Modulation Analysis of the Stochastic Camassa–Holm Equation with Pure Jump Noise

Abstract

We study the stochastic Camassa–Holm equation with pure jump noise. We prove that if the initial condition of the solution is a solitary wave solution of the unperturbed equation, the solution decomposes into the sum of a randomly modulated solitary wave and a small remainder. Moreover, we derive the equations for the modulation parameters and show that the remainder converges to the solution of a stochastic linear equation as amplitude of the jump noise tends to zero.

Appendix A. General Itô Formula for Marcus Canonical SDEs

First, we present the following Itô formula for the Marcus canonical SDEs.

Lemma A.1

(Itô formula, Brzeźniak and Manna (2019), Theorem B.2) Assume that U is a Hilbert space. Let X be a U-valued process given by

$$\begin{aligned} X(t)=X_0+\int _0^ta(X(s))ds+\int _0^t\int _{\mathbb {Z}}f(X(s-))\diamond dL(s),\;\;t\ge 0, \end{aligned}$$

where $a, f: U\rightarrow U$ are $\mathcal {F}_t$-adapted random mappings. Let V be a separable Hilbert space. Let $h: U\rightarrow V$ be a function of class $C^1$ such that the first derivative $h':U\rightarrow L(U; V)$ is $(p-1)$-Hölder continuous. Then for every $t > 0$, we have $\mathbb {P}$-a.s.

$$\begin{aligned} h(X(t))=&h(X_0)+\int _0^th'(X(s))(a(X(s)))ds\\&+\int _0^t\int _{\mathbb {Z}} [h(\Phi (1,z,X(s-)))-h(X(s-))]\tilde{\mathcal {N}}(ds,\textrm{d}z)\\&+\int _0^t\int _{\mathbb {Z}} [h(\Phi (1,z,X(s)))-h(X(s))-zh'(X(s))f(X(s))]\vartheta (\textrm{d}z)ds, \end{aligned}$$

where $\Phi (v):=\Phi (v,z,w)$ solves

$$\begin{aligned} \frac{d}{dt}\Phi (v)=zf(\Phi (v)), \end{aligned}$$

with initial condition $\Phi (0)=w.$

Next, we give the following general Itô formula for a pair of Marcus canonical SDEs.

Lemma A.2

(general Itô formula) Assume that U is a Hilbert space. Let X be a U-valued process and Y be a $\mathbb {R}$-valued process given by

$$\begin{aligned}&X(t,x)=X_0+\int _0^ta(X(s,x))ds+\int _0^t\int _{\mathbb {Z}}f(X(s-,x))\diamond dL(s),\\&Y(t)=Y_0+\int _0^tb(Y(s))ds+\int _0^t\int _{\mathbb {Z}}g(Y(s-)) \tilde{\mathcal {N}}(ds,\textrm{d}z). \end{aligned}$$

where $a, f: U\rightarrow U$ and $C^1$-class functions $b, g: \mathbb {R}\rightarrow \mathbb {R}$ are $\mathcal {F}_t$-adapted random mappings. Then the following formula holds

$$\begin{aligned}&X(t, Y(t))=X(0,Y_0)+\int _0^ta(X(s,Y(s)))ds\nonumber \\&+\int _0^tDX(s,Y(s))b(X(s,Y(s)))ds\nonumber \\&+\int _0^t\int _{\mathbb {Z}}(\Phi (1,z,X(s-, Y(s-)))-X(s-, Y(s-)))\tilde{\mathcal {N}}(ds,\textrm{d}z)\nonumber \\&+\int _0^t\int _{\mathbb {Z}}(\Phi (1,z,X(s-, Y(s-)+g(Y(s-)))-\Phi (1,z,X(s-, Y(s-))))\tilde{\mathcal {N}}(ds,\textrm{d}z)\nonumber \\&+\int _0^t\int _{\mathbb {Z}}(\Phi (1,z,X(s, Y(s)+g(Y(s))))-X(s, Y(s))\nonumber \\&-f(X(s,Y(s)))-DX(s,Y)g(Y))\vartheta (\textrm{d}z)ds, \end{aligned}$$

(A1)

where $\Phi (v)=\Phi (v,z,w_1)$ solves the following equations:

$$\begin{aligned} \frac{d}{dv}\Phi (v)=zf(\Phi (v)), \;\;\Phi (0)=w_1. \end{aligned}$$

Proof

The proof is similar to Theorem III.3.3 in Carmona and Nualart (1990) and Theorem 4.3 in [45].

Consider a sequence of mollifiers $h_n\in C^\infty _c(\mathbb {R}, \mathbb {R})$ given by $h_n(x) = nh(nx)$, where $h \in C^\infty _c(\mathbb {R}, \mathbb {R})$ supported on a unit ball $|x| \le 1, h(x) \ge 0$, and such that $\int _\mathbb {R}h(x)dx=1$.

Using Itô formula to $h_n(Y- x)$, we have

$$\begin{aligned}&h_n(Y(t)-x)=h_n(Y_0-x)+\int _0^th_n'(Y(s)-x)(b(Y(s)))ds\\&+\int _0^t\int _{\mathbb {Z}} [h_n(Y(s-)-x+g(Y(s-)))-h_n(Y(s-)-x)]\tilde{\mathcal {N}}(ds,\textrm{d}z)\\&+\int _0^t\int _{\mathbb {Z}} [h_n(Y(s)-x+g(Y(s)))-h_n(Y(s)-x)-zh_n'(Y(s)-x)g(Y(s))]\vartheta (\textrm{d}z)ds, \end{aligned}$$

Then, apply Itô product formula to $X(t,x)h_n(Y(t)-x)$ to yield

$$\begin{aligned}&X(t,x)h_n(Y(t)-x)=X_0h_n(Y_0-x) +\int _0^th_n(Y(s)-x)a(X(s,x))ds\nonumber \\&+\int _0^t\int _{\mathbb {Z}}h_n(Y(s)-x) [\Phi (1,z,X(s,x))-X(s,x)-zf(X(s,x))]\vartheta (\textrm{d}z)ds\nonumber \\&+\int _0^t\int _{\mathbb {Z}}X(s,x) [h_n(Y(s)-x+g(Y(s)))-h_n(Y(s)-x)-zh_n'(Y(s)-x)g(Y(s))]\vartheta (\textrm{d}z)ds\nonumber \\&+\int _0^t\int _{\mathbb {Z}} [\Phi (1,z,X(s,x))-X(s,x)][h_n(Y(s)-x+g(Y(s)))-h_n(Y(s)-x)]\vartheta (\textrm{d}z)ds\nonumber \\&+\int _0^tX(s,x)h_n'(Y(s)-x)(b(Y(s)))ds\nonumber \\&+\int _0^t\int _{\mathbb {Z}}h_n(Y(s-)-x)[\Phi (1,z,X(s-,x))-X(s-,x)]\tilde{\mathcal {N}}(ds,\textrm{d}z)\nonumber \\&+\int _0^t\int _{\mathbb {Z}}\Phi (1,z,X(s-,x)) [h_n(Y(s-)-x+g(Y(s-)))-h_n(Y(s-)-x)]\tilde{\mathcal {N}}(ds,\textrm{d}z), \end{aligned}$$

(A.2)

Taking the Lebesgue integrals on both sides of (A.2), letting $n\rightarrow \infty$ and following the proof in [45], we get (A1). $\square$