论文标题
一类随机部分微分方程的适合性，具有完全单调系数的lévy噪声的扰动
Well-posedness of a class of stochastic partial differential equations with fully monotone coefficients perturbed by Lévy noise
论文作者
论文摘要
在本文中，我们考虑以下类随机部分微分方程（SPDE）：\ begin {equination*} \ left \ {\ strign {aligned} \ mathrm {d} \ Mathbf {x}} t+\ mathrm {b}（t，\ \ mathbf {x}（t））\ mathrm {d} \ mathrm {w}（t）（t）+\ int _ {\ int {\ mathrm {z}}γ（ t，\ mathrm {d} z），\; t \ in [0，t]，\\ \ \ \ \ \ \ \ \ \ \ = \ boldsymbol {x} \ in \ mathbb {h}，\ end \ end {aligned} \ right。 \ Mathbb {h} \ subset \ Mathbb {V}^*$，其中mappings \ begin {align*} \ Mathrm {a}：[0，t] \ times \ times \ times \ times \ times \ mathbb {v} \ to \ to \ to \ to \ mathb {v} \ Mathbb {V} \ to \ Mathrm {l} _2（\ Mathbb {U}，\ Mathbb {h}），\Quadγ：[0，T] \ times \ times \ times \ Mathbb {v} \ times \ times \ times \ mathrm {z} {z} {z} $ \ MATHRM {l} _2（\ MATHBB {U}，\ MATHBB {H}）$是所有Hilbert-Schmidt操作员的空间$ \widetildeπ$是一个补偿时间均匀的泊松随机度量。这种SPDE涵盖了一大批的准线性SPDE和大量流体动态模型。在$ \ mathrm {a}的某些通用假设下，使用经典的faedo-galekin技术，一种紧凑的方法和Skorokhod代表定理的版本，我们证明了一个\ emph {probabilistic feel solution}的存在，＆emphemence and concorline and and p ro.我们使用经典的Yamada-Watanabe定理来获得\ emph {唯一的概率强溶液}的存在。最后，我们允许扩散系数$ \ mathrm {b}（t，\ cdot）$和跳跃噪声系数$γ（t，\ cdot，z）$都取决于$ \ mathbb {h} $ - norm-norm-norm-norm-norm and $ \ mathbb {v} $ - norm，这两个都可以依赖于该系数的辅助依赖于该系数。我们建立了全球可溶性结果。
In this article, we consider the following class of stochastic partial differential equations (SPDE): \begin{equation*} \left\{\begin{aligned}\mathrm{d} \mathbf{X}(t)&=\mathrm{A}(t,\mathbf{X}(t))\mathrm{d} t+\mathrm{B}(t,\mathbf{X}(t))\mathrm{d}\mathrm{W}(t)+\int_{\mathrm{Z}}γ(t,\mathbf{X}(t-),z)\widetildeπ(\mathrm{d} t,\mathrm{d} z),\; t\in[0,T],\\ \mathbf{X}(0)&=\boldsymbol{x} \in \mathbb{H},\end{aligned} \right.\end{equation*} with fully locally monotone coefficients in a Gelfand triplet $\mathbb{V}\subset \mathbb{H}\subset\mathbb{V}^*$, where the mappings \begin{align*} \mathrm{A}:[0,T]\times \mathbb{V}\to\mathbb{V}^*,\quad \mathrm{B}:[0,T]\times \mathbb{V}\to\mathrm{L}_2(\mathbb{U},\mathbb{H}), \quad γ:[0,T]\times\mathbb{V}\times\mathrm{Z}\to\mathbb{H}, \end{align*} are measurable, $\mathrm{L}_2(\mathbb{U},\mathbb{H})$ is the space of all Hilbert-Schmidt operators from $\mathbb{U}\to\mathbb{H}$, $\mathrm{W}$ is a $\mathbb{U}$-cylindrical Wiener process and $\widetildeπ$ is a compensated time homogeneous Poisson random measure. Such kind of SPDE cover a large class of quasilinear SPDE and a good number of fluid dynamic models. Under certain generic assumptions of $\mathrm{A},\mathrm{B}$ and $γ$, using the classical Faedo-Galekin technique, a compactness method and a version of Skorokhod's representation theorem, we prove the existence of a \emph{probabilistic weak solution} as well as \emph{pathwise uniqueness of solution}. We use the classical Yamada-Watanabe theorem to obtain the existence of a \emph{unique probabilistic strong solution}. Finally, we allow both diffusion coefficient $\mathrm{B}(t,\cdot)$ and jump noise coefficient $γ(t,\cdot,z)$ to depend on both $\mathbb{H}$-norm and $\mathbb{V}$-norm, which implies that both the coefficients could also depend on the gradient of solution. We establish the global solvability results.
