In this paper, we study the linearly damped stochastic differential equations, which have the invariants satisfying a linear differential equation whose coefficients are linear constant or time-dependent. A stochastic exponential integrator is proposed for linearly damped stochastic differential equations to preserve their intrinsic properties. Then, the conformal symplecticity of stochastic Hamiltonian systems with linearly damped term is studied. For linearly damped stochastic Hamiltonian systems, it is shown that the stochastic exponential integrator can exactly preserve conformal quadratic invariant and conformal symplecticity. The mean-square convergence order of the method is analyzed. Numerical tests present the good performance of the proposed stochastic exponential integrator in structure-preserving.
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This work is supported by the National Key R&D Program of China (No. 2017YFC1405600), the National Natural Science Foundation of China (Nos. 11501150 and 11701124) and the Natural Science Foundation of Shandong Province of China (No. ZR2017PA006). The authors would like to express their appreciation to the anonymous referees for their many valuable suggestions and for carefully correcting the preliminary version of the manuscript.
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