SOME RECENT PROGRESS ON STOCHASTIC HEAT EQUATIONS
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摘要
This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations(for example, stochastic heat equations)driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical(square integrable) solution(mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution; Feynman-Kac formula for the moments of the solution; and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.
This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations(for example, stochastic heat equations)driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical(square integrable) solution(mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution; Feynman-Kac formula for the moments of the solution; and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.
This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations(for example, stochastic heat equations)driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical(square integrable) solution(mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution; Feynman-Kac formula for the moments of the solution; and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.
引文
[1]Alberts T,Khanin K,Quastel J.The continuum directed random polymer.J Stat Phys,2014,154(1/2):305-326
[2]Amir G,Corwin I,Quastel J.Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions.Comm Pure Appl Math,2011,64(4):466-537
[3]Balan R,Chen L.Parabolic Anderson Model with space-time homogeneous Gaussian noise and rough initial condition.Journal of Theoretical Probability,2018,31:2216-2265
[4]Balan R,Jolis M,Quer-Sardanyons L.SPDEs with fractional noise in space with index H<1/2.Electron J Probab,2015,20(54):36 pp
[5]Balan R,Quer-Sardanyons L,Song J.H?lder continuity for the Parabolic Anderson Model with spacetime homogeneous Gaussian noise.Acta Mathematica Scientia,2019,39B(3):717-730.See also arXiv:1807.05420
[6]Bertini L,Cancrini N.The stochastic heat equation:Feynman-Kac formula and intermittence.J Statist Phys,1995,78(5/6):1377-1401
[7]Bezerra S,Tindel S,Viens F.Superdiffusivity for a Brownian polymer in a continuous Gaussian environment.Ann Probab,2008,36(5):1642-1675
[8]Biagini F,Hu Y,?ksendal B,Zhang T.Stochastic calculus for fractional Brownian motion and applications//Probability and its Applications(New York).London:Springer-Verlag London,Ltd,2008
[9]Carmona R,Lacroix J.Spectral Theory of Random Schr?dinger Operators//Probability and its Applications.Boston,MA:Birkh¨auser Boston,Inc,1990
[10]Carmona R A,Molchanov S A.Parabolic Anderson problem and intermittency.Mem Amer Math Soc,1994,108(518):viii+125
[11]Chen L,Dalang R C.Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions.Annals of Probability,2015,43:3006-3051
[12]Chen L,Dalang R C.H?lder-continuity for the nonlinear stochastic heat equation with rough initial conditions.Stoch Partial Differ Equ Anal Comput,2014,2(3):316-352
[13]Chen L,Hu G,Hu Y,Huang J.Space-time fractional diffusions in Gaussian noisy environment.Stochastics,2017,89(1):171-206
[14]Chen L,Hu Y,Kalbasi K,Nualart D.Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise.Probab Theory Related Fields,2018,171(1/2):431-457
[15]Chen L,Hu Y,Nualart D.Two-point correlation function and Feynman-Kac formula for the stochastic heat equation.Potential Anal,2017,46(4):779-797
[16]Chen L,Hu Y,Nualart D.Regularity and strict positivity of densities for the nonlinear stochastic heat equation.Memoirs of American Mathematical Society,2018(to appear).See also arXiv:1611.03909
[17]Chen L,Hu Y,Nualart D.Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd.Revised for Stochastic Processes and Appl
[18]Chen L,Huang J.Comparison principle for stochastic heat equation on Rd.Annals of Probability,2018,to appear
[19]Chen L,Kim K.Nonlinear stochastic heat equation driven by spatially colored noise:moments and intermittency.Acta Mathematica Scientia,2019,39B(3):645-668
[20]Chen X.Quenched asymptotics for Brownian motion in generalized Gaussian potential.Ann Probab,2014,42(2):576-622
[21]Chen X.Spatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise.Ann Probab,2016,44(2):1535-1598
[22]Chen X.Moment asymptotics for parabolic Anderson equation with fractional time-space noise:in Skorokhod regime.Ann Inst Henri Poincar Probab Stat,2017,53(2):819-841
[23]Chen X,Hu Y,Nualart D,Tindel S.Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise.Electron J Probab,2017,22(65):38 pp
[24]Chen X,Hu Y,Song J,Xing F.Exponential asymptotics for time-space Hamiltonians.Ann Inst Henri Poincar Probab Stat,2015,51(4):1529-1561
[25]Chen X,Phan T V.Free energy in a mean field of Brownian particles.Preprint
[26]Chernoff P R.Note on product formulas for operator semigroups.J Funct Anal,1968,2:238-242
[27]Chernoff P R.Product Formulas,Nonlinear Semigroups,and Addition of Unbounded Operators//Memoirs of the American Mathematical Society,No.140.Providence,RI:American Mathematical Society,1974
[28]Conus D,Joseph M,Khoshnevisan D,Shiu S-Y.Initial measures for the stochastic heat equation.Ann Inst Henri Poincar Probab Stat,2014,50(1):136-153
[29]Dalang R.Extending Martingale Measure Stochastic Integral with Applications to Spatially Homogeneous S.P.D.E’s.Electron J Probab,1999,4(6)
[30]Duncan T E,Hu Y,Pasik-Duncan B.Stochastic calculus for fractional Brownian motion.I.Theory.SIAMJ Control Optim,2000,38(2):582-612
[31]G¨artner J,Molchanov S A.Parabolic problems for the Anderson model.I.Intermittency and related topics.Comm Math Phys,1990,132(3):613-655
[32]G¨artner J,Molchanov S A.Parabolic problems for the Anderson model.II.Second-order asymptotics and structure of high peaks.Probab Theory Related Fields,1998,111(1):17-55
[33]Hairer M.Solving the KPZ equation.Ann of Math,2013,178(2):559-664
[34]Hille E,Phillips R S.Functional analysis and semi-groups.Third printing of the revised edition of1957//American Mathematical Society Colloquium Publications,Vol XXXI.Providence,RI:American Mathematical Society,1974
[35]Hu Y.Integral transformations and anticipative calculus for fractional Brownian motions.Mem Amer Math Soc,2005,175(825)
[36]Hu Y.Analysis on Gaussian space.Singapore:World Scientific,2017
[37]Hu Y.Heat equation with fractional white noise potentials.Appl Math Optim,2001,43:221-243
[38]Hu Y.Chaos expansion of heat equations with white noise potentials.Potential Anal,2002,16(1):45-66
[39]Hu Y.A class of SPDE driven by fractional white noise Leipzig.Stochastic processes,physics and geometry:new interplays,II.1999:317-325;CMS Conf Proc,29.Providence,RI:Amer Math Soc,2000
[40]Hu Y.Schr?dinger equation with Gaussian potential(To appear)
[41]Hu Y,Huang J,Nualart D,Tindel S.Stochastic heat equations with general multiplicative Gaussian noises:H?lder continuity and intermittency.Electron J Probab,2015,20(55)
[42]Hu Y,Huang J,Le K,Nualart D,Tindel S.Stochastic heat equation with rough dependence in space.Ann Probab,2017,45B(6):4561-4616
[43]Hu Y,Huang J,Le K,Nualart D,Tindel S.Parabolic Anderson model with rough dependence in space(To appear in Abel Proceedings)
[44]Hu Y,L?e K.A multiparameter Garsia-Rodemich-Rumsey inequality and some applications.Stochastic Process Appl,2013,123(9):3359-3377
[45]Hu Y,L?e K.Nonlinear Young integrals and differential systems in H?lder media.Trans Amer Math Soc,2017,369(3):1935-2002
[46]Hu Y,L?e K.Joint H?lder continuity of parabolic Anderson model.Acta Mathematics Scientia,2019,39B(3):764-780
[47]Hu Y,Liu Y,Tindel S.On the necessary and sufficient conditions to solve a heat equation with general Additive Gaussian noise.Acta Mathematics Scientia,2019,39B(3):669-690
[48]Hu Y,Lu F,Nualart D.Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H<1/2.Ann Probab,2012,40(3):1041-1068
[49]Hu Y,Nualart D.Stochastic heat equation driven by fractional noise and local time.Probab Theory Related Fields,2009,143(1/2):285-228
[50]Hu Y,Nualart D,Song J.Feynman-Kac formula for heat equation driven by fractional white noise.Ann.Probab,2011,39(1):291-326
[51]Hu Y,Nualart D,Zhang T.Large deviations for stochastic heat equation with rough dependence in space.Bernoulli,2018,24(1):354-385
[52]Hu Y,?ksendal B,Zhang T.General fractional multiparameter white noise theory and stochastic partial differential equations.Comm Partial Differential Equations,2004,29(1/2):123
[53]Hu Y,Yan J A.Wick calculus for nonlinear Gaussian functionals.Acta Math Appl Sin Engl Ser,2009,25(3):399-414
[54]Ikeda N,Watanabe S.Stochastic differential equations and diffusion processes.Second edition.NorthHolland Mathematical Library,24.Amsterdam:North-Holland Publishing Co;Tokyo:Kodansha,Ltd,1989
[55]Johnson G W,Lapidus M L.The Feynman integral and Feynman’s operational calculus.Oxford Mathematical Monographs.Oxford Science Publications.New York:The Clarendon Press,Oxford University Press,2000
[56]Kato T.Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups//Gohberg I,Kac M.Topics in Functional Analysis.London:Academic Press,1978
[57]Kilbas A A,Srivastava H M,Trujillo J J.Theory and applications of fractional differential equations.North-Holland Mathematics Studies,204.Amsterdam:Elsevier Science BV,2006
[58]Khoshnevisan D.Analysis of stochastic partial differential equations//CBMS Regional Conference Series in Mathematics,119.Published for the Conference Board of the Mathematical Sciences,Washington,DC.Providence,RI:the American Mathematical Society,2014:viii+116 pp
[59]K?nig W.The parabolic Anderson model.Random walk in random potential.Pathways in Mathematics.Birkh¨auser/Springer,[Cham],2016
[60]Memin J,Mishura Y,Valkeila E.Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion.Statist Probab Lett,2001,51:197-206
[61]Mueller C.Long-time existence for the heat equation with a noise term.Prob.Theory Rel Fields,1991,9:505-517
[62]Nualart D.The Malliavin calculus and related topics.Second edition.Probability and its Applications(New York).Berlin:Springer-Verlag,2006
[63]Peszat S,Zabczyk J.Stochastic evolution equations with a spatially homogeneous Wiener process.Stochastic Process Appl,1997,72(2):187-204
[64]Rhandi A.Dyson-Phillips expansion and unbounded perturbations of linear C0-semigroups.J Comput Appl Math,1992,44:339-349
[65]Trotter H F.On the product of semi-groups of operators.Proc Amer Math Soc,1959,10:545-551
[66]Vuillermot P-A.A generalization of Chernoff’s product formula for time-dependent operators.J Funct Anal,2010,259(11):2923-2938
[67]Vuillermot P-A,Wreszinski W F,Zagrebnov V A.A general Trotter-Kato formula for a class of evolution operators.J Funct Anal,2009,257:2246-2290
[68]Walsh B.An introduction to Stochastic Partial Differential Equations//Lecture Notes in Mathematics 1180.Springer-Verlag,1986:265-439
[69]Yosida K.Functional analysis.Reprint of the sixth edition(1980)//Classics in Mathematics.Berlin:Springer-Verlag,1995
[2]Amir G,Corwin I,Quastel J.Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions.Comm Pure Appl Math,2011,64(4):466-537
[3]Balan R,Chen L.Parabolic Anderson Model with space-time homogeneous Gaussian noise and rough initial condition.Journal of Theoretical Probability,2018,31:2216-2265
[4]Balan R,Jolis M,Quer-Sardanyons L.SPDEs with fractional noise in space with index H<1/2.Electron J Probab,2015,20(54):36 pp
[5]Balan R,Quer-Sardanyons L,Song J.H?lder continuity for the Parabolic Anderson Model with spacetime homogeneous Gaussian noise.Acta Mathematica Scientia,2019,39B(3):717-730.See also arXiv:1807.05420
[6]Bertini L,Cancrini N.The stochastic heat equation:Feynman-Kac formula and intermittence.J Statist Phys,1995,78(5/6):1377-1401
[7]Bezerra S,Tindel S,Viens F.Superdiffusivity for a Brownian polymer in a continuous Gaussian environment.Ann Probab,2008,36(5):1642-1675
[8]Biagini F,Hu Y,?ksendal B,Zhang T.Stochastic calculus for fractional Brownian motion and applications//Probability and its Applications(New York).London:Springer-Verlag London,Ltd,2008
[9]Carmona R,Lacroix J.Spectral Theory of Random Schr?dinger Operators//Probability and its Applications.Boston,MA:Birkh¨auser Boston,Inc,1990
[10]Carmona R A,Molchanov S A.Parabolic Anderson problem and intermittency.Mem Amer Math Soc,1994,108(518):viii+125
[11]Chen L,Dalang R C.Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions.Annals of Probability,2015,43:3006-3051
[12]Chen L,Dalang R C.H?lder-continuity for the nonlinear stochastic heat equation with rough initial conditions.Stoch Partial Differ Equ Anal Comput,2014,2(3):316-352
[13]Chen L,Hu G,Hu Y,Huang J.Space-time fractional diffusions in Gaussian noisy environment.Stochastics,2017,89(1):171-206
[14]Chen L,Hu Y,Kalbasi K,Nualart D.Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise.Probab Theory Related Fields,2018,171(1/2):431-457
[15]Chen L,Hu Y,Nualart D.Two-point correlation function and Feynman-Kac formula for the stochastic heat equation.Potential Anal,2017,46(4):779-797
[16]Chen L,Hu Y,Nualart D.Regularity and strict positivity of densities for the nonlinear stochastic heat equation.Memoirs of American Mathematical Society,2018(to appear).See also arXiv:1611.03909
[17]Chen L,Hu Y,Nualart D.Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd.Revised for Stochastic Processes and Appl
[18]Chen L,Huang J.Comparison principle for stochastic heat equation on Rd.Annals of Probability,2018,to appear
[19]Chen L,Kim K.Nonlinear stochastic heat equation driven by spatially colored noise:moments and intermittency.Acta Mathematica Scientia,2019,39B(3):645-668
[20]Chen X.Quenched asymptotics for Brownian motion in generalized Gaussian potential.Ann Probab,2014,42(2):576-622
[21]Chen X.Spatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise.Ann Probab,2016,44(2):1535-1598
[22]Chen X.Moment asymptotics for parabolic Anderson equation with fractional time-space noise:in Skorokhod regime.Ann Inst Henri Poincar Probab Stat,2017,53(2):819-841
[23]Chen X,Hu Y,Nualart D,Tindel S.Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise.Electron J Probab,2017,22(65):38 pp
[24]Chen X,Hu Y,Song J,Xing F.Exponential asymptotics for time-space Hamiltonians.Ann Inst Henri Poincar Probab Stat,2015,51(4):1529-1561
[25]Chen X,Phan T V.Free energy in a mean field of Brownian particles.Preprint
[26]Chernoff P R.Note on product formulas for operator semigroups.J Funct Anal,1968,2:238-242
[27]Chernoff P R.Product Formulas,Nonlinear Semigroups,and Addition of Unbounded Operators//Memoirs of the American Mathematical Society,No.140.Providence,RI:American Mathematical Society,1974
[28]Conus D,Joseph M,Khoshnevisan D,Shiu S-Y.Initial measures for the stochastic heat equation.Ann Inst Henri Poincar Probab Stat,2014,50(1):136-153
[29]Dalang R.Extending Martingale Measure Stochastic Integral with Applications to Spatially Homogeneous S.P.D.E’s.Electron J Probab,1999,4(6)
[30]Duncan T E,Hu Y,Pasik-Duncan B.Stochastic calculus for fractional Brownian motion.I.Theory.SIAMJ Control Optim,2000,38(2):582-612
[31]G¨artner J,Molchanov S A.Parabolic problems for the Anderson model.I.Intermittency and related topics.Comm Math Phys,1990,132(3):613-655
[32]G¨artner J,Molchanov S A.Parabolic problems for the Anderson model.II.Second-order asymptotics and structure of high peaks.Probab Theory Related Fields,1998,111(1):17-55
[33]Hairer M.Solving the KPZ equation.Ann of Math,2013,178(2):559-664
[34]Hille E,Phillips R S.Functional analysis and semi-groups.Third printing of the revised edition of1957//American Mathematical Society Colloquium Publications,Vol XXXI.Providence,RI:American Mathematical Society,1974
[35]Hu Y.Integral transformations and anticipative calculus for fractional Brownian motions.Mem Amer Math Soc,2005,175(825)
[36]Hu Y.Analysis on Gaussian space.Singapore:World Scientific,2017
[37]Hu Y.Heat equation with fractional white noise potentials.Appl Math Optim,2001,43:221-243
[38]Hu Y.Chaos expansion of heat equations with white noise potentials.Potential Anal,2002,16(1):45-66
[39]Hu Y.A class of SPDE driven by fractional white noise Leipzig.Stochastic processes,physics and geometry:new interplays,II.1999:317-325;CMS Conf Proc,29.Providence,RI:Amer Math Soc,2000
[40]Hu Y.Schr?dinger equation with Gaussian potential(To appear)
[41]Hu Y,Huang J,Nualart D,Tindel S.Stochastic heat equations with general multiplicative Gaussian noises:H?lder continuity and intermittency.Electron J Probab,2015,20(55)
[42]Hu Y,Huang J,Le K,Nualart D,Tindel S.Stochastic heat equation with rough dependence in space.Ann Probab,2017,45B(6):4561-4616
[43]Hu Y,Huang J,Le K,Nualart D,Tindel S.Parabolic Anderson model with rough dependence in space(To appear in Abel Proceedings)
[44]Hu Y,L?e K.A multiparameter Garsia-Rodemich-Rumsey inequality and some applications.Stochastic Process Appl,2013,123(9):3359-3377
[45]Hu Y,L?e K.Nonlinear Young integrals and differential systems in H?lder media.Trans Amer Math Soc,2017,369(3):1935-2002
[46]Hu Y,L?e K.Joint H?lder continuity of parabolic Anderson model.Acta Mathematics Scientia,2019,39B(3):764-780
[47]Hu Y,Liu Y,Tindel S.On the necessary and sufficient conditions to solve a heat equation with general Additive Gaussian noise.Acta Mathematics Scientia,2019,39B(3):669-690
[48]Hu Y,Lu F,Nualart D.Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H<1/2.Ann Probab,2012,40(3):1041-1068
[49]Hu Y,Nualart D.Stochastic heat equation driven by fractional noise and local time.Probab Theory Related Fields,2009,143(1/2):285-228
[50]Hu Y,Nualart D,Song J.Feynman-Kac formula for heat equation driven by fractional white noise.Ann.Probab,2011,39(1):291-326
[51]Hu Y,Nualart D,Zhang T.Large deviations for stochastic heat equation with rough dependence in space.Bernoulli,2018,24(1):354-385
[52]Hu Y,?ksendal B,Zhang T.General fractional multiparameter white noise theory and stochastic partial differential equations.Comm Partial Differential Equations,2004,29(1/2):123
[53]Hu Y,Yan J A.Wick calculus for nonlinear Gaussian functionals.Acta Math Appl Sin Engl Ser,2009,25(3):399-414
[54]Ikeda N,Watanabe S.Stochastic differential equations and diffusion processes.Second edition.NorthHolland Mathematical Library,24.Amsterdam:North-Holland Publishing Co;Tokyo:Kodansha,Ltd,1989
[55]Johnson G W,Lapidus M L.The Feynman integral and Feynman’s operational calculus.Oxford Mathematical Monographs.Oxford Science Publications.New York:The Clarendon Press,Oxford University Press,2000
[56]Kato T.Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups//Gohberg I,Kac M.Topics in Functional Analysis.London:Academic Press,1978
[57]Kilbas A A,Srivastava H M,Trujillo J J.Theory and applications of fractional differential equations.North-Holland Mathematics Studies,204.Amsterdam:Elsevier Science BV,2006
[58]Khoshnevisan D.Analysis of stochastic partial differential equations//CBMS Regional Conference Series in Mathematics,119.Published for the Conference Board of the Mathematical Sciences,Washington,DC.Providence,RI:the American Mathematical Society,2014:viii+116 pp
[59]K?nig W.The parabolic Anderson model.Random walk in random potential.Pathways in Mathematics.Birkh¨auser/Springer,[Cham],2016
[60]Memin J,Mishura Y,Valkeila E.Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion.Statist Probab Lett,2001,51:197-206
[61]Mueller C.Long-time existence for the heat equation with a noise term.Prob.Theory Rel Fields,1991,9:505-517
[62]Nualart D.The Malliavin calculus and related topics.Second edition.Probability and its Applications(New York).Berlin:Springer-Verlag,2006
[63]Peszat S,Zabczyk J.Stochastic evolution equations with a spatially homogeneous Wiener process.Stochastic Process Appl,1997,72(2):187-204
[64]Rhandi A.Dyson-Phillips expansion and unbounded perturbations of linear C0-semigroups.J Comput Appl Math,1992,44:339-349
[65]Trotter H F.On the product of semi-groups of operators.Proc Amer Math Soc,1959,10:545-551
[66]Vuillermot P-A.A generalization of Chernoff’s product formula for time-dependent operators.J Funct Anal,2010,259(11):2923-2938
[67]Vuillermot P-A,Wreszinski W F,Zagrebnov V A.A general Trotter-Kato formula for a class of evolution operators.J Funct Anal,2009,257:2246-2290
[68]Walsh B.An introduction to Stochastic Partial Differential Equations//Lecture Notes in Mathematics 1180.Springer-Verlag,1986:265-439
[69]Yosida K.Functional analysis.Reprint of the sixth edition(1980)//Classics in Mathematics.Berlin:Springer-Verlag,1995