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In a previous paper, we studied the ergodic properties of an Euler scheme of a stochastic differential equation with a Gaussian additive noise in order to approximate the stationary regime of such an equation. We now consider the case of multiplicative noise when the Gaussian process is a fractional Brownian motion with Hurst parameter H>1/2H>1/2 and obtain some (functional) convergence properties of some empirical measures of the Euler scheme to the stationary solutions of such SDEs.  相似文献   

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This paper is devoted to application of fractional multistep method in the numerical solution of fractional diffusion-wave equation. By transforming the diffusion-wave equation into an equivalent integro-differential equation and applying Lubich’s fractional multistep method of second order we obtain a scheme of order O(τα+h2)O(τα+h2) for 1?α?1.718321?α?1.71832 where αα is the order of temporal derivative and ττ and h denote temporal and spatial stepsizes. The solvability, convergence and stability properties of the algorithm are investigated and numerical experiment is carried out to verify the feasibility of the scheme.  相似文献   

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We study the problem (−Δ)su=λeu(Δ)su=λeu in a bounded domain Ω⊂RnΩRn, where λ   is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7n7 for all s∈(0,1)s(0,1) whenever Ω   is, for every i=1,...,ni=1,...,n, convex in the xixi-direction and symmetric with respect to {xi=0}{xi=0}. The same holds if n=8n=8 and s?0.28206...s?0.28206..., or if n=9n=9 and s?0.63237...s?0.63237.... These results are new even in the unit ball Ω=B1Ω=B1.  相似文献   

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In this paper, we consider Caputo type fractional differential equations of order 0<α<10<α<1 with initial condition x(0)=x0x(0)=x0. We introduce a technique to find the exact solutions of fractional differential equations by using the solutions of integer order differential equations. Generalization of the technique to finite systems is also given. Finally, we give some examples to illustrate the applications of our results.  相似文献   

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We prove the Malliavin regularity of the solution of a stochastic differential equation driven by a fractional Brownian motion of Hurst parameter H>0.5H>0.5. The result is based on the Fréchet differentiability with respect to the input function for deterministic differential equations driven by Hölder continuous functions. It is also shown that the law of the solution has a density with respect to the Lebesgue measure, under a suitable nondegeneracy condition.  相似文献   

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We study the fractional Schrödinger equations in R1+dR1+d, d?3d?3, of order d/(d−1)<α<2d/(d1)<α<2. Under the angular regularity assumption we prove linear and nonlinear profile decompositions which extend the previous results [9] to data without radial assumption. As applications we show blowup phenomena of solutions to mass-critical fractional Hartree equations.  相似文献   

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