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Game theoretical methods in PDEs. (English) Zbl 1322.35139

The authors describe how the solutions to certain PDEs of \(p\)-Laplacian type can be interpreted as limits of values of a specific Tug-of-War game, when the step-size \(\epsilon\) determining the allowed length of move of a token, decreases to 0. After explaining in Section 1, how linear elliptic equations arise in probability, they describe in Section 2 how \(p\)-harmonic functions can be approximated by functions known as \(p\)-harmonious functions defined by the mean value property. These \(p\)-harmonious functions have a probabilistic interpretation as values of Tug-of-War games with noise explained in Section 3. Using the observation that a sequence of random variables involving \(p\)-harmonious functions is supermartingale, they argue that the minimum gain of Player I and the maximimum loss of Player II in a Tug-of-War game with noise is equal to the \(p\)-harmonious function. In Section 5 the authors discuss the example of the proof of uniform convergence and the local Harnack inequality using the technique of assigning suitable strategies in a Tug-of-War game.

MSC:

35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
91A10 Noncooperative games
91A05 2-person games

References:

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[6] The cartoons in this paper were drawn by our colleague Kiumars Kaveh at the University of Pittsburgh · Zbl 1293.35350
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