ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

doi:10.1007/s10473-019-0304-5

Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (3): 669-690.doi: 10.1007/s10473-019-0304-5

• Articles • Previous Articles Next Articles

ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

Yaozhong HU¹, Yanghui LIU², Samy TINDEL²

1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada;
2. Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Received:2018-03-22 Revised:2018-12-04 Online:2019-06-25 Published:2019-06-27
Supported by:
Y. Hu is supported by an NSERC grant and a startup fund of University of Alberta; S. Tindel is supported by the NSF grant DMS1613163.

Abstract

Abstract: In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two different methods, respectively, based on variance computations and on path-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.

Key words: Stochastic heat equation, general Gaussian noise, L² solution, sufficient and necessary condition, Wong-Zakai approximation, pathwise solution, Hölder continuity, Besov space

CLC Number:

60G15

Yaozhong HU, Yanghui LIU, Samy TINDEL. ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE[J].Acta mathematica scientia,Series B, 2019, 39(3): 669-690.

Trendmd

References

[1] Bahouri H, Chemin J, Danchin R. Fourier Analysis and Nonlinear Partial Differential Equations. Springer, 2011
[2] Balan R M, Tudor C A. The stochastic heat equation with fractional-colored noise:existence of the solution. Latin Amer J Probab Math Stat, 2008, 4:57-87
[3] Chen X. Spatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise. Ann Probab, 2016, 44(2):1535-1598
[4] Chen X. Parabolic Anderson model with rough or critical Gaussian noise. 2018, preprint, 43pp
[5] Da Prato G, Zabczyk J. Stochastic equations in infinite dimensions//Encyclopedia of Mathematics and its Applications, 44. Cambridge:Cambridge University Press, 1992:xviii+454 pp
[6] Dalang R. Extending Martingale Measure Stochastic Integral with Applications to Spatially Homogeneous S.P.D.E's. Electron J Probab, 1999, 4(6):29pp
[7] Deya A. On a modeled rough heat equation. Probab Theory Related Fields, 2016, 166(1/2):1-65
[8] Foondun M, Khoshnevisan D. Intermittence and nonlinear parabolic stochastic partial differential equations. Electron J Probab, 2009, 14(21):548-568
[9] Garsia A M, Rodemich E, Rumsey H. A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ Math J, 1970, 20:565-578
[10] Gradinaru M, Nourdin I, Russo F, Vallois P. m-order integrals and generalized Itô's formula:the case of a fractional Brownian motion with Hurst index. Ann Inst H Poincaré Probab Statist, 2005, 41(4):781-806
[11] Hairer M. Solving the KPZ equation. Ann of Math, 2011, 178(2):559-664
[12] Hairer, M. A theory of regularity structures. Invent Math, 2014, 198(2):269-504
[13] Hu Y, Huang J, Nualart D, Tindel S. Stochastic heat equations with general multiplicative Gaussian noises:Hölder continuity and intermittency. Electron J Probab, 2015, 20(55):50pp
[14] Hu Y, Huang J, Lê K, Nualart D, Tindel S. Stochastic heat equation with rough dependence in space. Ann Probab, 2017, 45(6B):4561-4616
[15] 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
[16] Hu Y, Nualart D. Stochastic heat equation driven by fractional noise and local time. Probab Theory Related Fields, 2009, 143(1/2):285-328
[17] Hu Y, Nualart D, Song J. Feynman-Kac formula for heat equation driven by fractional white noise. Ann Probab, 2011, 39(1):291-326
[18] Mourrat J, Weber H. Global well-posedness of the dynamic Φ⁴ model in the plane. Ann Probab, 2017, 45(4):2398-2476
[19] Nualart D. The Malliavin Calculus and Related Topics. Second Edition. Probability and its Applications. Springer-Verlag, 2006
[20] Peszat S, Zabczyk J. Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Process Appl, 1997, 72(2):187-204
[21] Russo F, Vallois P. Forward, backward and symmetric stochastic integration. Probab Theory Related Fields, 1993, 97(3):403-421
[22] Russo F, Vallois P. Elements of stochastic calculus via regularization. Séminaire de Probabilités XL, 147- 185, Lecture Notes in Math, 1899. Springer, 2007
[23] Tudor C, Xiao Y. Sample paths of the solution to the fractional-colored stochastic heat equation. Stoch Dyn, 2017, (1):20pp

ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 9

Metrics

Comments

Recommended 0

[1]	Yaozhong HU. SOME RECENT PROGRESS ON STOCHASTIC HEAT EQUATIONS [J]. Acta mathematica scientia,Series B, 2019, 39(3): 874-914.
[2]	Yaozhong HU, Khoa LÊ. JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL [J]. Acta mathematica scientia,Series B, 2019, 39(3): 764-780.
[3]	Le CHEN, Kunwoo KIM. NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY [J]. Acta mathematica scientia,Series B, 2019, 39(3): 645-668.
[4]	Heyu LI, Xia CHEN. PRECISE MOMENT ASYMPTOTICS FOR THE STOCHASTIC HEAT EQUATION OF A TIME-DERIVATIVE GAUSSIAN NOISE [J]. Acta mathematica scientia,Series B, 2019, 39(3): 629-644.
[5]	Shuhong CHEN, Boling GUO. CLASSICAL SOLUTIONS OF GENERAL GINZBURG-LANDAU EQUATIONS [J]. Acta mathematica scientia,Series B, 2016, 36(3): 717-732.
[6]	Qixiang YANG, Yong DING. FAST ALGORITHM FOR CALDERÓN-ZYGMUND OPERATORS: CONVERGENCE SPEED AND ROUGH KERNEL [J]. Acta mathematica scientia,Series B, 2016, 36(2): 345-358.
[7]	BIAN Dong-Fen, GUO Bo-Ling. WELL-POSEDNESS IN CRITICAL SPACES FOR THE FULL COMPRESSIBLE MHD EQUATIONS [J]. Acta mathematica scientia,Series B, 2013, 33(4): 1153-1176.
[8]	YANG Zhan-Ying. BOUNDEDNESS OF CALDERÓN-ZYGMUND OPERATORS ON BESOV SPACES AND ITS APPLICATION [J]. Acta mathematica scientia,Series B, 2010, 30(4): 1338-1346.
[9]	Peng Lizhong, Yang Qixiang. PREDUAL SPACES FOR Q SPACES [J]. Acta mathematica scientia,Series B, 2009, 29(2): 243-250.