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March  2019, 18(2): 795-807. doi: 10.3934/cpaa.2019038

Stochastic parabolic Anderson model with time-homogeneous generalized potential: Mild formulation of solution

Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA

Received  February 2018 Revised  June 2018 Published  October 2018

A mild formulation for stochastic parabolic Anderson model with time-homogeneous Gaussian potential suggests a way of defining a solution to obtain its optimal regularity. Two different interpretations in the equation or in the mild formulation are possible with usual pathwise product and the Wick product: the usual pathwise interpretation is mainly discussed. We emphasize that a modified version of parabolic Schauder estimates is a key idea for the existence and uniqueness of a mild solution. In particular, the mild formulation is crucial to investigate a relation between the equations with usual pathwise product and the Wick product.

Citation: Hyun-Jung Kim. Stochastic parabolic Anderson model with time-homogeneous generalized potential: Mild formulation of solution. Communications on Pure and Applied Analysis, 2019, 18 (2) : 795-807. doi: 10.3934/cpaa.2019038
References:
[1]

R. H. Cameron and W. T. Martin, The orthogonal development of nonlinear functionals in a series of Fourier-Hermite functions, Ann. Math., 48 (1947), 385-392.  doi: 10.2307/1969178.

[2]

T. Coulhon and X. T. Duong, Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüs, Adv. Diff. Eq., 5 (2000), 343-368. 

[3]

M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs preprint, arXiv: 1210.2684v4. doi: 10.1017/fmp.2015.2.

[4]

M. Hairer, A theory of regularity structures, Invent. Math., 198 (2014), 269-504.  doi: 10.1007/s00222-014-0505-4.

[5]

M. Hairer and $\acute{E}.$ Pardoux, A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Japan, 67 (2015), 1551-1604.  doi: 10.2969/jmsj/06741551.

[6]

M. Hairer and C. Labb$\acute{e}$, A simple construction of the continuum parabolic Anderson model on $ \mathbb{R}^2 $, Electron. Commun. Probab., 20 (2015), 1-11.  doi: 10.1214/ECP.v20-4038.

[7]

M. Hairer and C. Labbé, Multiplicative stochastic heat equations on the whole space preprint, arXiv: 1504.07162v2. doi: 10.4171/JEMS/781.

[8]

Y. Hu, Chaos expansion of heat equations with white noise potentials, Potential Anal., 16 (2002), 45-66.  doi: 10.1023/A:1024878703232.

[9]

Y. Hu, J. Huang, D. Nualart and S. Tindel, Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency, Electron. J. Probab., 20 (2015), 50pp. doi: 10.1214/EJP.v20-3316.

[10]

H.-J. Kim and S. V. Lototsky, Time-homogeneous parabolic Wick-Anderson model in one space dimension: regularity of solution, Stochastics and Partial Differential Equations: Analysis and Computations, (2017), 1-33.  doi: 10.1007/s40072-017-0097-2.

[11]

H.-J. Kim and S. V. Lototsky, Heat equation with a geometric rough path potential in one space dimension: existence and regularity of solution preprint, arXiv: 1712.08196. doi: 10.1007/s40072-017-0097-2.

[12]

N. V. Krylov, An Analytic Approach to SPDEs, AMS, 1999. doi: 10.1090/surv/064/05.

[13]

W. Luo, Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations, Ph.D thesis, California Institute of Technology, 2006.

[14]

R. Mikulevicius and B. L. Rozovskii, On unbiased stochastic Navier-Stokes equations, Probab. Theory Related Fields, 154 (2012), 787-834.  doi: 10.1007/s00440-011-0384-1.

[15]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, R.I., 23 1968.

[16]

H. Uemura, Construction of the solution of 1-dimensional heat equation with white noise potential and its asymptotic behavior, Stochastic Anal. Appl., 14 (1996), 487-506.  doi: 10.1080/07362999608809452.

show all references

References:
[1]

R. H. Cameron and W. T. Martin, The orthogonal development of nonlinear functionals in a series of Fourier-Hermite functions, Ann. Math., 48 (1947), 385-392.  doi: 10.2307/1969178.

[2]

T. Coulhon and X. T. Duong, Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüs, Adv. Diff. Eq., 5 (2000), 343-368. 

[3]

M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs preprint, arXiv: 1210.2684v4. doi: 10.1017/fmp.2015.2.

[4]

M. Hairer, A theory of regularity structures, Invent. Math., 198 (2014), 269-504.  doi: 10.1007/s00222-014-0505-4.

[5]

M. Hairer and $\acute{E}.$ Pardoux, A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Japan, 67 (2015), 1551-1604.  doi: 10.2969/jmsj/06741551.

[6]

M. Hairer and C. Labb$\acute{e}$, A simple construction of the continuum parabolic Anderson model on $ \mathbb{R}^2 $, Electron. Commun. Probab., 20 (2015), 1-11.  doi: 10.1214/ECP.v20-4038.

[7]

M. Hairer and C. Labbé, Multiplicative stochastic heat equations on the whole space preprint, arXiv: 1504.07162v2. doi: 10.4171/JEMS/781.

[8]

Y. Hu, Chaos expansion of heat equations with white noise potentials, Potential Anal., 16 (2002), 45-66.  doi: 10.1023/A:1024878703232.

[9]

Y. Hu, J. Huang, D. Nualart and S. Tindel, Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency, Electron. J. Probab., 20 (2015), 50pp. doi: 10.1214/EJP.v20-3316.

[10]

H.-J. Kim and S. V. Lototsky, Time-homogeneous parabolic Wick-Anderson model in one space dimension: regularity of solution, Stochastics and Partial Differential Equations: Analysis and Computations, (2017), 1-33.  doi: 10.1007/s40072-017-0097-2.

[11]

H.-J. Kim and S. V. Lototsky, Heat equation with a geometric rough path potential in one space dimension: existence and regularity of solution preprint, arXiv: 1712.08196. doi: 10.1007/s40072-017-0097-2.

[12]

N. V. Krylov, An Analytic Approach to SPDEs, AMS, 1999. doi: 10.1090/surv/064/05.

[13]

W. Luo, Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations, Ph.D thesis, California Institute of Technology, 2006.

[14]

R. Mikulevicius and B. L. Rozovskii, On unbiased stochastic Navier-Stokes equations, Probab. Theory Related Fields, 154 (2012), 787-834.  doi: 10.1007/s00440-011-0384-1.

[15]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, R.I., 23 1968.

[16]

H. Uemura, Construction of the solution of 1-dimensional heat equation with white noise potential and its asymptotic behavior, Stochastic Anal. Appl., 14 (1996), 487-506.  doi: 10.1080/07362999608809452.

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