regularization properties. Finally we mention the recent paper [KCSW19], which shows non-explosion for a Hamiltonian ODE perturbed by additive noise and a suitable drift term, which preserves the Hamiltonian structure. Once non-explosion from any fixed x 0 is established, one could ask about finer properties, like global existence of a stochastic flow solving the SDE (1.1). We expect the answer to be negative. Indeed, in [LS11], the authors construct a (driftless) SDE by applying a similar transformation to bring ∞ into 0 and vice versa and show the lack of a stochastic flow solution of that SDE; see also [LS17] for a similar phenomenon for the example in [HM15a].

First proof
In the first proof, we apply the transformation Y = φ(X) to the SDE (1.1), by the specific form of σ we get an SDE for Y with irregular driftb(Y ) and additive noise. The assumptions on m and α guarantee that this SDE for Y admits a unique solution, whose law is equivalent to the Wiener measure (by Girsanov theorem). Since, for d ≥ 2, the Wiener measure does not see the 0 P-a.s., we conclude that Y does not hit 0 and hence X does not explode P-a.s.. Recall that τ is the explosion time of X, that is, on {τ < ∞}, it holds lim sup tրτ |X t | = ∞ P-a.s.. We have to show that τ = ∞ P-a.s.. For all t such that X t = 0, we call Y t = φ(X t ), where φ is defined as in (1.2). More precisely, to avoid the times when X hits 0 (and so Y is not defined), we show first that, P-a.s., X enters B R before exploding, then we use this fact to conclude. First part: We use the notation P x 0 to keep track of the initial condition x 0 . We define τ 0,R = inf{t ≥ 0 | X t ∈ B R }. We will show that (2.1) We start applying Itô formula to Y t = φ(X t ) for t < τ ∧ τ 0,R (this is possible because φ is smooth on B c R ). Assumption 1.1 on σ gives Since Y lives in B R −η , we can setb = 0 onB c R −η . Since b is locally Lipschitz, b is locally Lipschitz onB R −η \ {0}. Moreover, the assumption 1.1 for b gives, for some constant C, By assumption 1.1 on m and η, we have (η + 1 − m)/η > −1, so the driftb is in L p (R d ) for some p > d (and d ≥ 2). We are now in a position to apply [FF11, Theorem 1, Corollary 16]: the SDE (2.3) admits a global (strong) solutionỸ whose law is equivalent to the d-dimensional Wiener measure starting from φ(x 0 ). But the SDE (2.3) admits also a unique strong solution before exitingB R −η \ {0}, by the local Lipschitz property ofb, and hencẽ are equivalent, where W x 0 := W + φ(x 0 ) and ρ W x 0 and ρ W x 0 ,R −η are defined for W x 0 as in (2.2). Now, for d ≥ 2, for any x 0 , W + φ(x 0 ) does not hit 0 with probability 1, that is ρ W x 0 = ∞ P-a.s. (see e.g. [RY99, Chapter V, Proposition 2.7]). Hence we have ρ Y ≥ ρ Y,R −η P-a.s. and so τ ≥ τ 0,R P-a.s., that is (2.1). Second part: We recall that τ 0,R = inf{t ≥ 0 | X t ∈ B R } and we define recursively, for i nonnegative integer, that is the (i + 1)-th exit time from B R+1 and the (i + 1)-th hitting time of B R . The property (2.1) and the strong Markov property of X imply, via induction taking x 0 = X τ i,R+1 at each step, that τ ≥ τ i,R P-a.s. for every i. To conclude, it is enough to show that sup i τ i,R = ∞ P-a.s., that is, the sequence (τ i,R ) i has no finite accumulation point P-a.s.. For this, we note that b and σ are bounded in the annulusB R+2 \ B R , hence, callingτ i the first exit time fromB R+2 \B R after τ i,R+1 , a simple application of BDG inequality to the SDE (1.1) gives, for some C (independent of i and δ), for every δ > 0 and every i, It follows that, for all i, P-a.s., Hence, for every n < m positive integers, we have by conditioning We now choose δ > 0 such that p < 1. With this choice, P(lim inf i {τ i,R − τ i,R+1 < δ}) = 0. Now the event that τ i,R has a finite accumulation point is contained in lim inf i {τ i,R − τ i,R+1 < δ}, hence τ i,R has no finite accumulation point P-a.s.. The proof is complete.

Second proof
In the second proof, we show that (log |x|) α , with 0 < α < 1, is morally a Lyapunov function for the SDE (1.1). This implies not only non-explosion but also the existence of an invariant measure.
Second proof. We fix 0 < α < 1 and take a C 2 function V : R d → R such that, for a constant a > 0, and we show that such V is a Lyapunov function for the SDE (1.1), that is • V is nonnegative and inf |x|>r V (x) tends to ∞ as r → ∞ and • LV ≤ cV on R d for a constant c > 0, where L is the generator of the SDE (1.1).
The first condition is clearly satisfied. For the second condition, we write the SDE (1.1) in Itô form dX =b(X)dt + σ(X)dW, whereb is locally Lipschitz and This can be verified by a tedious computation of the Itô-Stratonovich correction of (1.1) or applying Itô formula (in Itô form) to and so for some constants c 1 , c 2 > 0. By the assumption 2η > m − 1, there exists r > 0 such that LV (x) is negative for all x outside B r . Therefore, since LV is locally bounded and V is bounded from below away from 0, the condition LV ≤ cV is satisfied on R d for a suitable c and so V is a Lyapunov function. Hence, by [Kha12, Theorem 3.5], there exists a global solution to the SDE (1.1). The proof is complete.
The existence of a Lyapunov function implies also the existence of an invariant distribution for the SDE (1.1): Proposition 3.1. Under assumptions 1.1, the SDE (1.1) in R d , d ≥ 2, admits a stationary solution (starting from a random initial condition X 0 ).