论文标题
sobolev空间理论，用于征收过程驱动的时间分数随机部分微分方程
A Sobolev space theory for the time-fractional stochastic partial differential equations driven by Levy processes
论文作者
论文摘要
我们提供了一个$ l_ {p} $ - 理论（$ p \ geq 2 $），用于时间 - 曲折随机部分微分方方程，该方程是由类型$ \ partial^α__{t} U = \ sum_ = \ sum_ = \ sum_ = \ sum_ {i，j = 1} = 1}^d a^d a^a^{ij} i^{ij} {ij} i} {j {j} { +f +\ sum_ {k = 1}^{\ infty} \ partial^β_{t} \ int_ {0}^{t}^{t}（\ sum_ {i = 1}^dμ^{ik} u_ {ik} u_ {x^i} +g^k）这里$ \ partial^α_t$和$ \ partial^β_t$是caputo分数衍生物，$α\ in（0,2），β\ in（0，α+1/p）$，$ \ {z^k_t：k_t：k_t：k = 1,2，\ cdots \ cdots \ cdots \ cdots \ cdots \ cdots \ cdots \ cdots \} $是独立的过程。系数是随机功能，具体取决于$（t，x）$。我们证明了唯一性和存在导致Sobolev空间，并获得了解决方案的最大规律性。
We present an $L_{p}$-theory ($p\geq 2$) for time-fractional stochastic partial differential equations driven by Lévy processes of the type $$ \partial^α_{t}u=\sum_{i,j=1}^d a^{ij}u_{x^{i}x^{j}} +f+\sum_{k=1}^{\infty}\partial^β_{t}\int_{0}^{t} (\sum_{i=1}^dμ^{ik} u_{x^i} +g^k) dZ^k_{s} $$ given with nonzero intial data. Here $\partial^α_t$ and $\partial^β_t$ are the Caputo fractional derivatives, $α\in (0,2), β\in (0,α+1/p)$, and $\{Z^k_t:k=1,2,\cdots\}$ is a sequence of independent Lévy processes. The coefficients are random functions depending on $(t,x)$. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity of the solution.
