论文标题
在整个空间中的分数和非局部抛物线平均野外游戏上
On fractional and nonlocal parabolic Mean Field Games in the whole space
论文作者
论文摘要
我们研究了由整个空间中大量非本地，分数和异常扩散驱动的平均现场游戏（MFGS）。这些非高斯的扩散是纯粹的跳跃莱维过程，其中一些$σ$ - 稳定的行为。其中包括$σ$ - 稳定过程和分数拉普拉斯扩散操作员$（ - δ）^{\fracσ2} $，金融中的钢化非对称过程，频谱单方面的过程以及不同订单的小额次要操作员的总和。我们的主要结果是MFG系统的经典解决方案的存在和唯一性，具有（1,2）$ in（1,2）$的订单$σ\订单的扩散运算符。我们考虑与局部和非本地耦合的整个空间中的抛物线方程。我们的证明使用纯PDE方法，并以狮子等的思想为基础。新成分是分数热内核估计值，分数Bellman的规律性结果，Fokker-Planck和耦合的平均野外游戏方程，以及（非常）在整个空间中分数Fokker-Planck方程的（非常）弱解的范围和紧凑性。我们的技术不需要时刻的假设，并且比Wasserstein使用较弱的拓扑。
We study Mean Field Games (MFGs) driven by a large class of nonlocal, fractional and anomalous diffusions in the whole space. These non-Gaussian diffusions are pure jump Lévy processes with some $σ$-stable like behaviour. Included are $σ$-stable processes and fractional Laplace diffusion operators $(-Δ)^{\fracσ2}$, tempered nonsymmetric processes in Finance, spectrally one-sided processes, and sums of subelliptic operators of different orders. Our main results are existence and uniqueness of classical solutions of MFG systems with nondegenerate diffusion operators of order $σ\in(1,2)$. We consider parabolic equations in the whole space with both local and nonlocal couplings. Our proofs uses pure PDE-methods and build on ideas of Lions et al. The new ingredients are fractional heat kernel estimates, regularity results for fractional Bellman, Fokker-Planck and coupled Mean Field Game equations, and a priori bounds and compactness of (very) weak solutions of fractional Fokker-Planck equations in the whole space. Our techniques requires no moment assumptions and uses a weaker topology than Wasserstein.
