Asymptotic mean value formulas, nonlocal space-time parabolic operators and anomalous tug-of-war games. (English) Zbl 1501.35435
Summary: The fractional heat operator \(( \partial_t - \Delta_x )^s\) and Continuous Time Random Walks (CTRWs) are interesting and sophisticated mathematical models that can describe complex anomalous systems. In this paper, we prove asymptotic mean value representation formulas for functions with respect to \(( \partial_t - \Delta_x )^s\) and we introduce new nonlocal, nonlinear parabolic operators related to a tug-of-war which accounts for waiting times and space-time couplings. These nonlocal, nonlinear parabolic operators and equations can be seen as nonlocal versions of the evolutionary infinity Laplace operator.
MSC:
| 35R11 | Fractional partial differential equations |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
fractional heat operatorReferences:
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