分数阶随机热方程的研究

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-09-24 DOI:10.30757/alea.v20-15
N. Schaeffer
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引用次数: 1

摘要

在这篇文章中,我们研究了一个具有二次非线性的$d$维随机非线性热方程(SNLH),该方程由分数时空白噪声强迫:\beart{equation*}\left{\bearth{array}{l}\partial_t u-\Delta u=\rho^2 u^2+\dot B\,\quad t\In[0,t]\,\,x\In\mathbb{R}^d\,\\u_0=\phi\。\end{array}\ right。\end{方程*}根据Hurst指数$H=(H_0,…,H_d)$$\in(0,1)^{d+1}$的范围,表现出两种类型的状态。特别地,我们证明了由Da-Prato-Debussche技巧得到的(SNLH)的局部适定性,当$2H_0+\sum_{i=1}^{d}H_i>d$。相反,当$2H_0+\sum_{i=1}时,(SNLH)更难处理^{d}H_i\leq d$。在这种情况下,由于时间相关的重整化,模型必须在Wick意义上进行解释。借助热半群的正则化效应,我们建立了全维$d\geq1的(SNLH)的局部适定性结果$
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Study of a fractional stochastic heat equation
In this article, we study a $d$-dimensional stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional space-time white noise: \begin{equation*} \left\{\begin{array}{l} \partial_t u-\Delta u= \rho^2 u^2 + \dot B \, , \quad t\in [0,T] \, , \, x\in \mathbb{R}^d \, ,\\ u_0=\phi\, . \end{array} \right. \end{equation*} Two types of regimes are exhibited, depending on the ranges of the Hurst index $H=(H_0,...,H_d)$ $\in (0,1)^{d+1}$. In particular, we show that the local well-posedness of (SNLH) resulting from the Da Prato-Debussche trick, is easily obtained when $2 H_0+\sum_{i=1}^{d}H_i>d$. On the contrary, (SNLH) is much more difficult to handle when $2H_0+\sum_{i=1}^{d}H_i \leq d$. In this case, the model has to be interpreted in the Wick sense, thanks to a time-dependent renormalization. Helped with the regularising effect of the heat semigroup, we establish local well-posedness results for (SNLH) for all dimension $d\geq1.$
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来源期刊
CiteScore
1.10
自引率
0.00%
发文量
48
期刊介绍: ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted. ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper. ALEA is affiliated with the Institute of Mathematical Statistics.
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