论文标题
与Kolmogorov一起进行的拔河
Tug-of-war with Kolmogorov
论文作者
论文摘要
我们介绍了一类新的变性非线性抛物线pdes $$（（（p-2）δ_ {\ infty，x}^n+δ_x） $（x，y，t）\ in \ mathbb r^m \ times \ mathbb r^m \ times \ mathbb r $，$ p \ in（1，\ infty）$，结合了Kolmogorov的经典PDE和正常化的$ P $ - laplace Operator。我们根据渐近平均值属性来表征解决方案，结果与噪声的某些拔河游戏分析有关。当控制可能步骤的大小的参数为零时，游戏的值函数将近似解决方案引入了所述的PDE。建立了粘度解决方案的存在和独特性。渐近平均值，相关的游戏和迪利奇问题的几何形状都反映了Kolmogorov类型的扩张家族和谎言组的家族，这使我们的环境与标准抛物线膨胀和欧几里得翻译的背景不同，适用于适用于热效应者和正常的Parabolity Infination Infination Infination Laplace Laplace laplace lapabice Laplace操作员。
We introduce a new class of strongly degenerate nonlinear parabolic PDEs $$((p-2)Δ_{\infty,X}^N+Δ_X)u(X,Y,t)+(m+p)(X\cdot\nabla_Yu(X,Y,t)-\partial_tu(X,Y,t))=0,$$ $(X,Y,t)\in\mathbb R^m\times \mathbb R^m\times \mathbb R$, $p\in (1,\infty)$, combining the classical PDE of Kolmogorov and the normalized $p$-Laplace operator. We characterize solutions in terms of an asymptotic mean value property and the results are connected to the analysis of certain tug-of-war games with noise. The value functions for the games introduced approximate solutions to the stated PDE when the parameter that controls the size of the possible steps goes to zero. Existence and uniqueness of viscosity solutions to the Dirichlet problem is established. The asymptotic mean value property, the associated games and the geometry underlying the Dirichlet problem, all reflect the family of dilation and the Lie group underlying operators of Kolmogorov type and this makes our setting different from the context of standard parabolic dilations and Euclidean translations applicable in the context of the heat operator and the normalized parabolic infinity Laplace operator.
