关于最近无穷维随机泛函微分方程研究的一些看法
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摘要
抽象或无穷维随机泛函微分方程是一个相当年幼的学科.本文基于作者近几年对该学科在学习和研究时所获得的一些经验和体会,对问题、方法、技巧、难点和将来工作中要注意的诸多方面提出自己的看法.本文的着眼点主要集中在分析和比较此学科的特点、问题的难点和在处理这些难点时经常使用的方法和工具,同时指出在使用这些方法时应该特别注意的方面.主要的目的是希望通过本文能吸引更多的人进入这个领域,抛砖引玉,从而推动这门学科的发展.
As a relatively new area in mathematics, the subject of abstract or infinite dimensional stochastic functional differential equations is still in its tender age at present. In this note, the author intends to show some of his personal views, mainly based on his own academic experience in this topic, about the current art of research. Special attention is paid to a statement of various methods, techniques and ideas in the undergoing development. What I am trying to do is to highlight the coherent characterization of various difficult problems,effective methods and useful toods frequently seen in this subject. A mild notice is also made on some possible pitfalls which an active researcher working in this area should take into account. It is hoped that, through this short work, some more valuable developments can be promoted and significant progress can be expected in the future.
As a relatively new area in mathematics, the subject of abstract or infinite dimensional stochastic functional differential equations is still in its tender age at present. In this note, the author intends to show some of his personal views, mainly based on his own academic experience in this topic, about the current art of research. Special attention is paid to a statement of various methods, techniques and ideas in the undergoing development. What I am trying to do is to highlight the coherent characterization of various difficult problems,effective methods and useful toods frequently seen in this subject. A mild notice is also made on some possible pitfalls which an active researcher working in this area should take into account. It is hoped that, through this short work, some more valuable developments can be promoted and significant progress can be expected in the future.
引文
1 Volterra V.Sulle equazioni integro differenziali della teoria dell’elasticit′a.Rend Accad Lincei,1909,18:295–301
2 Volterra V.Lecons sur les′equations int′egrales et les′equations int′egro-differentielles.Paris:Gauthier Villars,1913
3 Friedman A,Shinbrot M.Volterra integral equations in Banach spaces.Trans Amer Math Soc,1967,126:131–179
4 Real J.Stochastic partial differential equations with delays.Stochastics,1982,8:81–102
5 Barbu V,Bonaccorsi S,Tubaro L.Existence and asymptotic behavior for hereditary stochastic evolution equations.Appl Math Optim,2014,69:273–314
6 Bonaccorsi S,Fantozzi M.Infinite dimensional stochastic Volterra equations with dissipative nonlinearity.Dyn Syst Appl,2006,15:465–478
7 Caraballo T,Liu K,Truman A.Stochastic functional partial differential equations:Existence,uniqueness and asymptotic stability.Proc R Soc Lond Ser A,2000,456:1775–1802
8 Cox S,G′orajski M.Vector-valued stochastic delay equations—a semigroup approach.Semigroup Forum,2011,9:389 –411
9 Liu K.Stochastic retarded evolution equations:Green operators,convolutions and solutions.Stoch Anal Appl,2008,26 :624–650
10 Taniguchi T,Liu K,Truman A.Existence,uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces.J Differential Equations,2002,181:72–91
11 Zhang X C.Stochastic Volterra equations in Banach spaces and stochastic partial differential equation.J Funct Anal,2010,258:1361–1425
12 Bonaccorsi S,Tubaro L.Mittlag-Leffler function and stochastic linear Volterra equations of convolution type.Stoch Anal Appl,2003,20:61–78
13 Chow P,Liu K.Positivity and explosion of stochastic functional parabolic equations of retarded type.Stochastic Process Appl,2012,122:1709–729
14 Cl′ement P,Da Prato G.Some results on stochastic convolution arising in Volterra equations perturbed by noise.Rend Mat Accad Lincei,1996,3:147–153
15 Liu K.Regularity of stochastic linear retarded functional differential equations in Hilbert spaces.J Theoret Probab,2012,25:565–593
16 Bao J H,Truman A,Yuan C G.Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps.Proc R Soc Lond Ser A,2012,465:2111–2134
17 Caraballo T.Asymptotic exponential stability of stochastic partial differential equations with delay.Stochastics,1990,33 :27–47
18 Caraballo T,Liu K.Exponential stability of mild solutions of stochastic partial differential equations with delays.Stoch Anal Appl,1999,17:743–764
19 Jahanipur R.Stability of stochastic delay evolution equations with monotone nonlinearity.Stoch Anal Appl,2003,7:161 –181
20 Luo J W,Liu K.Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps.Stochastic Process Appl,2008,118:864–895
21 Liu K.On stability for a class of semilinear stochastic evolution equations.Stochastic Process Appl,1997,70:219–241
22 Luo J W.Fixed points and stability of neutral stochastic delay differential equations.J Math Anal Appl,2007,334:431 –440
23 Wan L,Duan J Q.Exponential stability of non-autonomous stochastic partial differential equations with finite memory.Statist Probab Lett,2008,78:490–498
24 Bierkens J.Pathwise stability of degenerate stochastic evolutions.Integral Equations Operator Theory,2010,23:1–27
25 Caraballo T,Robinson J C.Stabilisation of linear PDEs by Stratonovich noise.Systems Control Lett,2004,53:41–50
26 Dauer J,Mahmudov N.Controllability of stochastic semilinear functional differential equations in Hilbert spaces.J Math Anal Appl,2004,10:373–394
27 Liang F,Guo Z H.Asymptotic behavior for second order order stochastic evolution equations with memory.J Math Anal Appl,2014,115:1333–1350
28 Liu K.Necessary and sufficient conditions for exponential stability and ultimate boundedness of systems governed by stochastic partial differential equations.J Lond Math Soc,2000,62:311–320
29 Liu K.Stationary solutions of retarded Ornstein-Uhlenbeck processes in Hilbert spaces.Statist Probab Lett,2008,78:1775–1783
30 Liu K.Retarded stationary Ornstein-Uhlenbeck processes driven by L′evy noise and operator self-decomposability.Potential Anal,2010,33:291–312
31 Bierkens J,van Ganns O.Dissipativity of the delay semigroup.J Differential Equations,2014,257:2418–2429
32 Bierkens J,van Ganns O,Lunel S.Existence of an invariant measure for stochastic evolutions driven by an eventually compact semigroup.J Evol Equ,2009,9:771–786
33 Liu K.Existence of invariant measures of stochastic systems with delay in the highest order partial derivatives.Statist Probab Lett,2014,94:267–272
34 Bonaccorsi S,Fantozzi M.Large deviation principle for semilinear stochastic Volterra equations.Dyn Syst Appl,2004,9 :203–220
35 Liu K,Zhang T S.A large deviation principle of retarded Ornstein-Uhlenbeck processes driven by Levy noise.Stoch Anal Appl,2014,32:889–910
36 Flandoli F.Solution and control of a bilinear stochastic delay equation.SIAM J Control Optim,1990,28:936–949
37 Hale J K.Theory of Functional Differential Equations.New York:Springer-Verlag,1977
2 Volterra V.Lecons sur les′equations int′egrales et les′equations int′egro-differentielles.Paris:Gauthier Villars,1913
3 Friedman A,Shinbrot M.Volterra integral equations in Banach spaces.Trans Amer Math Soc,1967,126:131–179
4 Real J.Stochastic partial differential equations with delays.Stochastics,1982,8:81–102
5 Barbu V,Bonaccorsi S,Tubaro L.Existence and asymptotic behavior for hereditary stochastic evolution equations.Appl Math Optim,2014,69:273–314
6 Bonaccorsi S,Fantozzi M.Infinite dimensional stochastic Volterra equations with dissipative nonlinearity.Dyn Syst Appl,2006,15:465–478
7 Caraballo T,Liu K,Truman A.Stochastic functional partial differential equations:Existence,uniqueness and asymptotic stability.Proc R Soc Lond Ser A,2000,456:1775–1802
8 Cox S,G′orajski M.Vector-valued stochastic delay equations—a semigroup approach.Semigroup Forum,2011,9:389 –411
9 Liu K.Stochastic retarded evolution equations:Green operators,convolutions and solutions.Stoch Anal Appl,2008,26 :624–650
10 Taniguchi T,Liu K,Truman A.Existence,uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces.J Differential Equations,2002,181:72–91
11 Zhang X C.Stochastic Volterra equations in Banach spaces and stochastic partial differential equation.J Funct Anal,2010,258:1361–1425
12 Bonaccorsi S,Tubaro L.Mittlag-Leffler function and stochastic linear Volterra equations of convolution type.Stoch Anal Appl,2003,20:61–78
13 Chow P,Liu K.Positivity and explosion of stochastic functional parabolic equations of retarded type.Stochastic Process Appl,2012,122:1709–729
14 Cl′ement P,Da Prato G.Some results on stochastic convolution arising in Volterra equations perturbed by noise.Rend Mat Accad Lincei,1996,3:147–153
15 Liu K.Regularity of stochastic linear retarded functional differential equations in Hilbert spaces.J Theoret Probab,2012,25:565–593
16 Bao J H,Truman A,Yuan C G.Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps.Proc R Soc Lond Ser A,2012,465:2111–2134
17 Caraballo T.Asymptotic exponential stability of stochastic partial differential equations with delay.Stochastics,1990,33 :27–47
18 Caraballo T,Liu K.Exponential stability of mild solutions of stochastic partial differential equations with delays.Stoch Anal Appl,1999,17:743–764
19 Jahanipur R.Stability of stochastic delay evolution equations with monotone nonlinearity.Stoch Anal Appl,2003,7:161 –181
20 Luo J W,Liu K.Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps.Stochastic Process Appl,2008,118:864–895
21 Liu K.On stability for a class of semilinear stochastic evolution equations.Stochastic Process Appl,1997,70:219–241
22 Luo J W.Fixed points and stability of neutral stochastic delay differential equations.J Math Anal Appl,2007,334:431 –440
23 Wan L,Duan J Q.Exponential stability of non-autonomous stochastic partial differential equations with finite memory.Statist Probab Lett,2008,78:490–498
24 Bierkens J.Pathwise stability of degenerate stochastic evolutions.Integral Equations Operator Theory,2010,23:1–27
25 Caraballo T,Robinson J C.Stabilisation of linear PDEs by Stratonovich noise.Systems Control Lett,2004,53:41–50
26 Dauer J,Mahmudov N.Controllability of stochastic semilinear functional differential equations in Hilbert spaces.J Math Anal Appl,2004,10:373–394
27 Liang F,Guo Z H.Asymptotic behavior for second order order stochastic evolution equations with memory.J Math Anal Appl,2014,115:1333–1350
28 Liu K.Necessary and sufficient conditions for exponential stability and ultimate boundedness of systems governed by stochastic partial differential equations.J Lond Math Soc,2000,62:311–320
29 Liu K.Stationary solutions of retarded Ornstein-Uhlenbeck processes in Hilbert spaces.Statist Probab Lett,2008,78:1775–1783
30 Liu K.Retarded stationary Ornstein-Uhlenbeck processes driven by L′evy noise and operator self-decomposability.Potential Anal,2010,33:291–312
31 Bierkens J,van Ganns O.Dissipativity of the delay semigroup.J Differential Equations,2014,257:2418–2429
32 Bierkens J,van Ganns O,Lunel S.Existence of an invariant measure for stochastic evolutions driven by an eventually compact semigroup.J Evol Equ,2009,9:771–786
33 Liu K.Existence of invariant measures of stochastic systems with delay in the highest order partial derivatives.Statist Probab Lett,2014,94:267–272
34 Bonaccorsi S,Fantozzi M.Large deviation principle for semilinear stochastic Volterra equations.Dyn Syst Appl,2004,9 :203–220
35 Liu K,Zhang T S.A large deviation principle of retarded Ornstein-Uhlenbeck processes driven by Levy noise.Stoch Anal Appl,2014,32:889–910
36 Flandoli F.Solution and control of a bilinear stochastic delay equation.SIAM J Control Optim,1990,28:936–949
37 Hale J K.Theory of Functional Differential Equations.New York:Springer-Verlag,1977