INVARIANT MEASURE OF STOCHASTIC HIGHER ORDER KDV EQUATION DRIVEN BY POISSON PROCESSES∗

The current paper is devoted to stochastic damped higher order KdV equation driven by Poisson process. We establish the well-posedness of stochastic damped higher-order KdV equation, and prove that there exists an unique invariant measure for non-random initial conditions. Some discussion on the general pure jump noise case are also provided. Some numerical simulations of the invariant measure are provided to support the theoretical results. Mathematics Subject Classification. 60H15, 37L55. Received March 27, 2020. Accepted July 13, 2021.

Observe that ifū(t) solves Thus, u(t) solves the equation du + (∂ 2n+1 x u + λu)dt = 0. Denote S λ (t) by the solution operator of du + (∂ 2n+1 x u + λu)dt = 0, then we have S λ (t) = e −λt S 0 (t). Then the mild solution of equation (1.1) can be written as Let u 0 ∈ H n (R) be a deterministic condition,and u(t) be the solution of equation (1.1). For all B ∈ B(H n (R)), the transition probabilities of the equation can be defined by P t (u 0 , B) = P (u(t) ∈ B). For any function ξ ∈ In this paper, we will investigated the ergodicity of stochastic damped higher order KdV equation with pure jump noise. We prove the global well-posedness and the existence of invariant measure for stochastic damped higher order KdV equation with random initial value. Moreover, we obtain the ergodicity of stochastic damped higher order KdV equation with non-random initial conditions.
Comparing with stochastic damped KdV equation with Gaussian noise, the non-Gaussian noise driven higher Order KdV equation leads to the trajectories of the solutions are Càdlàg. To overtake the difficulty caused by dispersive term ∂ 2n+1 x u instead of the dissipative term u xx , motivated by the idea from [15], we establish the uniform estimates for L 2 norm and H n norm respectively, which are critical to obtain the existence of invariant measure. Moreover, we give some numerical simulations of the invariant measure to support the theoretical results.
The rest of paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we prove that the solutions in H n space are uniformly bounded and establish the global well-posedness. In Section 4, we establish the existence of invariant measure for stochastic damped stochastic damped higher order KdV equation. In Section 5, we prove that the invariant measure is unique for deterministic initial value. In Section 6, we discuss the general pure jump noise case,some numerical simulations of the invariant measure are provided to support the theoretical results.

Preliminaries
In this section, some basic concepts and some inequalities are provided, which plays the crucial role in establishing the main theorems.
Let Definition 2.1. Assume that (X, T ) is a polish space, Σ is a σ-algebra on X, M is a set of measures on Σ. M is said to be tight if for any ε > 0, there exists a compact set K ⊂ X such that |µ(K c ε )| < ε, for any µ ∈ M.  (b) there exist two constants c > 0 and γ > 0 and a real number r > 0 such that for all θ > 0, t ∈ [0, T − θ], and n ≥ 0 In the sequel, we impose the following hypothesis on the Poisson noise as (H1) For any u ∈ X(T ), there exists a constant C < ∞ such that 19]). There exist some monotontic increasingly continuous functionsC 1 (t) and a constant α > 0,∀h, g ∈ X(T ) such that

Lemma 2.4 ([19]
). There exist some monotonely increasingly continous functionsC 2 (t) such that Proof. Define a stopping time by τ N = inf{t : u(t) 2 L 2 ) > N }. By using Itô formula, we have Next, we will estimate Taking the expectation and using Burkholder-Davis-Gundy inequality, we have For I 2 (t ∧ τ N ), by using a direct calculation, we have From the above estimates, we have Therefore, we deduce The Gronwall's inequality leads to Similarly, applying Itô formula and Burkholder-Davis-Gundy inequality as well as Sobolev embeddings Lemmas to we also obtain We have completed the proof of Lemma 2.5.

Existence of mild solution of equation (1.1)
In this section, we will establish the well-posedness of (1.1) in the space X(T ) = D(0, T ; H n (R)). Proof. Let {τ n : n ∈ N} be a family of independent exponential distributed random variables with parameter ρ, and set Define {N (t) : t 0} as the counting process Then N (t) is a Poisson distributed random variable with parameter ρt for any fixed t > 0. Let {Y n : n ∈ N} be a family of independent, ν/ρ distributed random variables. Then Notice that N (t) = 0 on the interval [0, T 1 ), then the stochastic equation (1.1) can be rewritten as The mild solution of equation (3.1) can be represented by Define the operator F by Then it follows that Due to Lemma 2.3, Lemma 2.4, we obtain that there exists some constant C > 0 such that Thus, F maps X(T 1 ) into itself, and for any u 1 , u 2 ∈ X(T 1 ). Hence, the Banach Fixed point theorem guarantees that there exists a sufficient small T 1 > 0 such that the stochastic equation ( Next, we will consider the solution on [T 1 , T 2 ). Since a jump with size g(u(T 1 −), Y n ) occurs at time T 1 , we denote u 0 1 = u 1 (T 1 −) + g(u 1 (T 1 −), Y n ), and consider a second process on [T 1 , T 2 ) follows as Similarly, equation ( We have completed the proof of Theorem 3.1.

Existence of invariant measure
In this section, we will prove the existence of the invariant measure for equation (1.1). Let P t is a semigroup defined by the global solution of equation (1.1) on a Banach space X. It is well known that lim k→∞ P t ξ(u k 0 ) = P t ξ(u 0 ).
for any sequence u k 0 ∈ X, k ∈ N} with u k 0 − u 0 B → 0, k → ∞, for any t > 0 and ξ ∈ B b (X, R). Then P t is a strong Feller semigroup on B. By using Chebychev's inequality, we obtain that there exists some constant C(R 0 ) > 0 for v 0 − u 0 ≤ 1, Choosing a sufficient large R > 0 such that C(R0) Then we have .
Let τ 0 = τ , and define inductively a sequence of stopping times by whereū τ k (s) = s τ k Z S λ (t − s)g(u(s−), z)η(ds, dz), and X(τ k , s) is defined on [τ k , s]. Then τ k+1 − τ k and τ are independent with the same distribution. The law of large number gives that Therefore, P(τ n ≤ t) → 0 as n → ∞. Hence, there exists n > 0, n ∈ N such that

It follows that for any
Since v 0 converges to u 0 , then Thus, it holds that E[|ξ(u(t)) − ξ(v(t))|] → 0. the proof of Theorem 4.1 is complete.
Definition 4.2. Let (X, T ) be a topological space, and let {X n , n ∈ N}X 0 be a (X, T )-valued random variable, X n is said to converge to X 0 in distribution, if for any bounded continuous function F : X → R, Proof. Without loss of generality, we assume that t n is an increasing sequence, and denote by u n (t) the solution with u n (0) be the initial condition. Next, we will prove that u n (t) converges to ξ in distribution in L 2 loc (R). In fact, since Since the bounded sets in H n (R) is relatively compact in L 2 loc (R). Then there exists L 2 loc (R)-valued variable ξ and a subsequence u n (t n ) (we also denote it as u n (t n )) such that u n (t n ) converges to ξ in distribution in L 2 loc (R). Assume that {f i } is a set of smooth and compactly supported orthonormal basis of H n (R), then we have Therefore, Thus, it holds that ξ takes value in H n . Next we shall show the convergence in L 2 (R). Assume that {g i } is a set of smooth orthonormal basis with compactly support of H n (R), then we have Since E[ u n (t n ) 2 as M increases. Therefore, it follows that Therefore, u n (t n ) converges to ξ in distribution in L 2 (R). From the above arguments, we have Therefore u n (t n ) converges to ξ in distribution inH n (R), and {P tn (u n 0 , ·) : n ∈ N} is tight on {H n }. We have completed the proof of Theorem (4.3). Let R 0 = sup v∈K v H n + 1, we can choose a suitable R > 0 such that for any ε > 0, δ > 0 Define a stopping time τ as Let τ 0 = τ , Choose some proper N such that P (τ N ≤ 1) ≤ ε 2 . Let Then, we have on the interval A n {τ N 1}, and (2C(t)) N +1 v − v n H n < δ provided with n being sufficiently large. Therefore, have Therefore, it holds that lim  Proof. For any ε > 0, since {P n (0, ·) : n 0} is tight, then we can choose a compact set K ε ⊂ H n (R) such that sup 1], v ∈ K} is tight on{H n }, then by choosing a compact set We can deduce which implies that µ n (·) = 1 n n 0 P t (0, ·)dt, n = 1, 2, ... is tight on H n (R). We have completed the proof of Theorem 4.5. Proof. By using the Krylov-Bogoliubov Theorem and combing Theorem 4.1 with Theorem 4.5, we can obtain the existence of the invariant measures for the semigroup P t .
We have completed the proof of Theorem 4.6. and {(µ n P tn )(·) : n ∈ N + } = {µ n (·) : n ∈ N + } is tight for any deterministic initial condition since K is a close set,then K is compact. The Krein-Milman theorem yields that a convex compact set possesses extremal point. By using Theorem 3.2.7 in [32], we deduce that this extremal point is ergodic. Therefore, the equation (1.1) has an ergodic invariant measure. We have completed the proof of Theorem 4.6.

Discussion
In this section, we consider the following stochastic weakly damped higher-order KdV equation driven by pure jump noise du + (∂ 2n+1 where λ,η, ν are the same with that in the first section. To prove the existence of invariant measure for Markov semigroup generated by equation (6.1), we impose some assumption on noise: Hypothesis 6.1.
The mild solution of equation (6.1) can be written as Lemma 6.2. If u(t) ∈ X(T ) solves the equation (6.2), under the condition of hypothesis 6.1, there exists a constant C > 0,such that: With the similar scheme, we obtain the following result.
Theorem 6.4. Under the conditions of theorem 6.3, P t is a strong Feller semigroup on H n (R),and µ n (·) = 1 n n 0 P t (0, ·)dt, n = 1, 2, ... is tight on H n (R), hence, there exists invariant measures of equation (6.1). Furthermore, there exists an ergodic invariant measure with deterministic initial conditions. Remark 6.5. If ν(Z) = +∞, then u(t) is not square integrable, and the uniqueness of solution can not obtain. Hence, we could not prove the existence of invariant measure in this case.
Then we will give some numerical simulation of the invariant measure. To the end, we give the distribution of the solution 1 T M T M m=0 E [Φ (u(t m ))] by using the so-called the Monte Carl method as following, one can prove theoretically that it does have a unique invariant measure, which derive to the ergodicity [33].
We firstly use the the norm conservative finite difference scheme introduced by [2] to simulate the equation (1.1) to the and s≤t 1 A (∆L s (ω)) is a possion process with the parameter P ∆t. Now we set β = 0.01, λ = 0.5, and u 0 (x) = sin(x). The simulation of (1.1) driven by Poisson noise with g(u(t−), z) = 0.2u(t−)z is given in Figure 1. Figure 2 gives time changes of u(t, ·) L 2 x using different sample trajectories.   It can be clearly seen from Figure 2 that the decay rate of the equation is slowed down under the influence of noise. At the same time, as shown in Figure 3, it can be seen that for Φ(y) = exp(−|y| 2 ), the distribution of the solution 1 N +1 N n=0 E Φ U (n) tends to a measure µ as T → ∞. The above numerical simulation of stochastic damped KdV equation (Fig. 3) in the sense of E u(t, ·) L 2 x reveals that stochastic damped KdV equation driven by Poisson noise posses an unique ergodic invariant measure.