Random walks and random tug of war in the Heisenberg group
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Mathematische Annalen
Random walks and random tug of war in the Heisenberg group Marta Lewicka1 · Juan Manfredi1
· Diego Ricciotti2
Received: 4 December 2018 / Revised: 30 April 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We study mean value properties of p-harmonic functions on the first Heisenberg group H, in connection to the dynamic programming principles of certain stochastic processes. We implement the approach of Peres and Sheffield (Duke Math J 145(1):91– 120, 2008) to provide a game-theoretical interpretation of the sub-elliptic p-Laplacian; and of Manfredi et al. (Proc Am Math Soc 138(3):881–889, 2010) to characterize its viscosity solutions via asymptotic mean value expansions.
1 Introduction In this paper, we are concerned with mean value properties of p-harmonic functions on the Heisenberg group H, in connection to the dynamic programming principles of certain stochastic processes. More precisely, we develop asymptotic mean value expansions of the type:
N Average(v, r )(q) = v(q) + cr 2 H,p v(q) + o(r 2 )
as r → 0+,
(1.1)
Communicated by Y. Giga.
B
Juan Manfredi [email protected] Marta Lewicka [email protected] Diego Ricciotti [email protected]
1
Department of Mathematics, University of Pittsburgh, 139 University Place, Pittsburgh, PA 15260, USA
2
Department of Mathematics and Statistics, University of South Florida, 4202 E Fowler Ave, Tampa, FL 33620, USA
123
M. Lewicka et al. N of the p-H-Laplacian for the normalized version H,p H,p in (1.4)–(1.5), for any 1 < p < ∞. The “Average” denotes here a suitable mean value operator, acting on a given function v : H → R, on a setffl of radius r and centered at a point q ∈ H. This operator may be either “stochastic”: , or “deterministic”: 21 (sup + inf), or it may be given through various compositions or further averages of such types. The averaging set may be one of the following: the 3-dimensional Korányi ball Br (q); the 2-dimensional ellipse in the tangent plane Tq passing through q, whose horizontal projection coincides with the 2-dimensional Euclidean ball of radius r ; the 1-dimensional boundary of such ellipse; or the 3-dimensional Korányi ellipsoid that is the image of Br (q) under a suitable linear map. For particular expansions in (1.1), we study solutions to the boundary value problems for the related mean value equations, posed on a bounded domain D ⊂ H, with data F ∈ C(H):
Average(u , ) = u in D,
u = F on H \ D.
(1.2)
We identify the solution u as the value of a process with, in general, both random and deterministic components. The purely random component is related to the “stochastic” averaging part of the operator Average as described above, whereas the deterministic component is related to the “deterministic” part and can be interpreted as the Tug of War game. Recall that the Tug of War is a zero-sum, two-players game process, in which the position of the particle in D is shifted according to the deterministic strategies of the two players. The players take turns with equal probab
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