Solvability of Parabolic Anderson Equation with Fractional Gaussian Noise

Zhen-Qing Chen , Yaozhong Hu

Communications in Mathematics and Statistics ›› 2023, Vol. 11 ›› Issue (3) : 563 -582.

PDF
Communications in Mathematics and Statistics ›› 2023, Vol. 11 ›› Issue (3) : 563 -582. DOI: 10.1007/s40304-021-00264-5
Article

Solvability of Parabolic Anderson Equation with Fractional Gaussian Noise

Author information +
History +
PDF

Abstract

This paper provides necessary as well as sufficient conditions on the Hurst parameters so that the continuous time parabolic Anderson model $\frac{\partial u}{\partial t}=\frac{1}{2}\Delta +u{\dot{W}}$ on $[0, \infty )\times {{\mathbb {R}}}^d $ with $d\ge 1$ has a unique random field solution, where W(tx) is a fractional Brownian sheet on $[0, \infty )\times {{\mathbb {R}}}^d$ and formally $\dot{W} =\frac{\partial ^{d+1}}{\partial t \partial x_1 \cdots \partial x_d} W(t, x)$. When the noise W(tx) is white in time, our condition is both necessary and sufficient when the initial data u(0, x) is bounded between two positive constants. When the noise is fractional in time with Hurst parameter $H_0>1/2$, our sufficient condition, which improves the known results in the literature, is different from the necessary one.

Keywords

Stochastic heat equation / Fractional Brownian fields / Wiener chaos expansion / Random field solution / Necessary condition / sufficient condition / Moment bounds

Cite this article

Download citation ▾
Zhen-Qing Chen,Yaozhong Hu. Solvability of Parabolic Anderson Equation with Fractional Gaussian Noise. Communications in Mathematics and Statistics, 2023, 11(3): 563-582 DOI:10.1007/s40304-021-00264-5

登录浏览全文

4963

注册一个新账户 忘记密码

References

Funding

Simons Foundation

Canadian Network for Research and Innovation in Machining Technology

Natural Sciences and Engineering Research Council of Canada

AI Summary AI Mindmap
PDF

0

Accesses

0

Citation

Detail

Sections
Recommended

AI思维导图

/