带有奇异局部Hölder连续漂移项的随机阻尼波动方程与Euler-Bernoulli方程
Stochastic damped wave and Euler-Bernoulli equations with singular locally Hölder continuous drift
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- Università degli Studi di Parma（帕尔马大学）
- Università degli Studi dell’Insubria（因苏布里亚大学）
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中文总结 AI 辅助
该研究针对含奇异局部Hölder连续漂移项的随机微分方程，证明其满足弱唯一性与路径wise唯一性，覆盖1维随机阻尼波动方程及3维内的随机阻尼Euler-Bernoulli梁方程。
中文摘要 AI 辅助
设$U$和$H$为两个可分希尔伯特空间，$T>0$，我们考虑在希尔伯特空间$H$中演化的随机微分方程，形式为：$dX(t)=AX(t)dt+B(t,X(t))dt+GdW(t)$，$t\in[0,T]$，$X(0)=x \in H$，其中$A:D(A)\subseteq H\to H$是强连续半群$(e^{tA})_{t\geq0}$的无穷小生成元，$W=(W(t))_{t\geq0}$是定义在标准滤过概率空间$(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in [0,T]},\mathbb{P})$上的$U$-柱维纳过程，$G:U\to H$是线性有界算子，$B:[0,T]\times H\to H_\beta$是关于第二个变量的局部$\theta$-Hölder连续函数（对第一个变量一致），其中$\theta\in(0,1)$。此处$H_\beta$是包含$H$且具有连续嵌入的希尔伯特空间，因此漂移项是奇异的：它无法在环境空间$H$中明确定义。系数$\beta\geq0$衡量该奇异性的阶数，$\beta=0$对应漂移项取值于$H$的情况。我们证明，在系数满足适当假设时，方程(1)满足弱唯一性和路径wise唯一性。特别地，所假设的系数条件覆盖了1维随机阻尼波动方程，以及直至3维的随机阻尼Euler-Bernoulli梁方程，甚至包括双曲型情形。
英文摘要
Let $U$ and $H$ be two separable Hilbert spaces and $T>0$. We consider a stochastic differential equation which evolves in the Hilbert space $H$ of the form \begin{align} \label{SDEa} dX(t)=AX(t)dt+B(t,X(t))dt+GdW(t), \quad t\in[0,T], \quad X(0)=x \in H, \end{align} where $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup $(e^{tA})_{t\geq0}$, $W=(W(t))_{t\geq0}$ is a $U$-cylindrical Wiener process defined on a normal filtered probability space $(Ω,\mathcal{F},\{\mathcal{F}_t\}_{t\in [0,T]},\mathbb{P})$, $G:U\to H$ is a linear bounded operator and $B:[0,T]\times H\to H_β$ is a locally $θ$-Hölder continuous function with respect to the second variable, uniformly with respect to the first one, for some suitable $θ\in(0,1)$. Here, $H_β$ is a Hilbert space which contains $H$ with continuous embedding, so that the drift term is singular: it is not well-defined from the ambient space $H$ into itself. The coefficient $β\geq0$ measures the order of such a singularity and the case $β=0$ corresponds to a drift term with values in $H$. We prove that, under suitable assumptions on the coefficients, weak and pathwise uniqueness hold true for equation \eqref{SDEa}. In particular, the conditions assumed on the coefficients cover the stochastic damped wave equation in dimension $1$ and the stochastic damped Euler--Bernoulli beam equation up to dimension $3$, even in the hyperbolic case.
