关于粘性局部时的电报信号
On the telegrapher's signals of sticky local times
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- Ulm University（乌尔姆大学）
- Sapienza University of Rome（罗马第一大学）
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中文总结 AI 辅助
本文研究粘性布朗运动边界局部时的电报信号，给出分数阶电报方程解的概率表示，并刻画一类粘性布朗运动的局部时，发现粘性效应导致局部时路径高度不规则。
中文摘要 AI 辅助
我们考虑一类新的粘性布朗运动的边界迹过程，并研究边界局部时的电报信号。对于分数阶电报方程 \begin{align*} (\sigma/\eta) D^\alpha_t v(t,x) + D^{2\alpha}_t v(t,x) = \frac{\partial^2 v}{\partial x^2}(t,x), \quad t>0,\\, x \in \mathbb{R}, \quad \alpha \in (0,1] \end{align*} 我们提供了其在各向异性Sobolev空间中的概率表示，并将其与文献中已知的表示进行比较。随后，我们讨论了相关过程并给出了它们的路径表示。基于这些表示，我们引入了由非局部动态边界条件 \begin{align*} \eta D^\alpha_t \varpi(t,x) = - \sigma \partial_{\bf n} \varpi(t,x), \qquad t>0, \\; x \in \partial \Omega \end{align*} 控制的一大类粘性布朗运动在 $\Omega$ 上的局部时的刻画，其中 $D^\alpha_t$ 表示Caputo-Džrbašjan意义下的分数阶导数。我们主要关注区间 $[a,b]$ 以构建原型模型，然后为 $\mathbb{R}^d$ 中球上的分析奠定基础。在异常动力学下，波传播与扩散之间的平滑插值捕捉了底层粘性布朗运动在边界上经历显著延长捕获时间的行为。解 $v$ 在 $H^1(\mathbb{R})$ 中保持到 $t=0$ 的连续性，这对应于电报过程在初始时刻的均方连续性。然而，对于 $t>0$，严重的粘性效应导致随机轨迹经历长时间的捕获期，使得粘性布朗运动的局部时路径高度不规则且粗糙。在我们的构造中，布朗结构立即出现，而不仅仅是作为流体动力学极限。
英文摘要
We consider the boundary trace process of a new class of sticky Brownian motions and study the telegraph signals of the boundary local time. For the fractional telegraph equation \begin{align*} (σ/η) D^α_t v(t,x) + D^{2α}_t v(t,x) = \frac{\partial^2 v}{\partial x^2}(t,x), \quad t>0,\, x \in \mathbb{R}, \quad α\in (0,1] \end{align*} we provide a probabilistic representation of the solution in anisotropic Sobolev spaces and compare it with well-known representations in the literature. We subsequently discuss the associated processes and provide their pathwise representations. Based on these representations, we introduce a characterization of the local times for a wide class of sticky Brownian motions on $Ω$ governed by the non-local dynamic boundary condition \begin{align*} ηD^α_t \varpi(t,x) = - σ\partial_{\bf n} \varpi(t,x), \qquad t>0, \; x \in \partial Ω\end{align*} where $D^α_t$ denotes the fractional derivative in the Caputo-Džrbašjan sense. We focus mainly on the interval $[a,b]$ to construct a prototype model, and then lay the foundation for the analysis on balls in $\mathbb{R}^d$. The smooth interpolation between wave propagation and diffusion under anomalous dynamics captures the behaviour of the underlying sticky Brownian motion experiencing significantly prolonged trapping times on the boundary. The solution $v$ retains its continuity up to $t=0$ in $H^1(\mathbb{R})$, which corresponds to the mean-square continuity of the telegrapher's process at the initial instant. However, for $t>0$, the severe sticky effect causes the stochastic trajectories to undergo prolonged trapping periods, leading to highly irregular and rough paths for the local time of the sticky Brownian motion. In our construction the Brownian structure appears immediately, rather than only as a hydrodynamic limit.
