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Spatial asymptotic of the stochastic heat equation with compactly supported initial data. (English) Zbl 1431.60057

Summary: We investigate the growth of the tallest peaks of random field solutions to the parabolic Anderson models over concentric balls as the radii approach infinity. The noise is white in time and correlated in space. The spatial correlation function is either bounded or non-negative satisfying Dalang’s condition. The initial data are Borel measures with compact supports, in particular, include Dirac masses. The results obtained are related to those of D. Conus et al. [Ann. Probab. 41, No. 3B, 2225–2260 (2013; Zbl 1286.60060)] and X. Chen [Ann. Probab. 44, No. 2, 1535–1598 (2016; Zbl 1348.60092)] where constant initial data are considered.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
60F10 Large deviations
60G60 Random fields
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References:

[1] Chen, L., Huang, J.: Comparison principle for stochastic heat equation on \[\mathbb{R}^d\] Rd. Ann. Probab. (to appear) · Zbl 1433.60049
[2] Chen, X.: Quenched asymptotics for Brownian motion in generalized Gaussian potential. Ann. Probab. 42(2), 576-622 (2014) · Zbl 1294.60101 · doi:10.1214/12-AOP830
[3] Chen, X.: Spatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise. Ann. Probab. 44(2), 1535-1598 (2016) · Zbl 1348.60092 · doi:10.1214/15-AOP1006
[4] Chen, L., Hu, Y., Nualart, D.: Two-point correlation function and Feynman-Kac formula for the stochastic heat equation. Potential Anal. 46(4), 779-797 (2017) · Zbl 1367.60079 · doi:10.1007/s11118-016-9601-y
[5] Chen, X., Hu, Y., Nualart, D., Tindel, S.: Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise. Electron. J. Probab. 22, Paper No. 65, 38 (2017) · Zbl 1386.60135
[6] Chen, X., Hu, Y., Song, J., Xing, F.: Exponential asymptotics for time-space Hamiltonians. Ann. Inst. Henri Poincar’e Probab. Stat. 51(4), 1529-1561 (2015) · Zbl 1337.60201 · doi:10.1214/13-AIHP588
[7] Conus, D., Joseph, M., Khoshnevisan, D.: On the chaotic character of the stochastic heat equation, before the onset of intermitttency. Ann. Probab. 41(3B), 2225-2260 (2013) · Zbl 1286.60060 · doi:10.1214/11-AOP717
[8] Chen, X., V Phan, T.: Free energy in a mean field of Brownian particles (preprint) · Zbl 1404.60116
[9] Dalang, R.C.: Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4(6), 29 (1999). (electronic) · Zbl 0922.60056
[10] Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992) · Zbl 0761.60052 · doi:10.1017/CBO9780511666223
[11] Dalang, R.C., Quer-Sardanyons, L.: Stochastic integrals for spde’s: a comparison. Expos. Math. 29(1), 67-109 (2011) · Zbl 1234.60064 · doi:10.1016/j.exmath.2010.09.005
[12] Erd’elyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. III. Robert E. Krieger Publishing Co. Inc, Fla., Melbourne (1981). Based on notes left by Harry Bateman, Reprint of the 1955 original · Zbl 0542.33001
[13] Garsia, A.M., Rodemich, E., Rumsey Jr., H.: A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20, 565-578 (1970/1971) · Zbl 0252.60020
[14] Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269-504 (2014) · Zbl 1332.60093 · doi:10.1007/s00222-014-0505-4
[15] Hu, Y., Huang, J., Nualart, D., Tindel, S.: Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electron. J. Probab. 20(55), 50 (2015) · Zbl 1322.60113
[16] Huang, J., Lê, K., Nualart, D.: Large time asymptotics for the parabolic Anderson model driven by spatially correlated noise. Ann. Inst. Henri Poincar’e Probab. Stat. 53(3), 1305-1340 (2017) · Zbl 1372.60092 · doi:10.1214/16-AIHP756
[17] Hu, Y., Nualart, D.: Stochastic heat equation driven by fractional noise and local time. Probab. Theory Relat. Fields 143(1-2), 285-328 (2009) · Zbl 1152.60331 · doi:10.1007/s00440-007-0127-5
[18] Huang, J.: On stochastic heat equation with measure initial data. Electron. Commun. Probab. 22, Paper No. 40, 6 (2017) · Zbl 1379.60070
[19] Lê, K.: A remark on a result of Xia Chen. Statist. Probab. Lett. 118, 124-126 (2016) · Zbl 1375.60108 · doi:10.1016/j.spl.2016.06.004
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