Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2
Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2
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DOI：
10.1214/11-aop649
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发表时间：
2010-07
影响因子：
2.3
通讯作者：
Yaozhong Hu;F. Lu;D. Nualart
中科院分区：
文献类型：
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作者：
Yaozhong Hu;F. Lu;D. Nualart
本文建立了高斯噪声驱动下的随机偏微分方程的Feynman-Kac公式，该随机偏微分方程是关于时间的分数布朗运动，其Hurst参数H < 1/2.为了建立这样一个公式，我们引入并研究了一个非线性随机积分从给定的高斯噪声。为了证明Feynman-Kac积分的存在性，还需要证明非线性随机积分的指数可积性。然后，利用Malliavin积分的逼近技巧证明了Feynman-Kac积分是随机偏微分方程的弱解.
In this paper, a Feynman–Kac formula is established for stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter H < 1/2. To establish such a formula, we introduce and study a nonlinear stochastic integral from the given Gaussian noise. To show the Feynman–Kac integral exists, one still needs to show the exponential integrability of nonlinear stochastic integral. Then, the approach of approximation with techniques from Malliavin calculus is used to show that the Feynman–Kac integral is the weak solution to the stochastic partial differential equation.