Harnack inequality and gradient estimate for G-SDEs with degenerate noise

In this paper,the Harnack inequality for G-SDEs with degenerate noise is derived by the method of coupling by change of the measure.Moreover,for any bounded and continuous function f,the gradient estimate∣∇P_(t)f∣≤c(p,t)(Pt|f|p)^(1/p),p>1,t>0 is obtained for the associated nonlinear semigroup P¯t.As an application of the Harnack inequality,we prove the existence of the weak solution to degenerate G-SDEs under some integrable condition.Finally,an example is presented.All of the above results extend the existing ones in the linear expectation setting.

Harnack inequality and gradient estimate for functional G-SDEs with degenerate noise

In this paper,the Harnack and shift Harnack inequalities for functional G-SDEs with degenerate noise are derived by the method of coupling by change of measure.Moreover,the gradient estimate for the associated nonlinear semigroup P_(t) is obtained.All of the above results extend the existed results in linear expectation setting.

Well-posedness and exponential mixing for stochastic magneto-hydrodynamic equations with fractional dissipations

Consider d-dimensional magneto-hydrodynamic(MHD)equations with fractional dissipations driven by multiplicative noise.First,we prove the existence of martingale solutions for stochastic fractional MHD equations in the case of d=2,3 andα∧β〉0,whereα,βare the parameters of the fractional dissipations in the equation.Second,for d=2,3 andα∧β≥12+d4,we show the pathwise uniqueness of solutions and then obtain the existence and uniqueness of strong solutions using the Yamada-Watanabe theorem.Furthermore,we establish the exponential mixing property for stochastic MHD equations with degenerate multiplicative noise when d=2,3 andα∧β≥12+d4.