由分数噪声驱动的一类分数阶随机偏微分方程的光滑密度研究（英文）
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摘要
本文中,我们研究了由分数噪声驱动的一类分数阶随机偏微分方程,利用Malliavin分析技巧,证明了该类方程的适度解在任意固定的点(t,x)∈[0,T]×R具有光滑密度.
In this paper we consider a class of fractional stochastic partial differential equation driven by fractional noise. We prove that the solution admits a smooth density at any fixed point(t,x) ∈ [0, T] × R with T > 0 by using the techniques of Malliavin calculus.
In this paper we consider a class of fractional stochastic partial differential equation driven by fractional noise. We prove that the solution admits a smooth density at any fixed point(t,x) ∈ [0, T] × R with T > 0 by using the techniques of Malliavin calculus.
引文
[1]BALAN R M,TUDOR C A.Stochastic heat equation with multiplicative fractional-colored noise[J].J Theoret Probab,2010,23(3):834-870.
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[14]KOMATSU T.Markov processes associated with certain integro-differential operators[J].Osaka J Math,1973,10(2):271-303.
[15]KOMATSU T.Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms[J].Osaka J Math,1988,25(3):697-728.
[16]KOMATSU T.Uniform estimates for fundamental solutions associated with non-local Dirichlet forms[J].Osaka J Math,1995,32(4):833-860.
[17]NUALART D.The Malliavin Calculus and Related Topics[M].2nd ed.Berlin:Springer-Verlag,2006.
[18]HOH W.Pseudo differential operators with negative definite symbols of variable order[J].Rev Mat Iberoam,2000,16(2):219-241.
[19]JACOB N,LEOPOLD H G.Pseudo differential operators with variable order of differentiation generating feller semigroups[J].Integral Equations Operator Theory,1993,17(4):544-553.
[20]KIKUCHI K,NEGORO A.On Markov process generated by pseudodifferential operator of variable order[J].Osaka J Math,1997,34(2):319-335.
[2]BALAN R M.The stochastic wave equation with multiplicative fractional noise:a malliavin calculus approach[J].Potential Anal,2012,36(1):1-34.
[3]HU Y Z,NUALART D,SONG J.Feynman-Kac formula for heat equation driven by fractional white noise[J].Ann Probab,2011,39(1):291-326.
[4]LIU J F,YAN L T.Solving a nonlinear fractional stochastic partial differential equation with fractional noise[J].J Theoret Probab,2016,29(1):307-347.
[5]LIU J F,TUDOR C A.Analysis of the density of the solution to a semilinear SPDE with fractional noise[J].Stochastics,2016,88(7):959-979.
[6]MILLET A,SANZ-SOLE M.A stochastic wave equation in two space dimension:smoothness of the law[J].Ann Probab,1999,27(2):803-844.
[7]NUALART D,QUER-SARDANYONS L.Existence and smoothness of the density for spatially homogeneous SPDEs[J].Potential Anal,2007,27(3):281-299.
[8]NUALART D,QUER-SARDANYONS L.Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations[J].Stochastic Process Appl,2009,119(11):3914-3938.
[9]QUER-SARDANYONS L,SANZ-SOLE M.A stochastic wave equation in dimension 3:smoothness of the law[J].Bernoulli,2004,10(1):165-186.
[10]WALSH J B.An introduction to stochastic partial differential equations[M]//CARMONA R,KESTEN H,WALSH J B.Ecole d'Ete de Probabilites de Saint-Flour XIV-1984,Berlin:SpringerVerlag,1986:265-439.
[11]WU J L,XIE B.On a Burgers type nonlinear equation perturbed by a pure jump Levy noise in R~d[J].Bull Sci Math,2012,136(5):484-506.
[12]BASS R F.Uniqueness in law for pure jump Markov processes[J].Probab Theory Related Fields,1988,79(2):271-287.
[13]KOLOKOLTSOV V.Symmetric stable laws and stable-like jump-diffusions[J].Proc London Math Soc,2000,80(3):725-768.
[14]KOMATSU T.Markov processes associated with certain integro-differential operators[J].Osaka J Math,1973,10(2):271-303.
[15]KOMATSU T.Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms[J].Osaka J Math,1988,25(3):697-728.
[16]KOMATSU T.Uniform estimates for fundamental solutions associated with non-local Dirichlet forms[J].Osaka J Math,1995,32(4):833-860.
[17]NUALART D.The Malliavin Calculus and Related Topics[M].2nd ed.Berlin:Springer-Verlag,2006.
[18]HOH W.Pseudo differential operators with negative definite symbols of variable order[J].Rev Mat Iberoam,2000,16(2):219-241.
[19]JACOB N,LEOPOLD H G.Pseudo differential operators with variable order of differentiation generating feller semigroups[J].Integral Equations Operator Theory,1993,17(4):544-553.
[20]KIKUCHI K,NEGORO A.On Markov process generated by pseudodifferential operator of variable order[J].Osaka J Math,1997,34(2):319-335.