In this paper, an averaging principle for multidimensional, time dependent, stochastic differential equations (SDEs) driven by fractional Brownian motion and standard Brownian motion was established. ...We combined the pathwise approach with the Itô stochastic calculus to handle both types of integrals involved and proved that the original SDEs can be approximated by averaged SDEs in the manner of mean square convergence.
In this paper, we study distribution dependent stochastic differential equations driven simultaneously by fractional Brownian motion with Hurst index H>12 and standard Brownian motion. We first ...establish the existence and uniqueness theorem for solutions of the distribution dependent stochastic differential equations by utilising the Carathéodory approximation. Then under certain averaging condition, we show that the solutions of distribution dependent stochastic differential equations can be approximated by the solutions of the associated averaged distribution dependent stochastic differential equations in the sense of the mean square convergence.
In this article, we are concerned with averaging principle for stochastic hyperbolic–parabolic equations driven by Poisson random measures with slow and fast time-scales. We first establish the ...existence and uniqueness of weak solutions of the stochastic hyperbolic–parabolic equations. Then, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the stochastic wave equation is an average with respect to the stationary measure of the fast varying process. Finally, we derive the rate of strong convergence for the slow component towards the solution of the averaged equation.
This paper is concerned with stochastic differential equations (SDEs for short) with irregular coefficients. By utilising a functional analytic approximation approach, we establish the existence and ...uniqueness of strong solutions to a class of SDEs with critically irregular drift coefficients in a new critical Lebesgue space, where the element may be only weakly integrable in time. To be more precise, let b:0,T×Rd→Rd be Borel measurable, where T>0 is arbitrarily fixed and d⩾1. We consider the following SDEXt=x+∫0tb(s,Xs)ds+Wt,t∈0,T,x∈Rd, where {Wt}t∈0,T is a d-dimensional standard Wiener process. For p,q∈1,+∞), we denote by Cq(0,T;Lp(Rd)) the space of all Borel measurable functions f such that t1qf(t)∈C(0,T;Lp(Rd)). If b=b1+b2 such that |b1(T−⋅)|∈Cq(0,T;Lp(Rd)) with 2/q+d/p=1 and ‖b1(T−⋅)‖Cq(0,T;Lp(Rd)) is sufficiently small, and that b2 is bounded and Borel measurable, then we show that there exists a weak solution to the above equation, and if in addition limt↓0‖t1qb(T−t)‖Lp(Rd)=0, the pathwise uniqueness holds. Furthermore, we obtain the strong Feller property of the semi-group and the existence of density associated with the above SDE. Besides, we extend the classical results concerning partial differential equations (PDEs) of parabolic type with Lq(0,T;Lp(Rd)) coefficients to the case of parabolic PDEs with Lq∞(0,T;Lp(Rd)) coefficients, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Theorem 3.1). Our results extend Krylov-Röckner and Krylov's profound results of SDEs with singular time dependent drift coefficients 20,23 to the critical case of SDEs with critically irregular drift coefficients in a new critical Lebesgue space.
Due to the intrinsic link with (kinetic) nonlinear Fokker–Planck equations and many diverse applications, distribution dependent stochastic differential equations have been investigated intensively ...in recent years. The appearance of the probability distributions (or laws) of the random variables of solutions in the coefficients is a distinct feature of distribution dependent stochastic differential equations. In this paper, under certain averaging conditions, we establish a stochastic averaging principle for distribution dependent stochastic differential equations.
In this paper, we are concerned with multi-scale distribution-dependent stochastic differential equations driven by fractional Brownian motion (with Hurst index
H
>
1
2
) and standard Brownian ...motion, simultaneously. Our aim is to establish a large deviation principle for the multi-scale distribution-dependent stochastic differential equations. This is done via the weak convergence approach and our proof is based heavily on the fractional calculus.
In this paper, we established the Freidlin–Wentzell-type large deviation principles for first-order scalar conservation laws perturbed by small multiplicative noise. Due to the lack of the viscous ...terms in the stochastic equations, the kinetic solution to the Cauchy problem for these first-order conservation laws is studied. Then, based on the well-posedness of the kinetic solutions, we show that the large deviations holds by utilising the weak convergence approach.
In this paper, by utilising Besov space techniques, we establish the time averaging principle for a heat equation with fractional Laplace driven by a general stochastic measure μ which is assumed ...(only) to satisfy the σ-additivity in probability.
In this paper, we consider the extended stochastic Navier–Stokes equations with Caputo derivative driven by fractional Brownian motion. We firstly derive the pathwise spatial and temporal regularity ...of the generalized Ornstein–Uhlenbeck process. Then we discuss the existence, uniqueness, and Hölder regularity of mild solutions to the given problem under certain sufficient conditions, which depend on the fractional order α and Hurst parameter H. The results obtained in this study improve some results in existing literature.
Discovery of small molecule inhibitors targeting Mcl-1 (Myeloid cell leukemia 1) confronts many challenges. Based on the fact that Mcl-1 is mainly localized in mitochondria, we propose a new strategy ...of targeting mitochondria to improve the binding efficiency of Mcl-1 inhibitors. We report the discovery of complex 9, the first mitochondrial targeting platinum-based inhibitor of Mcl-1, which selectively binds to Mcl-1 with high binding affinity. Complex 9 was mainly concentrated in the mitochondria of tumor cells which led to an enhanced antitumor efficacy. Complex 9 induced Bax/Bak-dependent apoptosis in LP-1 cells and synergized with ABT-199 to kill ABT-199 resistant cells in multiple cancer models. Complex 9 was effective and tolerable as a single agent or in combination with ABT-199 in mouse models. This research work demonstrated that developing mitochondria-targeting Mcl-1 inhibitors is a new potentially efficient strategy for tumor therapy.
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