High order Anderson parabolic model driven by rough noise in space. (English) Zbl 1484.60020
Summary: In this paper, we concern the fourth parabolic model on \(\mathbb{R}\) driven by a multiplicative Gaussian noise which behaves like fractional Brownian motion in time and space with Hurst index \(\frac{ 1}{ 2}< H_0<1\) and \(0< H_1<\frac{1}{2}\), respectively. The existence and uniqueness of mild solution in Skorohod sense are proved, and the weak intermittency is obtained by estimating \(p\)th \((p\geq2)\) moment of the solution. Moreover, the Hölder continuity can be obtained for the time and space variable.
MSC:
| 60E10 | Characteristic functions; other transforms |
| 82B35 | Irreversible thermodynamics, including Onsager-Machlup theory |
| 60J76 | Jump processes on general state spaces |
| 60G22 | Fractional processes, including fractional Brownian motion |
References:
| [1] | Balan, R. M. and Conus, D., Intermittency for the wave and heat equations with fractional noise in time, Ann. Probab.44 (2016) 1488-1534. · Zbl 1343.60081 |
| [2] | Balan, R. M., Jolis, M. and Quer-Sardanyons, L., Intermittency for the hyperbolic Anderson model with rough noise in space, Stoch. Process. Appl.127 (2017) 2316-2338. · Zbl 1367.60077 |
| [3] | Baxendale, P. H. and Rozovskiĭ, B. L., Kinematic dynamo and intermittence in a turbulent flow, Geophys. Astrophys. Fluid Dynam.73 (1993) 33-60. |
| [4] | Bertini, L. and Cancrini, N., The stochastic heat equation: Feynman-Kac formula and intermittence, J. Statist. Phys.78 (1995) 1377-1401. · Zbl 1080.60508 |
| [5] | Biagini, F., Hu, Y., Øksendal, B. and Zhang, T., Stochastic Calculus for Fractional Brownian Motion and Applications (Springer-Verlag, 2008). · Zbl 1157.60002 |
| [6] | Bo, L., Jiang, Y. and Wang, Y., On a class of stochastic Anderson models with fractional noises, Stoch. Anal. Appl.26 (2008) 256-273. · Zbl 1136.60345 |
| [7] | Bo, L. and Wang, Y., Stochastic Cahn-Hilliard partial differential equations with Lévy space time white noises, Stoch. Dyn.6 (2006) 229-244. · Zbl 1098.60058 |
| [8] | Cardon-Weber, C., Cahn-Hilliard stochastic equation: Existence of the solution and of its density, Bernoulli7 (2001) 777-816. · Zbl 0995.60058 |
| [9] | Chakraborty, P., Chen, X., Gao, B. and Tindel, S., Quenched asymptotics for a 1-D stochastic heat equation driven by a rough spatial noise, Stoch. Process. Appl.130 (2020) 6689-6732. · Zbl 1454.60090 |
| [10] | Chen, X., Parabolic Anderson model with a fractional Gaussian noise that is rough in time, Ann. Inst. H. Poincaré Probab. Stat.56 (2020) 792-825. · Zbl 1434.60083 |
| [11] | Chen, L. and Dalang, R. C., Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions, Ann. Probab.43 (2015) 3006-3051. · Zbl 1338.60155 |
| [12] | Chen, L., Hu, Y., Kalbasi, K. and Nualart, D., Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise, Probab. Theory Related Fields171 (2018) 431-457. · Zbl 1391.60153 |
| [13] | Chen, X., Hu, Y., Nualart, D. and Tindel, S., Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise, Electron. J. Probab.2265 (2017) 1-38. · Zbl 1386.60135 |
| [14] | Dasgupta, A. and Kallianpur, G., Chaos decomposition of multiple fractional integrals and applications, Probab. Theory Related Fields115 (1999) 527-548. · Zbl 0952.60043 |
| [15] | Hu, Y., Heat equations with fractional white noise potentials, Appl. Math. Optim.43 (2001) 221-243. · Zbl 0993.60065 |
| [16] | Hu, Y., Chaos expansion of heat equations with white noise potentials, Potential Anal.16 (2002) 45-66. · Zbl 0993.60063 |
| [17] | Jolis, M., The Wiener integral with respect to second order processes with stationary increments, J. Math. Anal. Appl.366 (2010) 607-620. · Zbl 1188.60027 |
| [18] | Khoshnevisan, D., Analysis of Stochastic Partial Differential Equations (Amer. Math. Soc., 2014). · Zbl 1304.60005 |
| [19] | Nualart, D., The Malliavin Calculus and Related Topics, 2nd edn. (Springer-Verlag, 2006). · Zbl 1099.60003 |
| [20] | Pipiras, V. and Taqqu, M. S., Integration questions related to fractional Brownian motion, Probab. Theory Related Fields118 (2000) 251-291. · Zbl 0970.60058 |
| [21] | Song, J., Song, X. and Xu, F., Fractional stochastic wave equation driven by a Gaussian noise rough in space, Bernoulli26 (2020) 2699-2726. · Zbl 1459.60141 |
| [22] | Walsh, J. B., An introduction to stochastic partial differential equations, École d’Été de Probabilités de Saint-Flour, Vol. 1180 (Springer, 1986), pp. 265-439. · Zbl 0608.60060 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
