RANDOM TUG OF WAR GAMES FOR THE p-laplacian: 1 < p <

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1 RANDOM TUG OF WAR GAMES FOR THE p-laplacian: 1 < p < MARTA LEWICKA Abstract. We propose a new finite difference approximation to the Dirichlet problem for the homogeneous p-laplace equation posed on an N-dimensional domain, in connection with the Tug of War games with noise. Our game and the related mean-value expansion that we develop, superposes the deterministic averages 1 inf + sup taken over balls, with the stochastic averages ffl, taken over N-dimensional ellipsoids whose aspect ratio depends on N, p and whose orientations span all directions while determining inf / sup. We show that the unique solutions u ɛ of the related dynamic programming principle are automatically continuous for continuous boundary data, and coincide with the well-defined game values. Our game has thus the min-max property: the order of supremizing the outcomes over strategies of one player and infimizing over strategies of their opponent, is immaterial. We further show that domains satisfying the exterior corkscrew condition are game regular in this context, i.e. the family {u ɛ} ɛ 0 converges uniformly to the unique viscosity solution of the Dirichlet problem. 1. Introduction In this paper, we study the finite difference approximations to the Dirichlet problem for the homogeneous p-laplace equation p u = 0, posed on an N-dimensional domain, in connection to the dynamic programming principles of the so-called Tug of War games with noise. It is a well known fact that for u C R N there holds the following mean value expansion: r uy dy = ux + Bx,r N + ux + or as r 0 +. Indeed, an equivalent condition for harmonicity u = 0 is the mean value property, and thus ux provides the second-order offset from the satisfaction of this property. When we replace Bx, r by an ellipse Ex, r; α, ν = x + {y R N ; y, ν + α y y, ν ν < α r } with the radius r, the aspect ratio α > 0 and oriented along some given unit vector ν, we obtain: r uy dy = ux + ux + α 1 ux : ν + or. 1.1 Ex,r;α,ν N + Recalling the interpolation: p u = u p u + p u, 1. the formula 1.1 becomes: ffl Ex,r;α,ν uy dy = ux + r u p N+ pux + or, for the choice α = p 1 and ν = ux ux. To obtain the mean value expansion where the left hand side averaging does not require the knowledge of ux and allows for the identification of a p- harmonic function that is a priori only continuous, we need to, in a sense, additionally average over all equally probable vectors ν. This can be carried out by superposing: i the deterministic average 1 inf + sup, with ii the stochastic average ffl, taken over appropriate ellipses E whose aspect ratio depends on N, p and whose orientations ν span all directions while determining inf / sup in i. Date: October 5,

2 MARTA LEWICKA In fact, such construction can be made precise see Theorem.1, leading to the expansion: 1 inf + sup z Bx,r z Bx,r E z, γpr, αp z x r, z x uy dy z x N+ p 1 = ux + r u p p ux + or, p 1 with γ p = that is a fixed stochastic sampling radius factor, and with α p that is the aspect ratio in radial function of the deterministically chosen position z Bx, r. The value p 1 of α p varies quadratically from 1 at the center of Bx, r to a p = at its boundary. We will be concerned with the mean value expansions of the form 1.3, in connection with the specific Tug of War games with noise. This connection has been displayed in [11] by Peres and Scheffield, based on another interpolation property of p : p u = u p u 1 u + p 1 u, 1.4 which has first appeared in the context of the applications of p to image recognition in [5]. Indeed, the construction in [11] interpolates from: i the 1-Laplace operator 1 corresponding to the motion by curvature game studied by Kohn and Serfaty [6], to ii the -Laplacian corresponding to the pure Tug of War studied by Peres, Schramm, Scheffield and Wilson [10]. We remark that if one uses 1. instead of 1.4, one is lead to the games studied by Manfredi, Parviainen and Rossi [8], that interpolate from: classically corresponding [] to Brownian motion, to ; this approach however poses a limitation on the exponents p [,. The original game presented in [11] was a two-player, zero-sum game, stipulating that at each turn, position of the token is shifted by some vector σ within the prescribed radius r = ɛ > 0, by a player who has won the coin toss, which is followed by a further update of the position by a random noise vector. The noise vector is uniformly distributed on the codimension- sphere, centered at the current position, contained within the hyperplane that is orthogonal to the last player s move σ, and with radius proportional to σ with factor γ = N 1 p 1. We again interpret that γ interpolates from: i + at the critical exponent p = 1 that corresponds to choosing a direction line and subsequently determining its orientation, to ii 0 at the critical exponent p = that corresponds to not adding the random noise at all. In this paper we utilize the full N-dimensional sampling on ellipses E, rather than on spheres. Together with another modification taking into account the boundary data F, we achieve that the solutions of the dynamic programming principle at each scale ɛ > 0 are automatically continuous in fact, they inherit the regularity of F and coincide with the well-defined game values. Our game has thus the min-max property: the order of supremizing the outcomes over strategies of one player and infimizing over strategies of his opponent, is immaterial. This point has been left unanswered in the case of the game in [11], where the regularity even measurability of the possibly distinct game values was not clear. We also point out that in [1], the authors presented a variant of the [11] game where the deterministic / stochastic sampling takes place, respectively on: N 1-dimensional spheres, and N 1-dimensional balls within the orthogonal hyperplanes. They obtain the min-max property and continuity of solutions to the mean value expansion in their setting, albeit at the expense of much more complicated analysis, passing through measurable construction and comparison to game values. In our case the uniqueness and continuity follow directly, much like in the linear p = case where the N-dimensional averaging guarantees smoothness of harmonic functions. 1.3

3 RANDOM TUG OF WAR GAMES FOR THE p-laplacian The content and structure of the paper. In section, Theorem.1, we prove the validity of our main mean value expansion 1.3. In the following remarks we show how other expansions with a wider range of exponents, with sampling set degenerating at the boundary, or pertaining to the [11] codimension- sampling arise in the same analytical context. In Theorem 3.1 in section 3, we obtain the existence, uniqueness and regularity of solutions u ɛ to the dynamic programming principle 3.1 at each sampling scale ɛ, that can be seen as a finite difference approximation to the p-laplace Dirichlet problem with continuous boundary data F. In particular, each u ɛ is continuous up to the boundary, where it assumes the values of F. Then, in Theorem 3.3 we show that in case F is already a restriction of a p-harmonic function with non-vanishing gradient, the corresponding family {u ɛ } ɛ 0 uniformly converges to F at the rate that is of first order in ɛ. Our proof uses an analytical argument and it is based on the observation that for s sufficiently large, the mapping x x s yields the variation that pushes the p-harmonic function F into the region of p-subharmonicity. In the linear case p =, the quadratic correction s = suffices, otherwise we give a lower bound 3.8 for the admissible exponents s = sp, N, F. In section 4, we develop the probability setting for the Tug of War game modelled on 1.3 and 3.1. Then in Theorem 4.1, using a classical martingale argument, we show that this game has a value, coinciding with the unique, continuous solution u ɛ. In section 5 we address convergence of the family {u ɛ } ɛ 0. In view of its equiboundedness, it suffices to prove equicontinuity. We first observe, in Theorem 5.1, that this property is equivalent to the seemingly weaker property of equicontinuity at the boundary. Again, our argument is analytical rather than probabilistic, based on the translation and well-posedness of 3.1. We then define the game regularity of the boundary points, which turns out to be a notion equivalent to the aforementioned boundary equicontinuity. Definition 5., Lemma 5.4 and Theorem 5.5 mimic the parallel statements in [11]. We further prove in Theorem 7. that any limit of a converging sequence in {u ɛ } ɛ 0 must be the viscosity solution to the p-harmonic equation with boundary data F. By uniqueness of such solutions, we obtain the uniform convergence of the entire family {u ɛ } ɛ 0 in case of the game regular boundary. In section 6 we show that domains that satisfy the exterior corkscrew condition are game regular. The proof in Theorem 6. is based on the concatenating strategies technique and the annulus walk estimate taken from [11]. We expand the proofs and carefully provide the details omitted in [11], for the benefit of the reader less familiar with the probability techniques. Finally, let us mention that similar results and approximations, together with their gametheoretical interpretation, can be also developed in other contexts, such as: the obstacle problems, nonlinear potential theory in Heisenberg group or in other subriemannian geometries, Tug of War on graphs, the non-homogeneous problems, the parabolic problems, problems with non-constant coefficient p, and the fully nonlinear case of. 1.. Notation for the p-laplacian. Let D R N be an open, bounded, connected set. Given p 1,, consider the following Dirichlet integral: ˆ I p u = ux p dx for all u W 1,p D, D that we want to minimize among all functions u subject to some given boundary data. The condition for the vanishing of the first variation of I p, assuming sufficient regularity of u so that the divergence theorem may be used, takes the form: ˆ η div u p u dx = 0 for all η Cc D, D

4 4 MARTA LEWICKA which, by the fundamental theorem of Calculus of Variations, yields: p u = div u p u = 0 in D. 1.5 The operator in 1.5 is called the p-laplacian, the partial differential equation 1.5 is called the p-harmonic equation and its solution u is a p-harmonic function. It is not hard to compute: p u = u p u + u p, u = u p u u + p u :, u which is precisely 1.. The second term in parentheses above is called the -Laplacian: u u = u :. u Applying 1. to the effect that 1 u = u 1 u u and introducing it in 1. again, yields 1.4. Likewise, for every 1 < p < q < s < there holds: s q u p p u = s p u q q u + p q u s s u Acknowledgments. The author is grateful to Yuval Peres for many discussions about random Tug of War games. Support by the NSF grant DMS is acknowledged.. A mean value expansion for p For ρ, α > 0 and a unit vector ν R N, we denote by E0, ρ; α, ν the ellipse centered at 0, with radius ρ, and with aspect ratio α oriented along ν, namely: E0, ρ; α, ν = { y R N ; For x R N, we have the translated ellipse: y, ν α + y y, ν ν < ρ }. Ex, ρ; α, ν = x + E0, ρ; α, ν. Note that, when ν = 0, this formula also makes sense and returns the ball Ex, ρ; α, 0 = Bx, ρ centered at x and with radius ρ. Given a continuous function u : R N R, define the averaging operator: A u; ρ, α, ν x = uy dy = Ex,ρ;α,ν u x + ρy + ρα 1 y, ν ν dy. B0,1 In what follows, we will often use the above linear change of variables: B 1 0 y ρα y, ν ν + ρ y y, ν ν E0, ρ; α, ν. Theorem.1. Let u C R N. Given p 1,, we set the scaling factors: N + p 1 γ p = p 1, a p =. Then, for every x 0 R N such that ux 0 0, we have: 1 inf + sup A u; γ p r, 1 + a p 1 x x 0 x Bx 0,r x Bx 0,r r, x x 0 x x 0 x = ux 0 + r p 1 ux 0 p p ux 0 + or as r

5 RANDOM TUG OF WAR GAMES FOR THE p-laplacian 5 The coefficient in the rate of convergence or depends only on p, N and in increasing manner on ux 0, ux 0 and the modulus of continuity of u at x 0. The expression in the left hand side of the formula.1 should be understood as the deterministic average 1 inf + sup, on the ball Bx 0, r, of the function x f u x; x 0, r in: f u x; x 0, r = A u; γr, 1 + a 1 x x 0 r, x x 0 x x x 0 = u x + γry + B0,1 γa 1 y, x x 0 x x 0 dy, r. where γ = γ p and a = a p. We will frequently use the notation: S ɛ ux 0 = 1 inf + x Bx 0,r sup x Bx 0,r f u x; x 0, r. For each x Bx 0, r the integral quantity in. returns the average of u on the N- dimensional ellipse centered at x, with radius γr, and with aspect ratio r +a 1 x x 0 along r the orientation vector x x 0 x x 0. Equivalently, writing x = x 0 + ry, the value f u x; x 0, r is the average of u on the scaled ellipse x 0 +re y, γ; 1+a 1 y, y y. Since the aspect ratio changes smoothly from 1 to a as x x 0 decreases from 0 to r, the said ellipse coincides with the ball Bx 0, γr at x = x 0 and it interpolates as x x 0 r, to E x, γr; a, x x 0 x x 0. Figure 1. The sampling ellipses in the expansion.1, when r = 1. Proof of Theorem We write γ = N+ p 1 p 1 and a =, and consider the Taylor expansion of u at a fixed x Bx 0, ρ under the second integral in.. Observe that the first order increments are

6 6 MARTA LEWICKA linear in y, hence they integrate to 0 on B0, 1. These increments are of order r and we get: f u x; x 0, r ux : γry + = ux + 1 B0,1 = ux + γ ux : r y dy + a 1 B0,1 Recall now that: + Consequently,.3 becomes: γa 1 y, x x 0 x x 0 dy + or r a 1 r y, x x 0 dy B0,1 y dy = B0,1 f u x; x 0, r = ux + γ r + γ a 1 N + ux = f u x; x 0, r + or, y, x x 0 y dy x x 0 B0,1 x x 0 + or. y1 dy Id N = 1 B0,1 N + Id N. N + + a 1 x x 0 r N + where a further Taylor expansion of u at x 0 gives: ux : x x 0 + or f u x; x 0, r = ux 0 + ux 0, x x 0 + γ r N + ux γ a 1 N + + a 1 x x 0 r ux 0 : x x 0. N + The left hand side of.1 thus satisfies: 1 inf f ux; x 0, r + sup x Bx 0,r x Bx 0,r = 1 inf f u x; x 0, r + x Bx 0,r f u x; x 0, r sup x Bx 0,r f u x; x 0, r + or, Since on Bx 0, r we have: fu x; x 0, r = ux 0 + ux 0, x x 0 + Or, the assumption ux 0 0 implies that the continuous function f u ; x 0, r attains its extrema on the boundary Bx 0, r, provided that r is sufficiently small. This reasoning justifies that f u in.4 may be replaced by the quadratic polynomial function: f u x; x 0, r = ux 0 + γ r N + ux 0 + ux 0, x x γ a 1 ux 0 : x x 0. N +. We now argue that fu attains its extrema on Bx 0, r, up to error Or 3 whenever r is sufficiently small, precisely at the opposite boundary points x 0 + r ux 0 ux 0 and x 0 r ux 0 ux

7 RANDOM TUG OF WAR GAMES FOR THE p-laplacian 7 We recall the adequate argument from [11], for the convenience of the reader. After translating and rescaling, it suffices to investigate the extrema on B0, 1, of the functions: g r x = a, x + r A : x, where a R N is of unit length and A R N N sym. Fix a small r > 0 and let x max B0, 1 be a maximizer of g r. Writing g r x max g r a, we obtain: a, x max 1 + r A : a x max 1 r A a x 1 r A a x max. Thus there holds: a x max = a, x max 4r A a x max, and so finally: Since a, x max 1, we conclude that: max a x max 4r A. 0 g r x max g r a = a, x max a + r A : x max a r A : x max a r A a x max 8r A. Likewise, for a minimizer x min we have: 0 g r x min g r a 8r A. It follows that: 1 gr x min +g r x max 1 gr a + g r a 8r A, which proves the claim for the unscaled functions fu. 3. Concluding, we compute: 1 inf f u x; x 0, r + x Bx 0,r sup x Bx 0,r f u x; x 0, r = ux 0 + γ r N + ux 0 + r 1 + γ a 1 N + = ux 0 + γ r N + = ux 0 + r p 1 ux 0 + N + γ + a 1 ux 0 + p ux 0 = ux 0 + r ux 0 p p ux 0 + Or 3, p 1 ux 0 + Or 3 ux 0 + Or 3 + Or 3 where in the last step we used 1.. This completes the proof in view of.4..5 Remark.. We see that the arguments in the proof of Theorem.1 work for any γ, a > 0 that set the prefactor at ux 0 in.5 to p, namely: N + γ + a = p 1..6 Thus, for every γ, a > 0 satisfying.6, we obtain the following mean value property: 1 inf + sup A u; γr, 1 + a 1 x x 0 x Bx 0,r r, x x 0 x x x 0 x Bx 0,r = ux 0 + r ux 0 p p 1 a pux 0 + or.

8 8 MARTA LEWICKA In.1 we chose to split evenly with a = N+ γ γ 0, in agreement with the well-known expansion: 1 inf u + Bx 0,r sup Bx 0,r = p 1. When p, one can take a = 1 and u = ux 0 + r ux 0 + or. Remark.3. On the other hand, when p 1+ in.6, then a 0+ for any admissible choice of a. The critical choice a = 0 is the only one valid for every p 0, 1. It corresponds to varying the aspect ratio along the radius of Bx 0, r, from 1 to 0 rather than to a > 0 and taking the stochastic averaging domains to be the corresponding ellipses: E x, γr; 1 x x 0 r, x x 0, x x 0 whose radius γr is scaled by the factor γ = N+ p 1. At x = x 0, the ellipse above coincides with the ball Bx 0, γr, whereas as x x 0 r it degenerates to the N 1-dimensional ball: E x, γr; 0, x x 0 { = x + y R N ; y, x x 0 = 0 and y < γr }. x x 0 The resulting mean value expansion is then: 1 inf + x Bx 0,r sup x Bx 0,r = ux 0 + A u; γr, 1 x x 0 r, x x 0 x x x 0 r p 1 ux 0 p p ux 0 + or. Remark.4. In [11], instead of averaging on an N-dimensional ellipse, the average is taken on the N -dimensional sphere centered at x, with radius γ x x 0, and contained within the hyperplane perpendicular to x x 0. The radius of the sphere thus increases linearly from 0 to γr with a factor γ > 0, as x x 0 varies from 0 to r. This corresponds to evaluating on Bx 0, r the deterministic averages of:.7 f u x; x 0, r = u x + γ x x 0 Rxy dy. B N 1 0,1 Here, Rx SON is such that Rxe N = x x 0 x x 0, and BN 1 0, 1 stands for the N - dimensional sphere of unit radius, viewed as a subset of R N contained in the subspace R N 1 orthogonal to e N. Note that x Rx can be only locally defined as a C function. However, the argument as in the proof of Theorem.1, can still be applied to get: f u x; x 0, r = ux + 1 ux : γ x x 0 Rxy dy + or B N 1 0,1 γ = ux + x x 0 ux ux : x x 0 + or N 1 = ux 0 + ux 0, x x 0 + N 1 x x 0 ux γ ux 0 : x x 0 + or N 1 γ

9 RANDOM TUG OF WAR GAMES FOR THE p-laplacian 9 where we used the general formula ffl B d 0,1 y dy = 1 ffl d B d 0,1 y dyid d = 1 d Id d, so that: 1 Rxy dy = N 1 Rx Id N e N Rx T = 1 Id N x x 0. N 1 x x 0 B N 1 0,1 Calling fu the polynomial: f u x; x 0, r = ux 0 + ux 0, x x 0 + γ N 1 ux 0Id N + 1 γ N 1 the claim in Step of proof of Theorem.1 yields: 1 inf f u x; x 0, r + f u x; x 0, r x Bx 0,r = ux 0 + γ r N 1 sup x Bx 0,r ux 0 + ux 0 : x x 0, N 1 γ 1 ux 0 + Or 3. Clearly, there holds N 1 1 = p, precisely for the scaling factor γ = γ this case, we get the mean value expansion with the coefficient in.7: 1 inf + sup f u x; x 0, r x Bx 0,r x Bx 0,r = ux 0 + N 1 p 1 r p 1 ux 0 p p ux 0 + or. as in [11]. In 3. The dynamic programming principle and the first convergence theorem Let D R N be an open, bounded, connected domain and let F CR N be a bounded data function. We have the following: Theorem 3.1. For every ɛ 0, 1 there exists a unique Borel, bounded function u : R N R denoted further by u ɛ, automatically continuous, such that:.8 ux = d ɛ xs ɛ ux + 1 d ɛ x F x for all x R N. 3.1 The solution operator to 3.1 is monotone, i.e. if F F then the corresponding solutions satisfy: u ɛ ū ɛ. Moreover u CR N F CR N. Proof. 1. The solution u of 3.1 is a fixed point of the operator T ɛ v = d ɛ S ɛ v + 1 d ɛ F. Recall that: S ɛ vx = 1 inf + sup f v x + ɛz; x, ɛ z B0,1 z B0,1 3. where: f v x + ɛz; x, ɛ = vw dw. x+ɛez,γ p;1+a p 1 z, z z Observe that for a fixed ɛ and x, and given a bounded Borel function v : R N R, the average f v is continuous in z B0, 1. In view of continuity of the weight d ɛ and the data F, it is not hard to conclude that both T ɛ, S ɛ likewise return a bounded continuous function, so in particular the solution to 3.1 is automatically continuous. We further note that S ɛ and T ɛ are monotone, namely: S ɛ v S ɛ v and T ɛ v T ɛ v if v v. The solution u of 3.1 is obtained as the limit of iterations u n+1 = T ɛ u n, where we set u 0 const inf F. Since u 1 = T ɛ u 0 u 0 on R N, by monotonicity of T ɛ, the sequence

10 10 MARTA LEWICKA {u n } n=0 is nondecreasing. It is also bounded by F CR N and thus it converges pointwise to a bounded, Borel limit u : R N R. Observe now that: T ɛ u n x T ɛ ux S ɛ u n x S ɛ ux sup z B0,1 x+ɛez,γ p;1+a p 1 z, z z u n u w dw C ɛ ˆ D u n u w dw, where C ɛ is the lower bound on the volume of the sampling ellipses. By the monotone convergence theorem, it follows that the right hand side in 3.3 converges to 0 as n. Consequently, u = T ɛ u, proving existence of solutions to We now show uniqueness. If u, ū both solve 3.1, then define M = sup x R N ux ūx = sup x D ux ūx and consider any maximizer x 0 D, where ux 0 ūx 0 = M. By the same bound in 3.3 it follows that: M = ux 0 ūx 0 = d ɛ x 0 S ɛ ux 0 S ɛ ūx 0 sup f u ū x + ɛz; x, ɛ M, z B0,1 3.3 yielding in particular ffl Bx 0,γ pɛ u ū w dw = M. Consequently, Bx 0, γ p ɛ D M = { u ū = M} and hence the set D M is open in R N. Since D M is obviously closed and nonempty, there must be D M = R N and since u ū = 0 on R N \ D, it follows that M = 0. Thus u = ū, proving the claim. Finally, we remark that the monotonicity of S ɛ yields the monotonicity of the solution operator to 3.1. Remark 3.. It follows from 3.3 that the sequence {u n } n=1 in the proof of Theorem 3.1 converges to u = u ɛ uniformly. In fact, the iteration procedure u n+1 = T u n started by any bounded and continuous function u 0 converges uniformly to the uniquely given u ɛ. We further remark that if F is Lipschitz continuous then u ɛ is likewise Lipschitz, with Lipschitz constant depending in nondecreasing manner on the following quantities: 1/ɛ, F C D and the Lipschitz constant of F D. We prove the following first convergence result. Our argument will be analytical, although a probabilistic proof is possible as well, based on the interpretation of u ɛ in Theorem 4.1. Theorem 3.3. Let F C R N be a bounded data function that satisfies on some open set U, compactly containing D: p F = 0 and F 0 in U. 3.4 Then the solutions u ɛ of 3.1 converge to F uniformly in R N, namely: u ɛ F CD Cɛ as ɛ 0, 3.5 with a constant C depending on F, U, D and p, but not on ɛ. Proof. 1. We first note that since u ɛ = F on R N \ D by construction, 3.5 indeed implies the uniform convergence of u ɛ in R N. Also, by translating D if necessary, we may without loss of generality assume that B0, 1 U =. We now show that there exists s and ˆɛ > 0 such that the following functions: satisfy, for every ɛ 0, ˆɛ: v ɛ x = F x + ɛ x s v ɛ 0 and p v ɛ ɛs v ɛ p in D. 3.6

11 RANDOM TUG OF WAR GAMES FOR THE p-laplacian 11 Observe first the following direct formulas: x s = s x s x, x s = ss x s 4 x + s x s Id N, x s = ss + N x s. Fix x D and denote a = v ɛ x and b = F x. Then, by 3.4 we have: p v ɛ x = v ɛ x ɛ x p s + p ɛ x s : a a + p F x : a b a b v ɛ x p ɛs x s s + N + p 1 4 F x x. F x Above, we have also used the bound: x s : a = ss x s a a a, x x together with the straightforward estimate: a a b b follows by fixing a large exponent s that satisfies: s 3 N + p max y D { + s x s s x s, a b b. The claim 3.6 hence 4 y F y F y }, 3.8 so that the quantity in the last line parentheses in 3.7 is greater than 1, and further taking ɛ > 0 small enough to have: min D v ɛ > 0.. We claim that s and ˆɛ in step 1 can further be chosen in a way that for all ɛ 0, ˆɛ: v ɛ S ɛ v ɛ in D. 3.9 Indeed, a careful analysis of the remainder terms in the expansion.1 reveals that: where: ɛ v ɛ x S ɛ v ɛ x = p 1 v ɛx p p v ɛ x + R ɛ, s, 3.10 R ɛ, s C p ɛ osc Bx,1+γpɛ v ɛ + Cɛ 3. Above, we denoted by C p a constant depending only on p, whereas C is a constant depending only v ɛ and v ɛ, that remain uniformly bounded for small ɛ. Since v ɛ is the sum of the smooth on U function x ɛ x s, and a p-harmonic function F that is also smooth in virtue of its non vanishing gradient this is a classical result [9], we obtain that 3.10 and 3.6 imply 3.9 for s sufficiently large and taking ɛ appropriately small. 3. Let A be a compact set in: D A U. Fix ɛ 0, ˆɛ and for each x A consider: By 3.9 and 3.1 we get: φ ɛ x = v ɛ x u ɛ x = F x u ɛ x + ɛ x s. φ ɛ x = d ɛ xv ɛ x S ɛ u ɛ x + 1 d ɛ xv ɛ x F x d ɛ xs ɛ v ɛ x S ɛ u ɛ x + 1 d ɛ xv ɛ x F x d ɛ x sup y B0,1 f φɛ x + ɛy, x, ɛ + 1 dɛ x v ɛ x F x. 3.11

12 1 MARTA LEWICKA Define: M ɛ = max φ ɛ. A We claim that there exists x 0 A with d ɛ x 0 < 1 and such that φ ɛ x 0 = M ɛ. To prove the claim, define D ɛ = { x D; distx, D ɛ}. We can assume that the closed set D ɛ {φ ɛ = M ɛ } is nonempty; otherwise the claim would be obvious. Let D0 ɛ be a nonempty connected component of D ɛ and denote DM ɛ = Dɛ 0 {φ ɛ = M ɛ }. Clearly, DM ɛ is closed in Dɛ 0 ; we now show that it is also open. Let x DM ɛ. Since d ɛx = 1 from 3.11 it follows that: y x M ɛ = φ ɛ x sup A φ ɛ ; γ p ɛ + a p 1 y Bx,ɛ ɛ, y x y M ɛ. y x Consequently, φ ɛ M ɛ in Bx, γ p ɛ and thus we obtain the openness of DM ɛ in Dɛ 0. In particular, DM ɛ contains a point x Dɛ. Repeating the previous argument for x results in φ ɛ M ɛ in B x, γ p ɛ, proving the claim. 4. We now complete the proof of Theorem 3.3 by deducing a bound on M ɛ. If M ɛ = φ ɛ x 0 for some x 0 D with d ɛ x 0 < 1, then 3.11 yields: M ɛ = φ ɛ x 0 d ɛ x 0 M ɛ + 1 d ɛ x 0 v ɛ x 0 F x 0, which implies: M ɛ v ɛ x 0 F x 0 = ɛ x 0 s. On the other hand, if M ɛ = φ ɛ x 0 for some x 0 A \ D, then d ɛ x 0 = 0, hence likewise: M ɛ = φ ɛ x 0 = v ɛ x 0 F x 0 = ɛ x 0 s. In either case: max D u u ɛ max φ ɛ + Cɛ Cɛ where C = max x V x s is independent of ɛ. A symmetric argument applied to u after noting that u ɛ = u ɛ gives: min Du u ɛ Cɛ. The proof is done. D 4. The random Tug of War game modelled on.1 We now develop the probability setting related to the expansion Let Ω 1 = B0, 1 {1, } 0, 1 and define: Ω = Ω 1 N = { ω = {w i, s i, t i } i=1; w i B0, 1, s i {1, }, t i 0, 1 for all i N }. The probability space Ω, F, P is given as the countable product of Ω 1, F 1, P 1. Here, F 1 is the smallest σ-algebra containing all products D S B where D B0, 1 R N and B 0, 1 are Borel, and S {1, }. The measure P 1 is the product of: the normalised Lebesgue measure on B0, 1, the uniform counting measure on {1, } and the Lebesgue measure on 0, 1: P 1 D S B = D B0, 1 S B. For each n N, the probability space Ω n, F n, P n is the product of n copies of Ω 1, F 1, P 1. The σ-algebra F n is always identified with the sub-σ-algebra of F, consisting of sets A i=n+1 Ω 1 for all A F n. The sequence {F n } n=0, where F 0 = {, Ω}, is a filtration of F.. Given are two families of functions σ I = {σi n} n=0 and σ II = {σii n } n=0, defined on the corresponding spaces of finite histories H n = R N R N Ω 1 n : σ n I, σ n II : H n B0, 1 R N,

13 RANDOM TUG OF WAR GAMES FOR THE p-laplacian 13 assumed to be measurable with respect to the target Borel σ-algebra in B0, 1 and the domain product σ-algebra on H n. For every x 0 R N and ɛ 0, 1 we recursively define: { X ɛ,x 0,σ I,σ II n : Ω R N} n=0. For simplicity of notation, we often suppress some of the superscripts ɛ, x 0, σ I, σ II and write X n or X x 0 n, or X σ I,σ II n, etc instead of X ɛ,x 0,σ I,σ II n, if no ambiguity arises. Let: X 0 x 0, X n w1, s 1, t 1,..., w n, s n, t n ɛ σ n 1 I h n 1 + γ p w n + γ p a p 1 w n, σ n 1 I h n 1 σ n 1 I h n 1 for s n = 1 = x n 1 + ɛ σ n 1 II h n 1 + γ p w n + γ p a p 1 w n, σ n 1 II h n 1 σ n 1 II h n 1 for s n =, where x n 1 = X n 1 w1, s 1, t 1,..., w n 1, s n 1, t n 1 and h n 1 = x 0, x 1, w 1, s 1, t 1,..., x n 1, w n 1, s n 1, t n 1 H n 1. In this game, the position x n 1 is first advanced deterministically according to the two players strategies σ I and σ II by a shift ɛy B0, ɛ, and then randomly uniformly by a further shift in the ellipse ɛe 0, γ p ; 1 + a p 1 y, y y. The deterministic shifts are activated by the value of the equally probable outcomes: s n = 1 activates σ I and s n = activates σ II. 3. The auxiliary variables t n 0, 1 serve as a threshold for reading the eventual value from the prescribed boundary data. Let D R N be an open, bounded and connected set. Define the random variable τ ɛ,q 0,σ I,σ II : Ω N { } in: where: τ ɛ,x 0,σ I,σ II w1, s 1, t 1, w, s, t,... = min { n 1; t n > d ɛ x n 1 }, d ɛ x = 1 ɛ min { ɛ, distx, R N \ D } is the scaled distance from the complement of D. As before, we drop the superscripts and write τ instead of τ ɛ,x 0,σ I,σ II if there is no ambiguity. By an easy classical argument, τ is a stopping time relative to the filtration {F} n=0, satisfying: Pτ < = 1. Our game is thus terminated, with probability 1 d ɛ x n 1, when the position x n 1 reaches the ɛ-neighbourhood of D. The first player collects then from his opponent the payoff given by the data F at the stopping position. The incentive of the collecting player to maximize the outcome and of the disbursing player to minimize it, leads to the definition of the two game values below. 4. Given a continuous function F : R N R, define the functions: u ɛ Ix = sup inf E [F X ɛ,x,σ ] I,σ II, σ I σ II τ ɛ,x,σ I,σ II 1 u ɛ IIx = inf sup E [F X ɛ,x,σ ] I,σ II. σ II τ ɛ,x,σ I,σ II 1 σ I The main result in Theorem 4.1 will show that u ɛ I = uɛ II CRN coincide with the unique solution to the dynamic programming principle in section 3, modelled on the expansion.1. It is also clear that u ɛ I,II depend only on the values of F in the ɛ-neighbourhood of D. In section 5 we will prove that as ɛ 0, the uniform limit of u ɛ I,II that depends only on F D, is p-harmonic in D and attains F on D, provided that D is regular

14 14 MARTA LEWICKA Theorem 4.1. For every ɛ 0, 1, let u ɛ I, uɛ II be as in 4. and u ɛ as in Theorem 3.1. Then: u ɛ I = u ɛ = u ɛ II. Proof. 1. We drop the sub/superscript ɛ to ease the notation. To show that u II u, fix x 0 R N and η > 0. We first observe that there exists a strategy σ 0,II where σ n 0,II h n = σ n 0,II x n satisfies for every n 0 and h n H n : f u x n + ɛσ n 0,IIx n ; x n, ɛ inf f ux n + ɛz; x n, ɛ + η z B0,1 n Indeed, using the continuity of., we note that there exists δ > 0 such that: inf f ux + ɛz; x, ɛ inf f u x + ɛz; x, ɛ < η z B0,1 z B0,1 n+ for all x x < δ. Let {Bx i, δ} i=1 be a locally finite covering of RN. For each i = 1..., choose z i B0, 1 satisfying: inf z B01 f u x i + ɛz; x i, ɛ f u x i + ɛz i ; x i, ɛ < η. Finally, define: n+ σ n 0,IIx = z i for x Bx i, δ \ i 1 j=1 Bx j, δ. The piecewise constant function σ n 0,II is obviously Borel and it satisfies Fix a strategy σ I and consider the following sequence of random variables M n : Ω R: M n = u X n 1 τ>n + F X τ 1 1 τ n + η n. We show that {M n } n=0 is a supermartingale with respect to the filtration {F n} n=0. Clearly: E M n F n 1 = E u Xn 1 τ>n F n 1 + E F Xn 1 1 τ=n F n 1 + E F X τ 1 1 τ<n F n 1 + η n a.s. We readily observe that: E F X τ 1 1 τ<n F n 1 = F Xτ 1 1 τ<n. 1 τ=n = 1 τ n 1 tn>dɛx n 1, it follows that: E F X n 1 1 τ=n F n 1 = E 1tn>d ɛx n 1 F n 1 F Xn 1 1 τ n = 1 d ɛ x n 1 F X n 1 1 τ n a.s. Similarly, since 1 τ>n = 1 τ n 1 tn d ɛx n 1, we get in view of 4.3: E u X n 1 τ>n F n 1 = E u Xn F n 1 dɛ x n 1 1 τ n ˆ = u X n dp 1 d ɛ x n 1 1 τ n Ω 1 = 1 A u; γ p ɛ, 1 + a p 1 σ n 1 I, σn 1 I x n 1 + ɛσ n 1 I + A u; γ p ɛ, 1 + a p 1 σ n 1 S X n 1 + η n dɛ x n 1 1 τ n σ n 1 I 0,II, σn 1 0,II σ n 1 0,II a.s. 4.4 Further, writing x n 1 + ɛσ n 1 0,II d ɛ x n 1 1 τ n

15 RANDOM TUG OF WAR GAMES FOR THE p-laplacian 15 Concluding, by 3.1 the decomposition 4.4 yields: E M n F n 1 d ɛ x n 1 S X n dɛ x n 1 F X n 1 1 τ n + F X τ 1 1 τ n 1 + η n 1 = M n 1 a.s. 3. The supermartingale property of {M n } n=0 being established, we conclude that: ux 0 + η = E [ ] [ ] [ ] η M 0 E Mτ = E F Xτ 1 + τ. Thus: u II x 0 sup E [ F X σ ] I,σ II,0 τ 1 ux0 + η. σ I As η > 0 was arbitrary, we obtain the claimed comparison u II x 0 ux 0. For the reverse inequality ux 0 u I x 0, we use a symmetric argument, with an almost-maximizing strategy σ 0,I and the resulting submartingale M n = u X n 1 τ>n + F X τ 1 1 τ n η, along a given n yet arbitrary strategy σ II. The obvious estimate u I x 0 u II x 0 concludes the proof. 5. Convergence of u ɛ and game-regularity Towards checking convergence of the family {u ɛ } ɛ 0, we first show that its equicontinuity is implied by the equicontinuity at D. This last property will be, in turn, implied by the game-regularity condition, which in the context of stochastic Tug of War games has been introduced in [11]. Below, we present an analytical proof. A probabilistic argument could be carried out as well, based on a game translation argument. Let D R N be an open, bounded connected domain and let F CR N be a bounded data function. We have the following: Theorem 5.1. Let {u ɛ } ɛ 0 be the family of solutions to 3.1. Assume that for every η > 0 there exists δ > 0 and ˆɛ 0, 1 such that for all ɛ 0, ˆɛ there holds: u ɛ y 0 u ɛ x 0 η for all y 0 D, x 0 D satisfying x 0 y 0 δ. 5.1 Then the family {u ɛ } ɛ 0 is equicontinuous in D. Proof. For every small ˆδ > 0, define the open, bounded, connected set Dˆδ and the distance: Dˆδ = { q D; distq, R N \ D > ˆδ } and dˆδ ɛ q = 1 ɛ min{ɛ, distq, RN \ Dˆδ}. Fix η > 0. In view of 5.1 and since without loss of generality the data function F is constant outside of some large bounded superset of D in R N, there exists ˆδ > 0 satisfying: u ɛ x + z u ɛ x η for all x R N \ Dˆδ, w ˆδ, ɛ 0, ˆɛ. 5. Fix x 0, y 0 D with x 0 y 0 ˆδ ˆδ and let ɛ 0,. Consider the following function ũ ɛ CR N : ũ ɛ x = u ɛ x x 0 y 0 + η. Then, by 3.1 and recalling the definition of the principal averaging operator S ɛ, we get: S ɛ ũ ɛ x = S ɛ u ɛ x x 0 y 0 + η = u ɛ x x 0 y 0 + η = ũ ɛ x for all x Dˆδ. 5.3 because in Dˆδ there holds: distx x 0 y 0, R N \ D distx, R N \ D x 0 y 0 ˆδ ˆδ = ˆδ > ɛ.

16 16 MARTA LEWICKA It follows now from 5.3 that: ũ ɛ = dˆδ ɛ S ɛ ũ ɛ + 1 dˆδ ɛ ũɛ in R N. On the other hand, u ɛ itself similarly solves the same problem above, subject to its own data u ɛ on R N \Dˆδ. Since for every x R N \Dˆδ we have: ũ ɛ x u ɛ x = u ɛ x x 0 y 0 u ɛ x+η 0 in view of 5., the monotonicity property in Theorem 3.1 yields: u ɛ ũ ɛ in R N. Thus, in particular: u ɛ x 0 u ɛ y 0 η. Exchanging x 0 with y 0 we get the opposite inequality, and hence u ɛ x 0 u ɛ y 0 η, establishing the claimed equicontinuity of {u ɛ } ɛ 0 in D. Following [11], we say that a point x 0 D is game-regular if, whenever the game starts near x 0, one of the players has a strategy for making the game terminate still near x 0, with high probability. More precisely: Definition 5.. Consider the Tug of War game with noise in 4.1 and 4.. a We say that a point x 0 D is game-regular if for every η, δ > 0 there exist ˆδ 0, δ and ˆɛ 0, 1 such that the following holds. Fix ɛ 0, ˆɛ and x Bx 0, ˆδ; there exists then a strategy σ 0,I with the property that for every strategy σ II we have: P X ɛ,x,σ 0,I,σ II τ 1 Bx 0, δ 1 η. 5.4 b We say that D is game-regular if every boundary point x 0 D is game-regular. Remark 5.3. If condition b holds, then ˆδ and ˆɛ in part a can be chosen independently of x 0. Also, game-regularity is symmetric with respect to σ I and σ II. Lemma 5.4. Assume that for every bounded data F CR N, the family of solutions {u ɛ } ɛ 0 of 3.1 is equicontinuous in D. Then D is game-regular. Proof. Fix x 0 D and let η, δ 0, 1. Define the data function: F x = min { 1, x x 0 }. By assumption and since u ɛ x 0 = F x 0 = 0, there exists ˆδ 0, δ and ˆɛ 0, 1 such that: Consequently: u ɛ x < ηδ for all x Bx 0, ˆδ and ɛ 0, ˆɛ. sup inf E [ F X ɛ,x ] τ 1 = u ɛ σ I σ I x > ηδ, II and thus there exists σ 0,I with the property that: E [ F X ɛ,x,σ ] 0,I,σ II τ 1 > ηδ for every strategy σ II. Then: P X τ 1 Bx 0, δ 1 ˆ F X τ 1 dp < η, δ proving 5.4 and hence game-regularity of x 0. Theorem 5.5. Assume that D is game-regular. Then, for every bounded data F CR N, the family {u ɛ } ɛ 0 of solutions to 3.1 is equicontinuous in D. Ω

17 RANDOM TUG OF WAR GAMES FOR THE p-laplacian 17 Proof. In virtue of Theorem 5.1 it is enough to validate the condition 5.1. To this end, fix η > 0 and let δ > 0 be such that: F x F x 0 η 3 for all x 0 D and x Bx 0, δ. 5.5 By Remark 5.3 and Definition 5., we may choose ˆδ 0, δ and ˆɛ 0, δ such that for every ɛ 0, ˆɛ, every x 0 D and every x Bx 0, ˆδ, there exists a strategy σ 0,II with the property that for every σ I there holds: P X ɛ,x,σ I,σ 0,II τ 1 Bx 0, δ η F CR N +1 Let x 0 D and y 0 D satisfy x 0 y 0 ˆδ. Then: u ɛ y 0 u ɛ x 0 = u ɛ IIy 0 F x 0 sup E [ F X ɛ,y 0,σ I,σ 0,II τ 1 F x 0 ] σ I E [ F X ɛ,y 0,σ 0,I,σ 0,II τ 1 F x 0 ] + η 3, for some almost-supremizing strategy σ 0,I. Thus, by 5.5 and 5.6: ˆ u ɛ y 0 u ɛ x 0 F X τ 1 F x 0 dp {X τ 1 Bx 0,δ} ˆ + {X τ 1 Bx 0,δ} F X τ 1 F x 0 dp + η 3 η 3 + F CR N P X τ 1 Bx 0, δ + η 3 η. The remaining inequality u ɛ y 0 u ɛ x 0 > η is obtained by a reverse argument. 6. The exterior corkscrew condition as sufficient for game-regularity Definition 6.1. We say that a given boundary point x 0 D satisfies the exterior corkscrew condition provided that there exists µ 0, 1 such that for all sufficiently small r > 0 there exists a ball Bx, µr such that: The main result of this section is: Bx, µr Bx 0, r \ D. Theorem 6.. If x 0 D satisfies the exterior corkscrew condition, then x 0 is game-regular. Towards the proof, we first recall a useful result on concatenating strategies, which proposes a condition equivalent to the game-regularity criterion in Definition 5. a. This result has been proved with little detail in [11], we thus reprove it for the convenience of the reader. Let D H be an open, bounded, connected domain. Theorem 6.3. For a given x 0 D, assume that there exists θ 0 0, 1 such that for every δ > 0 there exists ˆδ 0, δ and ˆɛ 0, 1 with the following property. Fix ɛ 0, ˆɛ and choose an initial position x 0 Bx 0, ˆδ; there exists a strategy σ 0,II such that for every σ I we have: Then x 0 is game-regular. P n < τ X n Bx 0, δ θ

18 18 MARTA LEWICKA Proof. 1. Under condition 6.1, construction of an optimal strategy realising the arbitrarily small threshold η in 5.4 is carried out by concatenating the m optimal strategies corresponding to the achievable threshold η 0, on m concentric balls, where 1 η 0 m = 1 θ m 0 1 η. Fix η, δ > 0. We want to find ˆɛ and ˆδ such that 5.4 holds. Observe first that for θ 0 η the claim follows directly from 6.1. In the general case, let m {, 3,...} be such that: θ m 0 η. 6. Below we inductively define the radii {δ k } m k=1, together with the quantities {ˆδδ k } m k=1, {ˆɛδ k} m k=1 from the assumed condition 6.1. Namely, for every initial position in Bx 0, ˆδδ k in the Tug of War game with step less than ˆɛδ k, there exists a strategy σ 0,II,k guaranteeing exiting Bx 0, δ k before the process is stopped with probability at most θ 0. We set δ m = δ and find ˆδδ m 0, δ and ˆɛδ m 0, 1, with the indicated choice of the strategy σ 0,II,m. Decreasing the value of ˆɛδ m if necessary, we then set: δ m 1 = ˆδδ m 1 + γ p ˆɛδ m > 0. Similarly, having constructed δ k > 0, we find ˆδδ k 0, δ k and ˆɛδ k 0, ˆɛδ k+1 and define: Eventually, we call: δ k 1 = ˆδδ k 1 + γ p ˆɛδ k > 0. ˆδ = ˆδδ 1, ˆɛ = ˆɛδ 1. To show that the condition of game-regularity at x 0 is satisfied, we will concatenate the strategies {σ 0,II,k } m k=1 by switching to σ 0,II,k+1 immediately after the token exits Bx 0, δ k Bx 0, ˆδδ k+1. This construction is carried out in the next step.. Fix y 0 Bx 0, ˆδ and let ɛ 0, ˆɛ. Define the strategy σ 0,II : σ n 0,II = σ n 0,II x0, x 1, w 1, s 1, t 1,..., x n, w n, s n, t n for all n 0, separately in the following two cases. Case 1. If x k Bx 0, δ 1 for all k n, then we set: σ n 0,II = σ n 0,II,1 x0, x 1, w 1, s 1, t 1,..., x n, w n, s n, t n. Case. Otherwise, define: k =. { } kx 0, x 1,..., x n = max 1 k m 1; 0 i n q i B δk q 0 i =. { } min 0 i n; q i Bx 0, δ k. and set: σ n 0,II = σ n i 0,II,k+1 xi, x i+1, w i+1, s i+1, t i+1,..., x n, w n, s n, t n. It is not hard to check that each σ n 0,II : H n B0, 1 R N is Borel measurable, as required. Let σ I be now any opposing strategy. In the auxiliary Lemma 6.4 below we will show that: P n < τ X n Bq 0, δ k θ 0 P n < τ X n Bq 0, δ k 1 for all k =... m, 6.3 Consequently: P n < τ X n Bq 0, δ θ m 1 0 P n τ x n Bq 0, δ 1 θ m 0, which yields the result by 6. and completes the proof.

19 RANDOM TUG OF WAR GAMES FOR THE p-laplacian 19 The inductive bound 6.3 is quite straightforward; we produce a precise argument for the sake of the reader less familiar with probabilistic arguments: Lemma 6.4. In the context of the proof of Theorem 6.3, we have 6.3. Proof. 1. Denote: Ω = { n τ X n By 0, δ k 1 } Ω. Since: P n τ X n Bx 0, δ k P n τ X n Bx 0, δ k 1, it follows that if P Ω = 0 then 6.3 holds trivially. For P Ω > 0, we define the probability space Ω, F, P by: F = { A Ω; A F } and PA = PA P Ω for all A F. Define also the measurable space Ω fin, F fin, by setting Ω fin = n=1 Ω n and by taking F fin to be the smallest σ-algebra containing n=1 F n. Finally, consider the random variables: Y 1 : Ω Ω fin Y 1 {wn, s n, t n } n=1 = {wn, s n, t n } τ k n=1 Y : Ω Ω Y {wn, s n, t n } n=1 = {wn, s n, t n } n=τ k +1, where τ k is the following stopping time on Ω: τ k = min { n 1; X n Bx 0, δ k 1 }. We claim that Y 1 and Y are independent. Indeed, given n, m N and A 1 F n, A F m : P Y 1 A 1 = Pn A 1 {τ k = n} { ti d ɛ x i 1 }, i<n P Y A = P Ω Pm A P {Y 1 A 1 } {Y A } = P n A 1 {τ k = n} i<n { ti d ɛ x i 1 } P m A. This implies the following property equivalent to the claimed independence: P Ω P {Y 1 A 1 } {Y A } = P Y 1 A 1 P Y A. Consequently, Fubini s theorem yields for every random variable Z : Ω fin Ω R +, that is measurable with respect to the product σ-algebra of F fin and F: ˆ Z Y 1 ω, Y ω ˆ ˆ d Pω = Z Y 1 ω 1, Y ω d Pω d Pω Ω Ω Ω. We now apply 6.4 to the indicator random variable: Z {w i, s i, t i } n i=1, {w i, s i, t i } i=n+1 = 1 { n τ Xn {w i, s i, t i } i=1 Bx 0, δ k }, to the effect that: P n τ X n Bx 0, δ k ˆ = fω 1 d Pω 1, 6.5 Ω where for a given ω 1 = {w n, s n, t n } n=1 Ω, the integrand function f returns: fω 1 = P { w n, s n, t n } n=1 Ω; n τ X n {wi, s i, t i } τ k i=1, { w i, s i, t i } i=τ k +1 Bx0, δ k = P { w n, s n, t n } n=1 Ω; n τ X xτ k,σ I,σ 0,II,k n { wi, s i, t i } i=τ k +1 Bx0, δ k.

20 0 MARTA LEWICKA Since x τk Bx 0, ˆδδ k, by 6.1 it follows that: fω 1 = P n τ X xτ k,σ I,σ 0,II,k n Bx 0, δ k P Ω θ 0 P Ω for P-a.e. ω 1 Ω. In conclusion, 6.5 implies 6.3 and completes the proof. The proof of game-regularity in Theorem 6. will be based on the concatenating strategies technique in the proof of Theorem 6.3 and the analysis of the annulus walk below. Namely, one needs to derive an estimate on the probability of exiting a given annular domain D through the external portion of its boundary. It follows [11] that when the ratio of the annulus thickness and the distance of the initial token position from the internal boundary is large enough, then this probability may be bounded by a universal constant θ 0 < 1. When p N, then θ 0 converges to 0 as the indicated ratio goes to. Theorem 6.5. For given radii 0 < R 1 < R < R 3, consider the annulus D = B0, R 3 \ B0, R 1 R N. For every ξ > 0, there exists ˆɛ 0, 1 depending on R 1, R, R 3 and ξ, p, N, such that for every x 0 D B0, R and every ɛ 0, ˆɛ, there exists a strategy σ 0,II with the property that for every strategy σ I there holds: Here, v : 0, R is given by: { P X τ 1 B0, R 3 ɛ vt = vr vr 1 + ξ. 6.6 vr 3 vr 1 sgnp N t p N p 1 for p N log t for p = N, and { X n = X ɛ,x 0, σ I, σ 0,II n } n=0 and τ = τ ɛ,x 0, σ I, σ 0,II denote, as before, the random variables corresponding to positions and stopping time in the random Tug of War game on D. Proof. Consider the radial function u : R N \ {0} R given by ux = v x, where v is as in 6.7. Recall that: p u = 0 and u 0 in R N \ {0}. 6.8 Let ũ ɛ be the family of solutions to 3.1 with the data F provided by a smooth and bounded modification of u outside of the annulus B, R 3 \ B0, R 1. By Theorem 3.3, there exists a constant C > 0, depending only on p, u and D, such that: ũ ɛ u C D Cɛ as ɛ For a given x 0 D B R 0, there exists thus a strategy σ 0,II so that for every σ I we have: We now estimate: E [ ˆ ] u X τ 1 ux0 = E [ u X ɛ,x 0, σ I, σ 0,II τ 1 ] ux0 Cɛ. 6.9 { X τ 1 B0,R 3 ɛ} ˆ u X τ 1 dp + u X τ 1 dp ux 0 { X τ 1 B0,R 1 +ɛ} P X τ 1 B0, R 3 ɛ v R 3 ɛ + 1 P X τ 1 B0, R 3 ɛ v R 1 γ p ɛ vr, where we used the fact that v in 6.7 is an increasing function. Recalling 6.9, this implies: P X τ 1 B0, R 3 ɛ vr vr 1 γ p ɛ + Cɛ vr 3 ɛ vr 1 γ p ɛ. 6.10

21 RANDOM TUG OF WAR GAMES FOR THE p-laplacian 1 The proof of 6.6 is now complete, by continuity of the right hand side with respect to ɛ. By inspecting the quotient in the right hand side of 6.6 we obtain: Corollary 6.6. The function v in 6.7 satisfies, for any fixed 0 < R 1 < R : R p 1 a b lim R 3 lim M vr vr 1 vr 3 vr 1 = vmr 1 vr 1 vm R 1 vr 1 = p N 1 for 1 < p < N R 1 0 for p N, 1 for p = N 0 for p > N. Consequently, the estimate 6.6 can be replaced by: P X τ 1 B0, R 3 ɛ θ valid for any θ 0 > 1 R p 1 if p 1, N, and any θ 0 > 0 if p N, upon choosing R 3 sufficiently large with respect to R 1 and R. Alternatively, when p > N, the same bound with arbitrarily small θ 0 can be achieved by setting R = MR 1, R 3 = M R 1, with M large enough. R 1 p N The results of Theorem 6.5 and Corollary 6.6 are invariant under scaling, i.e.: Remark 6.7. The bounds 6.6 and 6.11 remain true if we replace R 1, R, R 3 by rr 1, rr, rr 3, the domain D by r D and ˆɛ by rˆɛ, for any r > 0. Figure. Positions of the concentric balls By 0, and Bx 0, in the proof of Theorem 6..

22 MARTA LEWICKA Proof of Theorem 6.. With the help of Theorem 6.5, we will show that the assumption of Theorem 6.3 is satisfied, with probability θ 0 < 1 depending only on p, N and µ 0, 1 in Definition 6.1. Namely, set R 1 = 1, R = µ and R 3 > R according to Corollary 6.6 a in order to have θ 0 = θ 0 p, N, R 1, R < 1. Further, set r = δ R 3 so that rr = δ µr 3. Using the corkscrew condition, we obtain: By 0, rr 1 Bx 0, δ µr 3 \ D, for some y 0 R N. In particular: x 0 y 0 < rr, so x 0 By 0, rr \ By 0, rr 1. It now easily follows that there exists ˆδ 0, δ with the property that: Bx 0, ˆδ By 0, rr \ By 0, rr 1. Finally, we observe that By 0, rr 3 Bx 0, δ because rr 3 + x 0 y 0 < rr 3 +rr < rr 3 = δ. Let ˆɛ/r > 0 be as in Theorem 6.5, applied to the annuli with radii R 1, R, R 3, in view of Remark 6.7. For a given x Bx 0, ˆδ and ɛ 0, ˆɛ, let σ 0,II be the strategy ensuring validity of the bound 6.11 in the annulus walk on y 0 + D. For a given strategy σ I there holds: { ω Ω; n < τ ɛ,x,σ I,σ 0,II ω { ω Ω; X ɛ,x,σ I,σ 0,II n ω Bx 0, δ The final claim follows by 6.11 and by applying Theorem 6.3. Xɛ,x, σ I, σ 0,II τ 1 ω By 0, rr 3 ɛ Remark 6.8. Using Corollary 6.6 b one can show that every open, bounded domain D R N is game-regular for p > N. The proof mimics the argument of [11] for the process based on the mean value expansion.8, so we omit it. 7. Uniqueness and identification of the limit in Theorem 5.5 Let F CR N be a bounded data function and let D be open, bounded and game-regular. In virtue of Theorem 5.5 and the Ascoli-Arzela theorem, every sequence in the family {u ɛ } ɛ 0 of solutions to 3.1 has a further subsequence converging uniformly to some u CR N and satisfying u = F on R N \ D. We will show that such limit u is in fact unique. Recall first the definition of the p-harmonic viscosity solution: Definition 7.1. We say that u C D is a viscosity solution to the problem: if the latter boundary condition holds and if: i for every x 0 D and every φ C D such that: p u = 0 in D, u = F on D, 7.1 φx 0 = ux 0, φ < u in D \ {x 0 } and φx 0 0, 7. there holds: p φx 0 0, ii for every x 0 D and every φ C D such that: φx 0 = ux 0, φ > u in D \ {x 0 } and φx 0 0, there holds: p φx 0 0. Theorem 7.. Assume that the sequence {u ɛ } ɛ J,ɛ 0 of solutions to 3.1 with a bounded data function F CR N, converges uniformly as ɛ 0 to some limit u CR N. Then u must be the viscosity solution to 7.1. } }.

23 RANDOM TUG OF WAR GAMES FOR THE p-laplacian 3 Proof. 1. Fix x 0 D and let φ be a test function as in 7.. We first claim that there exists a sequence {x ɛ } ɛ J D, such that: lim x ɛ = x 0 and u ɛ x ɛ φx ɛ = min u ɛ φ. 7.3 ɛ 0,ɛ J To prove the above, for every j N define η j > 0 and ɛ j > 0 such that: η j = min D\Bx 0, 1 j u φ and u ɛ u C D 1 η j for all ɛ ɛ j. D Without loss of generality, the sequence {ɛ j } j=1 is decreasing to 0 as j. ɛ ɛ j+1, ɛ j ] J, let x ɛ Bx 0, 1 j satisfy: u ɛ q ɛ φq ɛ = min u ɛ φ. Bx 0, 1 j Now, for Observing that the following bound is valid for every q D \ Bx 0, 1 j, proves 7.3: u ɛ q φq uq φq u ɛ u C D η j 1 η j u ɛ u C D u ɛ q 0 φq 0 min Bx 0, 1 j u ɛ φ.. Since by 7.3 we have: φx u ɛ x + φx ɛ u ɛ x ɛ for all x D, it follows that: S ɛ φx ɛ φx ɛ S ɛ ux ɛ + φx ɛ u ɛ x ɛ φx ɛ = 0, 7.4 for all ɛ small enough to guarantee that d ɛ x ɛ = 1. On the other hand,.1 yields: S ɛ φx ɛ φx ɛ = ɛ p 1 φx ɛ p p φx ɛ + oɛ, for ɛ small enough to get φx ɛ 0. Combining the above with 7.4 gives: p φx ɛ o1. Passing to the limit with ɛ 0, ɛ J establishes the desired inequality p φx 0 0 and proves part i of Definition 7.1. The verification of part ii is done along the same lines. Since the viscosity solutions u C D of 7.1 are unique see a direct proof in [7, Appendix, Lemma 4.], in view of Theorem 7. and Theorem 5.5 we obtain: Corollary 7.3. Let F CR N be a bounded data function and let D be open, bounded and game-regular. The family {u ɛ } ɛ 0 of solutions to 3.1 converges uniformly in D to the unique viscosity solution of 7.1. References [1] Arroyo, A., Heino, J. and Parvianinen, M., Tug-of-war games with varying probabilities and the normalized px-laplacian, Commun. Pure Appl. Anal. 163, , 017. [] Doob, J.L., Classical potential theory and its probabilistic counterpart, Springer-Verlag, New York, [3] Heinonen, J., Kilpeläinen, T. and Martio, O., Nonlinear potential theory of degenerate elliptic equations, Dover Publications, 006. [4] Kallenberg, O., Foundations of modern probability, Probability and Its Applications, nd edition, Springer, 00. [5] Kawohl, B., Variational versus PDE-based approaches in mathematical image processing, CRM Proceedings and Lecture Notes 44, , 008. [6] Kohn, R.V. and Serfaty, S., A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math. 593, , 006.