Weak averaging of semilinear stochastic differential equations with almost periodic coefficients

Kamenski, Mikhail; Mellah, Omar; Fitte, Paul Raynaud de

doi:10.1016/j.jmaa.2015.02.036

Cited by 65 publications

(37 citation statements)

References 10 publications

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“…Proof of Theorem 4.1 The proof of the existence and uniqueness of a mild solution to (4.1) in CUB R, L 2 (P, H 2 ) is the same as that of Theorem 3.1 in [31] or Theorem 3.3.1 in [37]. For the almost automorphy part, let (γ ′ n ) be a sequence in R. Since f and g are almost automorphic, there exists a subsequence (γ n ) and functions f :…”

Section: Pseudo Almost Automorphic Solutions To Stochastic Differentimentioning

confidence: 94%

Almost automorphy and various extensions for stochastic processes

Bedouhene

Challali

Mellah

et al. 2015

Journal of Mathematical Analysis and Applications

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We compare different modes of pseudo almost automorphy and variants for stochastic processes: in probability, in quadratic mean, or in distribution in various senses. We show by a counterexample that squaremean (pseudo) almost automorphy is a property which is too strong for stochastic differential equations (SDEs). Finally, we consider two semilinear SDEs, one with almost automorphic coefficients and the second one with pseudo almost automorphic coefficients, and we prove the existence and uniqueness of a mild solution which is almost automorphic in distribution in the first case, and pseudo almost automorphic in distribution in the second case.

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Section: Pseudo Almost Automorphic Solutions To Stochastic Differentimentioning

confidence: 94%

Almost automorphy and various extensions for stochastic processes

Bedouhene

Challali

Mellah

et al. 2015

Journal of Mathematical Analysis and Applications

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“…Lemma Let

g : double-struckR \to double-struckR

be a continuous function such that, for every

t \in double-struckR

0 \leq g false(t false) \leq α false(t false) + β_{1} true \int_{- \infty}^{t} e^{- δ_{1} false(t - s false)} g false(s false) d s + . . . + β_{n} true \int_{- \infty}^{t} e^{- δ_{n} false(t - s false)} g false(s false) d s,

for some locally integrable function

α : double-struckR \to double-struckR

, and for some constants β 1 ,…, β n ≥ 0, and some constants δ 1 ,…, δ n > β , where

β = \sum_{i = 1}^{i = n} β_{i}

. We assume that the integrals in the right‐hand side of are convergent.…”

Section: Resultsmentioning

confidence: 99%

Weyl almost periodic solutions to abstract linear and semilinear equations with Weyl almost periodic coefficients

Bedouhene

Ibaouene

Mellah

et al. 2018

Math Methods in App Sciences

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In this work, we study the existence and uniqueness of bounded Weyl almost periodic solution to the abstract differential equation uthat generates an exponentially stable C 0 -semigroup on X and ∶ R → X is a Weyl almost periodic function. We also investigate the semilinear problems of the form u ′ (t) = Au(t) + f (t, u(t)).KEYWORDS linear and semilinear differential equation, mild solution, Weyl almost periodic 9546

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“…We thus have

0true E ∥ x_{n} (t) - \overset{x}{true} {(t) false∥}^{2} \leq α_{n} + \frac{6 K^{2} L}{ω} \int_{- \infty}^{t} e^{- ω false(t - s false)} E ∥ x_{n} (s) - \overset{x}{true} {(s) false∥}^{2} d s + 6 K^{2} L_{1} \int_{- \infty}^{t} e^{- 2 ω false(t - s false)} E {false∥ x_{n} (s) - \overset{x}{true} (s) false∥}^{2} d s

for a sequence

α_{n}

such that

{trueprefixlim}_{n \to \infty} α_{n} = 0

. By a variant of Gronwall's lemma in (Lemma 3.3) and

\frac{6 K^{2} L}{ω^{2}} + \frac{6 K^{2} L_{1}}{ω} < 1

, it follows that

E false∥ x_{n} false(t false) - \tilde{x} {false(t false) ∥}^{2} \to 0, as n \to \infty for each t \in R .

Since

x true(t + s_{n} true)

has the same distribution as

x_{n} false(t false)

, it follows that

x true(t + s<...>

…”

Section: Almost Automorphic Solutions For Non‐linear Stochastic Diffementioning

confidence: 99%

“…Hence it is a unique way to study some stochastic periodicity in distribution for stochastic differential equations. Almost periodic solutions in distribution and fundamental averaging results for stochastic differential equations were studied in . Almost automorphic stochastic processes and their generalizations have been investigated only since 2010 starting with .…”

Section: Introductionmentioning

confidence: 99%

Almost automorphic solutions for stochastic differential equations driven by fractional Brownian motion

Chen¹,

Yang

2018

Mathematische Nachrichten

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This paper concerns a class of stochastic differential equations driven by fractional Brownian motion. The existence and uniqueness of almost automorphic solutions in distribution are established provided the coefficients satisfy some suitable conditions. To illustrate the results obtained in the paper, a stochastic heat equation driven by fractional Brownian motion is considered. 1 K E Y W O R D Salmost automorphic solutions, fractional Brownian motion, stochastic differential equations M S C ( 2 0 1 0 ) 34C27, 60E05, 60H10, 93C30 INTRODUCTIONIt is well known that the recurrence of stochastic processes introduced by Kolmogorov in 1930's [24] is similar to the recurrence of dynamical systems introduced by Birkhoff in 1910's [5]. In dynamical systems, the almost automorphy is a class of basic recurrence introduced by Bochner [6], which is a generalization of almost periodicity [7]. Almost automorphy functions and almost automorphic solutions for differential systems have been studied by many authors; see Veech [34], N'Gu'er'ekata [17], Johnson [22], N'Gu'er'ekata [18], Ding, Liang and Xiao [11][12][13], Shen and Yi [31] and N'Gu'er'ekata [19], for more details about these topics and recent developments. Accordingly, the concept of distributionally almost automorphy for stochastic processes was introduced in [14,15].Many authors investigated almost periodic or pseudo almost periodic solutions for stochastic differential equations; see literature, for example, [1,3,8,20,32]. Mellah and De Fitte [28] gave some counterexamples to mean square almost periodicity of solutions for some stochastic differential equations with almost periodic coefficients. Hence it is a unique way to study some stochastic periodicity in distribution for stochastic differential equations. Almost periodic solutions in distribution and fundamental averaging results for stochastic differential equations were studied in [23]. Almost automorphic stochastic processes and their generalizations have been investigated only since 2010 starting with [16]. There are some papers investigating almost automorphy in distribution [14,15,27]. In these paper, almost automorphic solutions in distribution of stochastic differential equations driven by Gaussian processes or non-Gaussian processes were studied. In [2], the history of almost automorphic stochastic processes and their generalizations were given, and the existence and uniqueness of pseudo almost automorphic mild solutions for stochastic differential equations were also obtained.Recently, the definition and some properties of almost automorphic random functions in probability has been introduced by Ding, Deng and N'Guérékata [10]. Mathematische Nachrichten. 2019;292:983-995.

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Weak averaging of semilinear stochastic differential equations with almost periodic coefficients

Cited by 65 publications

References 10 publications

Almost automorphy and various extensions for stochastic processes

Almost automorphy and various extensions for stochastic processes

Weyl almost periodic solutions to abstract linear and semilinear equations with Weyl almost periodic coefficients

Almost automorphic solutions for stochastic differential equations driven by fractional Brownian motion

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