Stochastic partial differential equations with gradient driven by space-time fractional noises

Abstract

We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition. We prove the strong existence and uniqueness and joint Hölder continuity of the solution to the SPDEs.

Keywords

Stochastic partial differential equation (SPDE) / fractional noise / uniqueness / strong solution / Hölder continuity

References

[1] Bo L, Jiang Y, Wang Y. Stochastic Cahn-Hilliard equation with fractional noise. Stoch Dyn, 2008, 8(4): 643–665

[2] Dong Z, Wang F, Xu L. Irreducibility and asymptotics of stochastic burgers equation driven by α-stable processes. Potential Anal, 2020, 52(3): 371–392

[3] Guasoni P. No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math Finance, 2006, 16(3): 569–582

[4] Gyöngy I. Existence and uniqueness results for semilinear stochastic partial differential equations. Stochastic Process Appl, 1998, 73(4): 271–299

[5] Hairer M, Voss J. Approximations to the stochastic Burgers equation. J Nonlinear Sci, 2011, 12(6): 897–920

[6] Hu Y. Heat equations with fractional white noise potentials. Appl Math Optim, 2001, 43: 221–243

[7] Hu Y, Jiang Y, Qian Z. Stochastic partial differential equations driven by space-time fractional noises. Stoch Dyn, 2019, 18(6): 1950012 (34 pp)

[8] Hu Y, Nualart D, Xu F. Central limit theorem for an additive functional of the fractional Brownian motion. Ann Probab, 2014, 42(1): 168–203

[9] Jiang Y, Wei T, Zhou X. Stochastic generalized Burgers equations driven by fractional noise. J Differential Equations, 2012, 252(2): 1934–1961

[10] Kou S C. Stochastic modeling in nanoscale biophysics: subdiffusion within proteins. Ann Appl Stat, 2008, 2(2): 501–535

[11] Kunita H. Stochastic Flows and Stochastic Differential Equations.Cambridge: Cambridge Univ Press, 1990

[12] Mandelbrot B, Van Ness J. Fractional Brownian motions, fractional noises and applications. SIAM Rev, 1968, 10(4): 422–437

[13] Mémin J, Mishura Y, Valkeila E. Inequalities for moments of Wiener integrals with respect to a fractional Brownian motion. Statist Probab Lett, 2001, 51(2): 197–206

[14] Mitoma I. An 1-dimensional inhomogeneous Langevin equation. J Funct Anal, 1985, 61: 342–359

[15] Mohammed S, Zhang T. Stochastic Burgers equation with random initial velocities: a Malliavin calculus approach. SIAM J Math Anal, 2013, 45(4): 2396–2420

[16] Mytnik L, Wachtel V. Multifractal analysis of superprocesses with stable branching in dimension one. Ann Probab, 2015, 43: 2763–2809

[17] Nualart D, Ouknine Y. Regularization of quasilinear heat equations by a fractional noise. Stoch Dyn, 2004, 4(2): 201–221

[18] Odde D J, Tanaka E M, Hawkins S S, Buettner H M. Stochastic dynamics of the nerve growth cone and its microtubules during neurite outgrowth. Biotechnol Bioeng, 1996, 50(4): 452–461

[19] Rosen J. Joint continuity of the intersection local times of Markov processes. Ann Probab, 1987, 15: 659–675

[20] Shiga T. Two contrasting properties of solutions for one-dimensional stochastic partial differential equation. Canad J Math, 1994, 46(2): 415–437

[21] Wang F, Wu J, Xu L. Log-Harnack inequality for stochastic Burgers equations and applications. J Math Anal Appl, 2011, 384(1): 151–159

[22] Xiong J. Super-Brownian motion as the unique strong solution to an SPDE. Ann Probab, 2013, 41: 1030{1054

[23] Xiong J, Yang X. Strong existence and uniqueness to a class of nonlinear SPDEs driven by Gaussian colored noises. Statist Probab Lett, 2017, 129: 113–129

RIGHTS & PERMISSIONS

Higher Education Press