Dynamics of a degenerately damped stochastic Lorenz-Steno system

Little seems to be known about the sensi- 1 tivity of steady states for stochastic systems. This pa- 2 per discusses such dynamics of a degenerately damped 3 stochastic Lorenz-Stenflo model. Precisely, the solution 4 is proved to be a nice diffusion via the Lie bracket tech- 5 nique and non-trivial Lyapunov functions. The finite- 6 ness of the expected positive recurrence time entails the 7 existence problem. On the other hand, a cut-off func- 8 tion is constructed to show the non-existence result via 9 proof by contradiction. For other interesting cases, the 10 expected recurrence time is shown to be infinite. 11


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To describe the low-frequency, short-wavelength acousticgravity perturbations in the atmosphere, Stenflo [1] has derived a four-dimensional continuous-time dynamical system, given by where x, y, z, w are state variables of the so-called Lorenz-namical behaviors such as boundedness [3,4], periodic- 23 ity [5,6], bifurcation [7][8][9], synchronization [10], chaotic 24 and hyperchaotic dynamics [11][12][13][14] and influences by 25 Lévy noise [15]. 26 Notice that the geometric parameter b is strictly 27 positive as in the derivation of (1), but it will tend to 28 zero under sufficiently large generalized Rayleigh num-29 ber. On the other hand, the so-called Homogeneous 30 Rayleigh-Bénard (HRB) system was established with 31 b ≤ 0 appearing in the temperature equation [16,17]. 32 Indeed, the similar degeneracy effect was observed in a 33 certain zero Prandlt limit to model mantle convection 34 [18,19]. Therefore, it is natural to investigate the cor-  It is well-known that an arbitrary small additive 47 noise can stabilize an explosive ordinary differential 48 equation (ODE) [20,21]. If, in addition, the correspond- 49 ing Markov process admits an invariant probability mea-50 sure, it corresponds to the so-called noise-induced sta-51 bilization problem. In this respect, there is already con-52 siderable interest in studying on stationary state, stable 53 oscillations, and related work [22][23][24][25][26][27][28][29][30][31][32]. 54 Motivated by the aforementioned discussion, we are interested in the stochastic Lorenz-Stenflo system where B i , i = 1, 2, 3, 4 are independent, standard Brow-55 nian motions, κ i ≥ 0, i = 1, 2, 3, 4 represent the intensity 56 of random noise and other parameters are in confor-57 mity with the ones in system (1). To ensure system (2) 58 is genuinely stochastic, we require that at least one κ i 59 is positive. 60 In the absence of noise, we know that the solutions  The goal of this paper is to apply the approaches in 67 [31,32] to solve the noise-induced stabilization problem 68 of (1) with additive Brownian noise. More precisely, 69 we first state the philosophy in proving the existence a 70 unique invariant probability measure to Markov transi-71 tion semigroup generated by (2). In fact, the first step is  Let M nk be an n × k real matrix and consider the following Itô stochastic differential equation where F = (F 1 , . . . , F n ) ∈ C 2 (R n ; R n ), G = (G 1 , . . . , G k ) ∈ C 2 (R n ; M nk ), and B t = (B 1 t , . . . , B k t ) T represents a standard k-dimensional Brownian motion on a filtered probability space (w, F , {F t } t≥0 , P). Given a function V ∈ C 2 (R n ; R), the infinitesimal generator of the process along V for system (3) is defined by In general, the smoothness of F and G can not guarantee the existence of global solution to (3), but one can define a local unique pathwise solution, which is denoted by X t = X(0, x;t) under the initial condition X 0 = x. We next introduce a stopping time where τ n = inf{t ≥ 0 : |X t | ≥ n} for n ∈ N + . Thus there is a unique solution X t , for all times t < τ, P-almost surely.
Herein, τ stands for the explosion time of the process X t and by which X t is said to be non-explosive if So if X t is non-explosive, it can generate a Markov process and its transition probability measure is defined as Denote by B the Borel σ -field of subsets of R n , the Markov transition semigroup satisfies for the bounded, B-measurable functions V : R n → R, 88 where E denotes the corresponding expectation. Indeed, 89 a positive measure π is invariant for P t if πP t = π for 90 all t ≥ 0. An invariant measure π for P t is invariant 91 probability measure for P t provided that π(R n ) = 1.

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To our purpose, we next introduce two concepts to

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Definition 2 Assume that the solution X t to (3) is non-100 explosive and further satisfies 101 (i) F ∈ C ∞ (R n ; R n ) and G ∈ C ∞ (R n ; M nk ); (ii) the operators L , L * , L ±∂ t , L * ±∂ t are hypo-elliptic on the respective domains R n , R n , R n ×R + , R n ×R + , 104 where L * is the formal adjoint of L with respect to the L 2 (R n ; dx) inner product;

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Then X t is called a nice diffusion.

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On the other hand, for two (smooth) vector fields the Lie bracket of them is defied by It allows us to introduce the following notations: In particular, when W depends polynomially on the components of X for any X ∈ R n , one further denotes n(X,W ) := max j=1,...,n deg(p j ) where p j (λ ) := W j (λ X).
Thus, for any collection of vector fields G on R n , let and {U 1 , . . . ,U N } ⊂ G .
Since we are only interested in the situation that G is independent of X and F is a polynomial, and further let by which, one lets . We next extract some criteria from [32], which are for p, q > 0. Then X t is non-explosive.

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(ii) Assume that X t is non-explosive and there is a function V ∈ C 2 (R n ; [0, +∞)), a compact set K ⊆ R n and constants p, q > 0 such that Lemma 3 [32, Theorem 2.6] Let F is a polynomial and G is X-independent. If the solution X t to (3) is nonexplosive and Then X t is a nice diffusion. for P t .

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(ii) P t has an invariant probability measure if and only

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In the sequel, we will denote by X t = (x t , y t , z t , w t ) 145 the solution to (2). We always assume ∑ 4 i=1 κ 2 i = 0 to 146 guarantee that system (2) is genuinely stochastic. Proof To prove the first assertion, we take and the corresponding infinitesimal generator reads Thus the required condition in Lemma 1(i) is reached 152 when taking V = H with Young's inequality.

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So it remains to show that system (2) satisfies the spanning condition in Lemma 3, then it turns out that X t is a nice diffusion. To this aim, using the Lie bracket, we have By considering F, G 1 , G 2 , G 3 , G 4 as vectors, we can get Therefore we can verify that n(G 1 , F) = 1, then we have And on the other hand, from n(G 3 , G ′ 1 ) = 1 it follows which satisfy the required spanning condition and therefore the proof of Theorem 1 is complete. ⊓ ⊔ 3.2 Existence of invariant probability measure 158 We will be led in the sequel to consider the degenerately damped situation, where b = 0. Thus (2) reduces to Our task here is to construct a suitable Lyapunov function resulting in a globally finite expected return subject to some compact set. To this purpose, we take whose detailed construction is put in the Appendix (for 159 the sake of reader's convenience).

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Proof Before proceeding any further, we emphasize that less explicitly stated otherwise. Also, we let X ′ := x 2 + 166 y 2 + w 2 and X ′′ := |x||z| 1 3 for the sake of simplicity.

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Regarding Lemma 1(ii), we begin by observing that where ∇ κ = (κ 1 ∂ x , κ 2 ∂ y , κ 3 ∂ z , κ 4 ∂ w ) and M , ψ 1 and ψ 2 are as in (22), (27) and (29) in the Appendix, respectively. To estimate each term in (6), we proceed to their derivatives fall on the cut-off functions θ 1 and θ 2 . In fact, it follows that for θ 1 where again C > 0 is independent of R 0 , R 1 , R 2 and R 3 . 168 We now are ready to expand ψ 1 M (θ 1 ) as Since on R 1 one has Using the estimates of derivatives on θ i , we have Hence, we obtain We next turn to estimate In the sequel, we focus on the cut-off terms involving ψ 2 . In this respect, we expand ψ 2 M (θ 2 ) as Notice that each term in (12) is supported on the set {X ′′ ≤ 2R 1 }, and therefore the estimate (30) applies.
Then it leads to
Then, choose R 1 > 1 such that and R 2 ≥ R 0 such that Finally, choose R 3 such that With these parameter selections and referring back to (32), (33) we therefore have Dynamics of a degenerately damped stochastic Lorenz-Stenflo system Consequently, (23) follows 171 with p = 2κ and q = 4κ.

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Finally, it remains to check the non-negativity of V . Notice that our selection of the parameters R 0 , R 1 , R 2 , R 3 and of n 1 , n 2 was made independent of the value κ 0 (see (34)). Moreover, by (8) and (30), we have respectively. Thus having fixed R 0 , R 1 , R 2 , R 3 , n 1 , n 2 and referring back to (34) we have which can be always positive for every (x, y, z, w) ∈ R 4 by choosing large enough κ 0 . Therefore, the proof of Theorem 2 is now complete. ⊓ ⊔

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In this subsection, we devote ourselves to the non-existence 174 issue. Our results are slightly different for b = 0 and 175 b < 0, so we state them separately.

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Proof Assume that there is an invariant probability measure µ of (5) and let (x, y, z, w) have law µ. Thus there exists an increasing sequence of integers Based on the construction and properties of F N as defined by (35) in Appendix, we apply Itô's formula to which further implies

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Proof We proceed it in four steps to construct V 1 and 181 V 2 satisfying the conditions in Lemma 2.

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(2) If ζ ∈ 2 3 π, B , then Ψ ′ (ζ ) > 0 and Ψ ′′ (ζ ) < 0. Thus, from (19) and (21) follows Dynamics of a degenerately damped stochastic Lorenz-Stenflo system 9 Step 4. Let us verify that the assumptions of Lemma 2 are satisfied with V 1 and V 2 . Clearly, (iv) follows from the construction of V 1 and V 2 and (ii) is sure due to the fact that lim |(x,y,z,w)|→∞ H(x, y, z, w) = ∞. As for (i), using the fact that z → V 1 (x, y, z, w) is increasing for large z and (x, y, w) → V 1 (x, y, z, w) is non-increasing. This finishes the proof based on Lemma 2. It is notoriously difficult to check that the infinitesimal generator of (5) leads to by which our immediate goal to acquire the inequality for some constants p, q > 0 and some compact set K ⊆ R 4 .

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We first choose the following Lyapunov functioñ H(x, y, z, w) = 1 2 where κ 0 > 0 is large enough so thatH ≥ 0. Since the required inequality (23) is sure on the set where |(x, y, w)| := x 2 + y 2 + w 2 is large. More specifically, let the region with a sufficiently large R 0 ≥ 0, that is We next pay attention to the situation that x 2 + y 2 + w 2 ≤ R 0 and |z| is large. For this purpose, we consider the scaling transformation where λ ≫ 1 and α ∈ (0, 1). Applying T λ to the generator M yields Obviously, the dynamics of (26) are twofold. First, if α ∈ (0, 1 3 ), the dominant term in (26) is −λ 1−α xz∂ y , which allows us to consider the dominant equatioṅ Indeed, it suggests us to seek a function ψ 1 such that where the constant n 1 > 2κ. Thus We now pause to define a region where R 0 , R 1 , R 3 ≥ 1 are large constants to be determined below. Note that, on R 1 , the following estimates are satisfied. So it is immediate that where C = C(a, c, r, κ 1 , κ 3 ) is independent of R 0 , R 1 , R 2 and n 1 . Thus, for sufficiently large R 1 depending on R 0 , we obtain M (ψ 1 ) ≤ − 1 2 n 1 on the region R 1 .
To our goal, we define the region Similarly, we are interested in finding a function ψ 2 that solves Clearly, a particular solution to the above equation is implying the estimate Thus we obtain  for i = 1, 2, so that the inequality (23) is sure where K := {x 2 + y 2 + w 2 ≤ R 0 , |z| ≤ R 3 }.