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Asymptotic behaviour of time fractional stochastic delay evolution equations with tempered fractional noise

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    *Corresponding author

This work was supported by the National Natural Science Foundation of China under grant 41875084

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  • This paper is concerned with stochastic delay evolution equations driven by tempered fractional Brownian motion (tfBm) $ B_Q^{\sigma, \lambda}(t) $ with time fractional operator of order $ \alpha\in (1/2+\sigma, 1) $, where $ \sigma\in (-1/2, 0) $ and $ \lambda>0 $. First, we establish the global existence and uniqueness of mild solutions by using the new established estimation of stochastic integrals with respect to tfBm. Moreover, based on the relations between the stochastic integrals with respect to tfBm and fBm, we show the continuity of mild solutions for stochastic delay evolution equations when tempered fractional noise is reduced to fractional noise. Finally, we analyze the stability with general decay rate (including exponential, polynomial and logarithmic stability) of mild solutions for stochastic delay evolution equations with tfBm and time tempered fractional operator.

    Mathematics Subject Classification: Primary: 34K50, 60G22; Secondary: 34K20, 34K37.


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