In this paper, we focus on the exponential stability of stochastic differential equations driven by fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/2, 1). Based on the generalized Itô formula and representation of the fBm, some sufficient conditions for exponential stability of a class of SDEs with additive fractional noise are given. Besides, we present a criterion on the exponential stability for the fractional Ornstein-Uhlenbeck process with Markov switching. A numerical example is provided to illustrate our results.1. Introduction. To characterize the continuous dynamical system changes with the discrete state, the following stochastic differential equations (SDEs) have been developedwhere {r t } t≥0 is a Markov chain taking values in S = {1, 2, ..., N } and {B t } t≥0 is a standard Brownian motion. The process {X t , r t } is called a switching diffusion or a diffusion with switching. In the past thirty years, stability of stochastic hybrid systems has been considered extensively. For example, Yuan and Mao [30] consider the moment exponential stability of stochastic hybrid delayed systems with Lévy noise in mean square. Mao [18] discusses the exponential stability of general nonlinear stochastic hybrid systems. Some sufficient conditions for asymptotic stability in distribution of SDEs with Markovian switching are given by Yuan and Mao [29]. Most recently, Tan [27] focuses on the exponential stability of fractional stochastic systems with distributed delay driven by fractional Brownian motion. There are lots of work having been dedicated to Markovian switching. See [24,3,7,17,32,23] and so forth. It is well known that if H > 1/2, {B H t } t≥0 exhibits long range dependence and self-similarity. Because of these properties, {B H t } t≥0 has been suggested as a useful tool in many fields, especially mathematical finance, network traffic analysis and pricing of weather derivatives. For example, fBm is used to model the dynamics of temperature in [6], and in [25], it is used to model the electricity prices in 2010 Mathematics Subject Classification. Primary: 60H10, 60G22; Secondary: 93E15, 93C30.

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Valine-glutamine genes (VQ) acted as transcription regulators and played the important roles in plant growth and development, and stress tolerance through interacting with transcription factors and other co-regulators. In this study, sixty-one VQ genes containing the FxxxVQxxTG motif were identified and updated in the Nicotiana tobacum genome. Phylogenetic analysis indicated that NtVQ genes were divided into seven groups and genes of each group had highly conserved exon-intron structure. Expression patterns analysis firstly showed that NtVQ genes expressed individually in different tobacco tissues including mixed-trichome (mT), glandular-trichome (gT), and nonglandular-trichome (nT), and the expression levels were also distinguishing in response to methyl jasmonate (MeJA), salicylic acid (SA), gibberellic acid (GA), ethylene (ETH), high salinity and PEG stresses. Besides, only NtVQ17 of its gene family was verified to have acquired autoactivating activity. This work will not only lead a foundation on revealing the functions of NtVQ genes in tobacco trichomes but also provided references to VQ genes related stress tolerance research in more crops.

In this paper, the exponential stability of stochastic differential equations driven by multiplicative fractional Brownian motion (fBm) with Markovian switching is investigated. The quasi-linear cases with the Hurst parameter H ∈ (1/2, 1) and linear cases with H ∈ (0, 1/2) and H ∈ (1/2, 1) are all studied in this work. An example is presented as a demonstration.