Pullback Attractors for Stochastic Young Differential Delay Equations

Abstract

We study the asymptotic dynamics of stochastic Young differential delay equations under the regular assumptions on Lipschitz continuity of the coefficient functions. Our main results show that, if there is a linear part in the drift term which has no delay factor and has eigenvalues of negative real parts, then the generated random dynamical system possesses a random pullback attractor provided that the Lipschitz coefficients of the remaining parts are small.

Appendix

1.1 Young Integrals

For $[a,b]\subset {\mathbb {R}}$, denote by ${\mathcal {C}}([a,b],{\mathbb {R}}^d)$ the space of all continuous functions $y: [a,b]\rightarrow {\mathbb {R}}^d$, equipped with the sup norm

$$\begin{aligned} \Vert y\Vert _{\infty ,[a,b]} = \sup _{t\in [a,b]} \Vert y(t)\Vert , \end{aligned}$$

in which $\Vert \cdot \Vert$ is the Euclide norm of a vector in ${\mathbb {R}}^d$. Also, for $0<\beta \le 1$ denote by ${\mathcal {C}}^{\beta }([a,b],{\mathbb {R}}^d)$ the Banach space of all Hölder continuous paths $y:[a,b]\rightarrow {\mathbb {R}}^d$ with exponential $\beta$, equipped with the norm

$$\begin{aligned} \Vert y\Vert _{\infty ,\beta ,[a,b]}:= & {} \Vert y\Vert _{\infty ,[a,b]} + \left| \! \left| \! \left| y\right| \! \right| \! \right| _{\beta ,[a,b]},\ \text {where}\nonumber \\ \left| \! \left| \! \left| y\right| \! \right| \! \right| _{\beta ,[a,b]}:= & {} \sup _{a\le s<t\le b} \frac{\Vert y(t)-y(s)\Vert }{(t-s)^{\beta }}< \infty . \end{aligned}$$

(4.47)

One can easily prove for any $a\le s\le t\le u\le b$ that

$$\begin{aligned} \left| \! \left| \! \left| y\right| \! \right| \! \right| _{\beta ,[s,u]} \le \left| \! \left| \! \left| y\right| \! \right| \! \right| _{\beta ,[s,t]}+\left| \! \left| \! \left| y\right| \! \right| \! \right| _{\beta ,[t,u]}. \end{aligned}$$

Note that the space ${\mathcal {C}}^{\beta }([a,b],{\mathbb {R}}^d)$ is not separable. However, the closure of ${\mathcal {C}}^{\infty }([a,b],{\mathbb {R}}^d)$ denoted by ${\mathcal {C}}^{0,\beta }([a,b],{\mathbb {R}}^d)$ is a separable space (see [19, Theorem 5.31, p. 96]), which can be defined as

$$\begin{aligned} {\mathcal {C}}^{0,\beta }([a,b],{\mathbb {R}}^d) := \Big \{x \in {\mathcal {C}}^{\beta }([a,b],{\mathbb {R}}^d) \quad | \quad \lim \limits _{h \rightarrow 0} \quad \sup _{a\le s<t\le b, |t-s|\le h} \frac{\Vert x(t)-x(s)\Vert }{(t-s)^{\beta }} = 0 \Big \}. \end{aligned}$$

It is worth to mention that for $\beta <\alpha$, ${\mathcal {C}}^{\alpha }([a,b],{\mathbb {R}}^d)$ is a subspace of ${\mathcal {C}}^{0,\beta }([a,b],{\mathbb {R}}^d)$ and moreover, the embedding operator

$$\begin{aligned} id: {\mathcal {C}}^{\alpha }([a,b],{\mathbb {R}}^d) \rightarrow {\mathcal {C}}^{\beta }([a,b],{\mathbb {R}}^d) \end{aligned}$$

is compact (see [19, Proposition 5.28, p. 94]).

Now we recall that for $y \in {\mathcal {C}}^{\beta }([a,b],{\mathbb {R}}^{d\times k})$ and $x \in {\mathcal {C}}^{\nu }([a,b],{\mathbb {R}}^k)$ with $\beta +\nu >1$. Then the Young integral $\int _a^by(t)dx(t)$ exists (see [25]) and satisfies the Young-Loeve estimate [19, Theorem 6.8, p. 116],

$$\begin{aligned} \left\| \int _s^ty(u)dx(u) - y(s)[x(t)-x(s)]\right\| \le K (t-s)^{\beta +\nu }\left| \! \left| \! \left| x \right| \! \right| \! \right| _{\nu ,[s,t]} \left| \! \left| \! \left| y \right| \! \right| \! \right| _{\beta ,[s,t]},\quad \forall a \le s\le t \le b, \end{aligned}$$

where $K:= \frac{1}{1-2^{1-(\beta +\nu )}}$. Hence

$$\begin{aligned} \left\| \int _s^ty(u)dx(u)\right\| \le (t-s)^\nu \left| \! \left| \! \left| x\right| \! \right| \! \right| _{\nu ,[s,t]} \left( \Vert y(s)\Vert +K(t-s)^{\beta }\left| \! \left| \! \left| y\right| \! \right| \! \right| _{\beta ,[s,t]} \right) . \end{aligned}$$

(4.48)

1.2 Tempered Variables

Let $(\Omega ,{\mathcal {F}},{\mathbb {P}})$ be a probability space equipped with an ergodic metric dynamical system $\theta$, which is a ${\mathbb {P}}$ measurable mapping $\theta : {\mathbb {T}} \times \Omega \rightarrow \Omega$, ${\mathbb {T}}$ is either ${\mathbb {R}}$ or ${\mathbb {Z}}$, and $\theta _{t+s} = \theta _t \circ \theta _s$ for all $t,s \in {\mathbb {T}}$. Recall that a random variable $\rho : \Omega \rightarrow [0,\infty )$ is called tempered if

$$\begin{aligned} \lim \limits _{t \rightarrow \pm \infty } \frac{1}{t} \log ^+ \rho (\theta _{t}x) =0,\quad \text {a.s.} \end{aligned}$$

(4.49)

which, as shown in [20, p. 220], [22], is equivalent to the sub-exponential growth

$$\begin{aligned} \lim \limits _{t \rightarrow \pm \infty } e^{-c |t|} \rho (\theta _{t}x) =0\quad \text {a.s.}\quad \forall c >0. \end{aligned}$$

Note that our definition of temperedness corresponds to the notion of temperedness from above given in [3, Definition 4.1.1(ii)].

Lemma 4.9

• (i) If $h_1, h_2\ge 0$ are tempered random variables then $h_1+h_2$ and $h_1h_2$ are tempered random variables.

• (ii) If $h_1\ge 0$ is a tempered random variable, $h_2\ge 0$ is a measurable random variable and $h_2\le h_1$ almost surely, then $h_2$ is a tempered random variable.

• (iii) Let $h_1$ be a nonnegative measurable function. If $\log ^+ h_1\in L^1$ then $h_1$ is tempered.

Proof

(i):

See [3, Lemma 4.1.2, p. 164].

(ii):

Immediate from the definition of tempered random variable, formula (4.49).

(iii):

See [3, Proposition 4.1.3, p. 165].

$\square$

Lemma 4.10

(i) Let $a : \Omega \rightarrow [0,\infty )$ be a random variable, $\log (1+a(\cdot ))\in L^1$ and ${\hat{a}} := E\log (1+a(\cdot )) = \int _\Omega \log (1+a(\cdot )) d{\mathbb {P}}$. Let $\lambda > {\hat{a}}$ be an arbitrary fixed positive number. Put

$$\begin{aligned} b(x) := \sum _{k=1}^\infty e^{-\lambda k} \prod _{i=0}^{k-1} (1+ a(\theta _{-i} x)). \end{aligned}$$

Then $b(\cdot )$ is a nonnegative almost everywhere finite and tempered random variable.

(ii) Let $c : \Omega \rightarrow [0,\infty )$ be a tempered random variable, and $\delta >0$ be an arbitrary fixed positive number. Put

$$\begin{aligned} d(x) := \sum _{k=1}^\infty e^{-\delta k} c(\theta _{-k} x). \end{aligned}$$

Then $d(\cdot )$ is a nonnegative almost everywhere finite and tempered random variable.

Proof

(i) Put $b_n(x) := \sum _{k=1}^n e^{-\lambda k} \prod _{i=0}^{k-1} (1+ a(\theta _{-i} x))$. Then $b_n(\cdot )$, $n\in {\mathbb {N}}$, is an increasing sequence of nonnegative random variable, hence converges to the nonnegative random variable $b(\cdot )$. Since $\log (1+a(\cdot ))\in L^1$, by Birkhoff ergodic theorem there exists a $\theta$-invariant set $\Omega '\subset \Omega$ of full measure such that for all $x\in \Omega '$ we have $\lim _{n\rightarrow \pm \infty } (\sum _{i=0}^{n-1}\log (1+ a(\theta _{-i} x))/n = {\hat{a}}$. Hence given any fixed $\delta >0$ for all n big enough, $\prod _{i=0}^{n-1} (1+ a(\theta _{-i} x)) < \exp ({\hat{a}} +\delta )n$. Consequently, since $\lambda > {\hat{a}}$ the sequence $b_n(\cdot )$, $n\in {\mathbb {N}}$ tends to limit $b(\cdot )$, which is finite almost surely.

Now we show that $b(\cdot )$ is tempered. For $m\in {\mathbb {N}}$ and $x\in \Omega '$ we obtain

$$\begin{aligned} b(\theta _{-m} x)= & {} \sum _{k=1}^\infty e^{-\lambda k} \prod _{i=0}^{k-1} (1+ a(\theta _{-i} \theta _{-m} x)) = \sum _{k=1}^\infty e^{-\lambda k} \prod _{j=m}^{k+m-1} (1+ a(\theta _{-j} x))\\\le & {} e^{\lambda m} \sum _{l=1+m}^\infty e^{-\lambda l} \prod _{j=0}^{l-1} (1+ a(\theta _{-j} x)) \le e^{\lambda m} \sum _{l=1}^\infty e^{-\lambda l} \prod _{j=0}^{l-1} (1+ a(\theta _{-j} x)) = e^{\lambda m} b(x). \end{aligned}$$

This implies that $\limsup _{m\rightarrow \infty } \frac{1}{m} \log ^+ b(\theta _{-m} x) \le \lambda$. By virtue of [3, Proposition 4.1.3(i), p. 165] and [23, Lemma 4, Corollary 4], for all $x\in \Omega '$ we have

$$\begin{aligned} \limsup _{m\rightarrow \infty } \frac{1}{m} \log ^+ b(\theta _{-m} x) = \limsup _{m\rightarrow -\infty } \frac{1}{-m} \log ^+ b(\theta _{-m} x)= 0, \end{aligned}$$

which proves that $b(\cdot )$ is tempered.

(ii) Put $d_n(x) := \sum _{k=1}^n e^{-\delta k} c(\theta _{-k} x)$. Then $d_n(\cdot )$, $n\in {\mathbb {N}}$, is an increasing sequence of nonnegative random variable, hence converges to the nonnegative random variable $d(\cdot )$. By temperedness of $c(\cdot )$ we can find a measurable set ${{\tilde{\Omega }}}\subset \Omega$ of full measure such that for all $x\in {{\tilde{\Omega }}}$ there exists $n_0(x) >0$ such that for all $n\ge n_0(x)$ we have $c(\theta _{-n} x) \le e^{n\delta /2}$. Hence $d_n(x)$, $n\in {\mathbb {N}}$, is an increasing sequence of positive numbers tending to finite value d(x). Thus $d(\cdot )$ is finite almost everywhere. Furthermore, for $m\in {\mathbb {N}}$ and $x\in {{\tilde{\Omega }}}$,

$$\begin{aligned} d(\theta _{-m} x)= & {} \sum _{k=1}^\infty e^{-\delta k} c(\theta _{-k} \theta _{-m} x) = \sum _{l=m+1}^\infty e^{-\delta (l-m)} a(\theta _{-l} x)\\\le & {} e^{\delta m} \sum _{l=1}^\infty e^{-\delta l} c(\theta _{-l} x) = e^{\delta m} d(x). \end{aligned}$$

This implies that $\limsup _{m\rightarrow \infty } \frac{1}{m} \log ^+ d(\theta _{-m} x) \le \delta$. Similar to (i) above, $d(\cdot )$ is tempered. $\square$

1.3 Gronwall Lemma

Lemma 4.11

(Continuous Gronwall Lemma) Let $[t_0,T]$ be an interval on ${\mathbb {R}}$. Assume that $u(\cdot ), a(\cdot ): [t_0,T] \rightarrow {\mathbb {R}}^+$ are positive continuous functions and $\beta >0$ is a positive number, such that

$$\begin{aligned} u(t) \le a(t) + \int _{t_0}^t \beta u(s) ds, \quad \forall t \in [t_0,T]. \end{aligned}$$

Then the following inequality holds

$$\begin{aligned} u(t) \le a(t) + \int _{t_0}^t a(s) \beta e^{\beta (t-s)} ds, \quad \forall t \in [t_0,T]. \end{aligned}$$

Proof

See [2, Lemma 6.1, p 89]. $\square$

Lemma 4.12

(Discrete Gronwall Lemma) Let a be a non negative constant and $u_n, \alpha _n,\beta _n$ be nonnegative sequences satisfying for all $n\in {\mathbb {N}}$, $n\ge 0$, the equalities

$$\begin{aligned} u_n\le a + \sum _{k=0}^{n-1} \alpha _ku_k + \sum _{k=0}^{n-1} \beta _k. \end{aligned}$$

Then for all $n\in {\mathbb {N}}$, $n\ge 1$, the following inequalities hold

$$\begin{aligned} u_n\le \max \{a,u_0\}\prod _{k=0}^{n-1} (1+\alpha _k) + \sum _{k=0}^{n-1}\beta _k\prod _{j=k+1}^{n-1}(1+\alpha _j). \end{aligned}$$

(4.50)

Proof

See [16]. $\square$