Introduction to Random Tug-of-WarGames and PDEs

Juan J. Manfredi

1 Introduction

The fundamental contributions of Kolmogorov, Ito, Kakutani, Doob, Hunt,Levy, and many others have shown the profound and powerful connectionbetween classical linear potential theory and probability theory. The ideabehind the classical interplay is that harmonic functions and martingalesshare a common cancelation property that can be expressed by using meanvalue properties. In these lectures, we will see how this approach turns outto very useful in the nonlinear theory as well.

The objective of this course is to provide an introduction to the connectionbetween the theory of stochastic tug-of-war games and non-linear equations ofp-Laplacian type in the Euclidean and discrete cases. These notes will providethe student with background to read [16] and [17]. Most of the material ebased on the joint papers [11–13] with Mikko Parviainen and Julio Rossi, andon the 2010 doctoral thesis of Alexander Sviridov [18]. I am grateful to Alexfor correcting many misprints and for suggesting changes that have improvedthe readability of the manuscript.

2 Probability Background

We present a quick introduction to Doob’s Optimal Sampling Theorem andto Kolmogorov’s construction of infinite product of measures. We follow thepresentation in the book [21]. We refer to [21] for the basic probabilitydefinitions and the proofs. However, we have chosen to emphasize certain

J.J. Manfredi (�)Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260e-mail: manfredi@pitt.edu

J. Lewis et al., Regularity Estimates for Nonlinear Elliptic and ParabolicProblems, Lecture Notes in Mathematics 2045, DOI 10.1007/978-3-642-27145-8 3,© Springer-Verlag Berlin Heidelberg 2012

133

134 J.J. Manfredi

details in the proofs that are likely to help student of analysis going throughthis material for the first time.

We are given a set Ω endowed with a σ-algebra F and probability measureP on F . The triplet (Ω,F ,P) is a probability space.

Definition 2.1. If G ⊂ F is a sub-σ-algebra of F and f ∈ L1(Ω,P) theconditional expectation of f given G is the only G-measurable function g =E[f |G] such that ∫

A

f dP =

∫A

g dP

for all sets A ∈ G.Example 2.2. Suppose that we have a finite partition Ω = A1∪A2 ∪· · ·∪An

where all the Ai ∈ F have positive probability P(Ai) > 0. Let G the σ-algebragenerated by this partition. For f ∈ L1(Ω,P), the function g = E[f |G] has aconstant value gi on each Ai given by

gi =

∫Ai

f dP

The notion of (discrete) martingale will be key to our developments later on.

Definition 2.3. Let Fi ⊂ Fi+1 ⊂ · · · ⊂ F be a filtration of σ algebras ofF and Xi : Ω �→ F be an Fi-measurable random variable (or Fi-measurablefunction.)

1. The sequence of random variables {Xi} is a martingale if Xi=E[Xi+1|Fi].2. The sequence of random variables {Xi} is a submartingale if Xi ≤

E[Xi+1|Fi].3. The sequence of random variables {Xi} is a supermartingale if Xi ≥

E[Xi+1|Fi].

These relations are supposed to hold a.e. with respect to P.

When needed we will make explicit the σ-algebras by writing {Xi,Fi}. If {Xi}is a martingale, we have E[Xi+1] = E[Xi] = · · · = E[X1] = c. The randomvariables Yi+1 = Xi+1 −Xi are the martingales differences. We clearly haveE[Yi] = 0.

Example 2.4. If X ∈ L1(Ω,P), the sequence Xi = [X |Fi] is a martingale. Itturns that this is the most general L1-martingale (see [21].)

Example 2.5. Let {Yi} be a collection of independent random variables withmean zero E[Yi] = 0. Set Fi = σ(Y1, Y2, . . . , Yi). Then, the sequence

Xi = c+i∑

j=1

Yj

is a martingale.

Introduction to Random Tug-of-War Games and PDEs 135

The following lemma is a consequence of Jensen’s inequality for conditionalexpectations.

Lemma 2.6. Let {Xi,Fi} be a martingale and 1 ≤ p < ∞. Then {|Xi|p,Fi}is a submartingale provided that E[|Xi|p] < ∞.

We present Doob’s weak type 1-1 inequality for finite martingales. The simpleproof contains all the ingredients of the more general cases.

Theorem 2.7. Let {Xi}ni=1 be a martingale and l > 0. Then we have

P

(ω : sup

1≤i≤n|Xi(ω)| ≥ l

)≤ 1

l

∫{sup |Xi|>l}

|Xn| dP

≤ 1

l

∫Ω

|Xn| dP.

Proof. The maximum random variable S is

S(ω) = sup{|Xi(ω) : 1 ≤ i ≤ n}.

For each 1 ≤ i ≤ n define the set describing the first time we go above l,

Ei = {ω : |X1(ω)| < l, |X1(ω)| < l, . . . , |Xi−1(ω)| < l, |Xi(ω)| ≥ l}

and the set when the maximum is above l

E = {ω : S(ω) ≥ l}.

We have a disjoint union

E =

n⋃1

Ei

and the basic estimate

P(Ei) ≤ 1

l

∫Ei

|Xi| dP. (1)

Since |Xi| is a submartingale we have E [ |Xn| | Fi] ≥ |Xi| for a.e. ω. SinceEi ∈ Fi we then have

E [χEi |Xn| | Fi] = χEiE [ |Xn| | Fi] ≥ χEi |Xi|.

Taking expectations we get

∫Ei

|Xn| dP ≥∫Ei

|Xi| dP

136 J.J. Manfredi

and using (1) we get

P(Ei) ≤ 1

l

∫Ei

|Xn| dP.

A well-known argument using distribution functions (see [21]) gives themaximal theorem

Corollary 2.8. (Doob’s maximal theorem) Let {Xi}ni=1 be a martingale and1 < p < ∞. For S = sup{|Xi : 1 ≤ i ≤ n} we have

E[Sp] ≤ (p

p− 1)pE[|Xn|p].

In addition to martingales, stopping times play a crucial role in stochasticgames.

Definition 2.9. The random variable τ : Ω �→ N∪{0,∞} is a stopping timewith respect to the filtration {Fi} if

{ω : τ(ω) ≤ n} ∈ Fn for all n ≥ 0.

Example 2.10. The following are examples of stopping times. The readerwould benefit from checking in detail that they are indeed stopping times.

1. τ(ω) = k, where k is constant.2. If τ is a stopping time and f is monotone such that f(t) ≥ t, then τ ′ = f ◦τ

is a stopping time.3. If τ1 and τ2 are stopping times so are max {τ1, τ2} and min {τ1, τ2}.We see then that the truncated stopping time τn = min{τ, n} is also astopping time, so that we have the following useful fact.

Corollary 2.11. Every stopping time is the limit of an increasing sequenceof bounded stopping times

We next formalize the notion of information available up to time τ .

Definition 2.12. Let τ be a stopping time respect to the filtration {Fi}.Set

Fτ = {A : A ∈ F and A ∩ {ω : τ(ω) ≤ n} ∈ Fn for all n}Lemma 2.13. (Basic properties of Fτ )

1. Fτ is a σ-field.2. If τ(ω) = k constant, then Fτ = Fk.3. τ1 ≤ τ2 =⇒ Fτ1 ⊂ Fτ2 .4. τ is Fτ measurable.5. {ω : τ(ω) < ∞} ∈ Fτ .

Introduction to Random Tug-of-War Games and PDEs 137

The last part follows from {ω : τ(ω) < ∞} =⋃

n{ω : τ(ω) ≤ n}. We stop amartingale by defining Xτ as follows

Xτ (ω) = Xτ(ω)(ω).

Lemma 2.14. The stopped martingale Xτ is Fτ measurable on the set whereτ is finite {ω : τ(ω) < ∞}.Proof. We need to check that for λ ∈ R we have

Eλ = {ω : τ(ω) < ∞, Xτω(ω) > λ} ∈ Fτ ,

which follows from the expression

Eλ =⋃n

({Xn(ω) > λ} ∩ {τ(ω) = n}

)

The cornerstone of the applications of martingale to partial differentialequations is Doob’s Optional Sampling Theorem.

Theorem 2.15. Let {Xn} be a submartingale with respect to the filtra-tion {Fn}. Let 0 ≤ τ1 ≤ τ2 be bounded stopping times. Then we have

E [Xτ2 | Fτ1 ] ≥ Xτ1 .

If {Xn} is supermartingale we get instead

E [Xτ2 | Fτ1 ] ≤ Xτ1 ,

and If {Xn} is martingale we get the equality

E [Xτ2 | Fτ1 ] = Xτ1 ,

By making τ1 equals to zero we get the following corollary.

Corollary 2.16. Let {Xn} be a martingale. For any stopping time we haveE[Xτ ] = E[X0].

Proof. Let us prove the theorem in the martingale case with τ1 = τ andτ2 = k. We need to establish that E[Xk | Fτ ] = Xτ , or equivalently that forall A ∈ Fτ we have ∫

A

Xk dP =

∫A

Xτ dP. (2)

Set Ei = {ω : τ(ω) = i} and decompose Ω = ∪k1Ei as a disjoint union.

Note that if A ∈ Fτ then A ∩ Ei ∈ Fi. We then have

138 J.J. Manfredi

∫A∩Ei

Xk dP =

∫A∩Ei

Xi dP =

∫A∩Ei

Xτ dP,

where the first equality follows by the martingale property and the secondby the definition of Ei. By adding in i we obtain the required (2).

We now change gears and consider the product of probability spaces.Suppose that we have two probability spaces (Ω1,B1,P1) and (Ω2,B2,P2).A rectangle is a set A = A1 ×A2 where A1 ∈ B1 and A2 ∈ B2. We denote byF the field of all finite disjoint unions of rectangles. The product σ-algebrais the σ algebra generated by F

B = B1 × B2 = σ(F).

We define the product probability P = P1 × P2 on B as follows:

(i) P(A1 ×A2) = P1(A1) · P2(A2) for rectangles A1 ×A2.(ii) Extend P to F as finitely additive measure.(iii) P is in fact countably additive on F .(iv) Extend P to B by using Caratheodory’s theorem.

A finite product of probability spaces is defined similarly. We next considerinfinite products. We are given probability measures Pn on (Rn,B(Rn)),where B(Rn)) is the Borel σ-algebra in R

n. The projection πn : Rn+1 �→ R

n

is given byπn(x1, x2, . . . , xn, xn+1) = (x1, x2, . . . , xn).

The family {Pn}n is consistent if for all n all rectangles A1 × . . . × An wehave

Pn+1(A1 × . . .×An × R) = Pn(A1 × . . .×An).

We also write this equation as Pn+1π−1n = Pn and say that the marginal

probability of Pn+1 on Rn is Pn.

From now on Ω will be the infinite cartesian product

Ω = R∞ = {ω : ω = (xn)n∈N}.

A cylinder with base A ∈ B(Rn) is a set of the form

C = {ω : (x1, x2, . . . , xn) ∈ A}.

The set of all cylinders form a field F . They generate a σ-algebra

Σ = σ(F).

Introduction to Random Tug-of-War Games and PDEs 139

Theorem 2.17. (Kolmogorov) Let {Pn} be a consistent family of probabil-ities in (Rn,B(Rn)). Then, there exists a unique probability in (Ω,Σ) suchthat

P π−1n = Pn.

That is, for all rectangles A1 ×A2 × . . .×An we have

P(A1 ×A2 × . . . An × R . . .× R× . . .) = Pn(A1 ×A2 × . . . An).

Proof. For a cylinder C ∈ F with base A ∈ B(Rn) define

P(C) = Pn(A).

The first observation is that P is well-defined in F by the consistencyhypothesis. The key point is to establish the P is countably additive on F .Then we can extend P to Σ by the Caratheodory’s procedure.

We need to show that if Bn ∈ F , Bn+1 ⊂ Bn, and ∩nBn = ∅, then

limn→∞P(Bn) = 0.

The proof is by contradiction. Suppose that for some δ > 0 we have thatP(Bn) > δ for all n ∈ N. Since Bn is a cylinder we write Bn = π−1

n (An).Suppose for the moment that all basis An are compact and write

Bn = An1 ×An

2 × . . .×Ann × R× R . . .

Bn+1 = An+11 ×An+1

2 × . . .×An+1n ×An+1

n+1 × R× R . . .

We see that for each j we have a nested sequence of compact subsets An+1j ⊂

Anj . By the finite intersection property we get

A∞j =

∞⋂n=1

Anj �= ∅.

But then we would get a contradiction since

∞⋂n=1

Bn = A∞1 ×A∞

2 × . . . �= ∅.

In the general case An is only a Borel set. For each n select a compact setKn ⊂ An such that

Pn(An \Kn) ≤ δ

2n+2.

140 J.J. Manfredi

Write Dn = π−1n (Kn) ⊂ Bn and observe that En =

⋂nj=1 Dj = π−1

n (Fn) forsome compact set Fn. We then have

P(Bn \ En) = P(Bn \ ∩n

j=1Dj

)

= P

⎛⎝ n⋃

j=1

(Bn \Dj)

⎞⎠

≤ P

⎛⎝ n⋃

j=1

(Bj \Dj)

⎞⎠

≤n∑

j=1

P (Bj \Dj))

≤n∑

j=1

P (Aj \Kj))

≤n∑

j=1

δ

2j+2

≤ δ

4.

From which we deduce that

P(En) ≥ P(Bn)− P(Bn \ En)

≥ δ − δ

4

≥ δ

2,

reducing the problem to the compact case.

Let see how Kolmogorov’s theorem can be used to give a quick constructionof the Lebesgue measure.

Consider the simplest case of ternary trees. We follow the formalismdeveloped in [5]. A directed tree T with regular 3-branching consists of theempty set ∅ as the top vertex, 3 sequences of length 1 with terms chosen fromthe set X = {0, 1, 2}, 9 sequences of length 2 with terms chosen from the setX2 = {0, 1, 2},. . . , 3r sequences of length r with terms chosen from the etXr = {0, 1, 2} and so on. A vertex br at level r is labeled by a sequence ofdigits d1d2 . . . dr, where dj ∈ X for all 1 ≤ j ≤ r. A branch of T is an infinitesequence of vertices, each followed by one of its immediate successors. Wedenote a branch b starting at the vertex b1 as follows b = (b1, b2, . . . , br, . . .).

Introduction to Random Tug-of-War Games and PDEs 141

The collection of all branches forms the boundary of the tree T and is denotedby ∂T . A branch b determines a real number in the interval [0, 1] by meansof the ternary expansion

g(b) =

∞∑k=1

bk3k

. (3)

Note that the set of all branches that start at a given vertex b = d1d2 . . . dris the ternary interval Ib = [0.d1d2 . . . dr, 0.d1d2 . . . dr + 3−r] of length 3−r,where the expansions are in base 3. Note also that the classical Cantor setC is the subset of ∂T formed by branches that don’t go through any vertexlabeled 1.

Let Pn be the uniform probability in Xn. The family {Pn} is a consistentfamily of probabilities, so that by Theorem 2.17 there exists a uniqueprobability in X∞ such that

P(A1 ×A2 × . . . An × X . . .× X× . . .) = Pn(A1 ×A2 × . . . An).

If we take A1 = {d1}, A2 = {d1},..., Ar = {dr} we get

P({b} ×X . . .×X . . .) = Pr({b}) = 3−r = |Ib|

Corollary 2.18. Consider the probability spaces (X∞, Σ,P) and ([0, 1],B,L),where L is the Lebesgue measure in the interval [0, 1]. Let g be the ternaryexpansion mapping (3). Then we have

P g−1 = L.

3 The p-Laplacian Gambling House

Start with a set X endowed with a σ-algebra B. Decompose

X = X ∪ Y

as a disjoint union of two non-empty sets X and Y . We shall call X theinterior and Y the boundary. For each point x ∈ X we have a nonempty setS(x) ⊂ X of successors of x. For points y ∈ Y we require that S(y) = {y}.Moreover, the set S(x) comes equipped with a probability measure supportedin S(x) denoted by μ(x). For points y ∈ Y on the boundary we have thatμ(y) = δy.

We are given non-negative numbers α and β so that α + β = 1 and apay-off function F : Y �→ R.

At every point x ∈ X we have a family of probability measures Γ (x) in(X,B) given by

142 J.J. Manfredi

Γ (x) ={α2(δxI + δxII ) + β μ(x) : xI , xII ∈ S(x)

}(4)

To play a Tug-of-War game with noise starting at a point x0 ∈ X, choose aprobability γ0[x0] ∈ Γ (x0). The next position x1 ∈ S(x0) is selected accordingto γ0[x0]. Once x0 and x1 are chosen, we pick a probability γ1[x0, x1] ∈ Γ (x1)to determine the next game position x2 ∈ S(x1). In this manner we determinea particular history

x = (x0, x1, x2, . . . ) ∈ X× X× · · · × X× · · · = X∞.

The game ends when we reach the boundary Y since once xj ∈ Y we havexj+1 ∈ S(xj) = {xj}. We write

τ(x) = inf{k : xk ∈ Y }

for the first time we hit the boundary with the understanding that τ(x) = ∞if the boundary is never reached. If the game ends at a point y ∈ Y thepay-off value is F (y).

We now apply a variant of the Kolmogorov’s construction. Let us denoteby Bj the product σ-algebra in Xj and by B∞ the σ-algebra in X∞ generatedby the cylinder sets

A0 ×A1 × · · · × Aj × X× X× · · · ,

where Ak ∈ Bk for k = 0, 1, . . . , j.We define a sequence of probability measures Px0,k

σ on (Xk,Bk) uniquelydetermined by the following properties:

(i) Px0,1σ = γ0[x0],

(ii) Px0,k+1σ has marginal probability P

x0,kσ on (Xk,Bk), and

(iii) Px0,k+1σ has conditional probabilities γk[x0, x1, . . . , xk−1] on the fibers

(x0, x1, . . . , xk−1)×X; that is, for every rectangle (A0 ×A1 × . . .×Ak)in (Xk+1,Bk+1) we have

Px0,k+1σ (A0×A1×. . .×Ak)=

∫(A0×A1×...×Ak−1)

γk[x0, x1, . . . , xk−1](Ak) dPx0,kσ

Under these conditions an extension of Kolmogorov’s construction due toTulcea [19, 20] shows that there exists a unique probability measure P

x0σ =

limk→∞ Px0,kσ in (X∞,B∞) with transition probabilities

Px0σ ({xj+1 ∈ A} | Bj+1) = γj [x0, x1, . . . , xj ] (5)

See Chap. 4 in [21] for more details.

Introduction to Random Tug-of-War Games and PDEs 143

We call the collection of probability measures

σ = (γ0[x0], γ1[x0, x1] . . . , γk[x0, x1 . . . xk], . . .)

a strategy.This formalism, coming from [14], is equivalent to the presentation in

[16]. In this paper a strategy S is a collection of mappings σj : Xj+1 �→ X

indicating the next move xj+1 = σj(x0, x1, . . . , xj) given the partial history(x0, x1, . . . , xj). A pair of strategies SI and SII and a starting point determinea family of measures

{Px0

SI ,SII}x0∈X

that describe the game played under this pair of strategies. That is, theplayers choose either xI or xII to move there in case they win the cointoss. Their choices determine the probability measures γ[x0, x1, . . . , xk] given(x0, x1, . . . , xk−1) and vice versa. Player I will try to choose points xI tomaximize the pay-off while player II will try to choose points xII to minimizethe pay-off. Each pair of strategies (SI , SII), SI for player I and SII for playerII as in [16], determine a strategy in this sense and vice versa. We write

σ = (SI , SII)

Having fixed a strategy σ and assuming, as we do from now on, that thegame ends a.s.

Px0σ (τ(x) < ∞) = 1, (6)

we average with respect to Px0σ to obtain the expected pay-off for the Tug-

of-War game starting at x0

uσ(x0) = Ex0σ [F (xτ )]. (7)

To write down the mean value property satisfied by uσ we condition on thefirst move using (5) with j = 0.

Lemma 3.1. ([14], Chap. 2) The value function uσ(x) satisfies the meanvalue property

uσ(x) =α

2

(uσ[xI ](xI) + uσ[xII ](xII)

)+ β

∫S(x0)

uσ[y](y) dμ(y) (8)

Here the conditional strategy σ[y0] is defined as follows for y0 ∈ S(x0)

σ[y0] = (γ1[x0, y0], γ2[x0, y0, y1] . . . , γk[x0, y0, y1 . . . yk], . . .)

so that Py0

σ[y0]is the conditional distribution of (x2, x3, . . . ) given that x1 = y0.

144 J.J. Manfredi

Let us stop and consider the particular case when α = 0 and β = 1. Inthis case –the linear case– the strategies are irrelevant since Γ (x) is alwaysμ(x) so that there is only one family of measures {Px0}x0∈X. We recover theclassical mean value formula

u(x) =

∫S(x)

u(y) dμ(y).

But the case of interest to us is when we have α �= 0. In this case the valuefunction for player I is

uI(x) = supSI

infSII

Exσ[F (xτ )]

and for player IIuII(x) = sup

SII

infSI

Exσ[F (xτ )].

Player I lets Player II choose a strategy, presumably to decrease Ex0σ [F (xτ )],

and then do as best a possible. Notice that we always have

uI(x) ≤ uII(x) for all x ∈ X.

It turns out that in many cases the game has a value; that is

uI(x) = uII(x) for all x ∈ X, (9)

and that this function satisfies a version of the Mean Value Property (3.1)given by

u(x) =α

2

(sup

y∈S(x)

u(y) + infy∈S(x)

u(y)

)+ β

∫S(x)

u(y) dμ(y). (10)

Equation (10) is the Dynamic Programming Principle or DPP for short. Next,we will present two scenarios in which all the details above have been workedout.

4 p-harmonious Functions

Consider a bounded Lipschitz domain Ω ⊂ Rn and fix ε > 0. To prescribe

boundary values, let us denote the compact boundary strip of width ε by

Γε = {x ∈ Rn \Ω : dist(x, ∂Ω) ≤ ε}.

Introduction to Random Tug-of-War Games and PDEs 145

Let X = Ω with the Borel σ-algebra, X = Ω \Γε and Y = Γε. The successorsof x are S(x) = Bε(x) = {y ∈ R

n : |y− x| ≤ ε} and the measure μ(x) is justthe Lebesgue measure restricted to S(x) and normalized so that μ(x)(S(x)) =1. As it will be clear later on, we take α and β to be

α =p− 2

p+ n, and β =

2 + n

p+ n. (11)

Notice that since α ≥ 0 we necessarily have p ≥ 2.We are given a bounded Borel pay-off function F : Γε → R and play the

Tug-of-War game with parameters α and β and obtain value functions uεI and

uεII , where we have chosen to emphasize the dependence on the step size ε.

The following results are from [12]:

Theorem 4.1. The value functions uεI and uε

II are p-harmonious in Ω withboundary values F : Γε → R; that is, they both satisfy

uε(x) =α

2

{supBε(x)

uε + infBε(x)

uε

}+ β

∫Bε(x)

uε dy for every x ∈ Ω,

(12)

anduε(x) = F (x), for every x ∈ Γε.

The existence of p-harmonious functions with given boundary values isobtained by playing the Tug-of-War games with noise. Uniqueness follows byusing martingales, although the equation is not linear. This was first provedto the best of my knowledge in [16] for p = ∞.

For finite p whether the original Tug-of-War game with noise described in[17] has a value is an open problem. For our modified version of the p-gamewe do have a value. The key is to judiciously choose strategies so that we canbring martingales into play.

Lemma 4.2. (Key Lemma) Let vε be p-harmonious such that F ≤ vεon Γε. Player I chooses an arbitrary strategy SI and player II chooses astrategy S0

II that almost minimizes vε,

vε(xk) ≤ infy∈Bε(xk−1)

vε(y) + η2−k.

Then Mk = vε(xk) + η2−k is a supermartingale for any η > 0 and uεI ≤ vε.

We can now see how the inequality at the boundary literally walks into theinterior by using Doob’s optional stopping theorem for martingales

146 J.J. Manfredi

uεI(x0) = sup

SI

infSII

ExSI ,SII

[F (xτ )]

≤ supSI

Ex0

SI ,S0II[vε(xτ ) + η2−τ ]

≤ supSI

Ex0

SI ,S0II[Mτ ]

≤ supSI

M0 = vε(x0) + η

An extension of the above technique gives the uniqueness of the valuefunction.

Theorem 4.3. [12] The game has a value. That is uεI = uε

II.

Most importantly for our purposes is the fact the p-harmonious functionssatisfy the Strong Comparison Principle:

Theorem 4.4. [12] Let Ω ⊂ Rn be a bounded domain and let uε and vε

be p-harmonious with boundary data Fu ≥ Fv in Γε. Then if there exists apoint x0 ∈ Ω such that uε(x0) = vε(x0), it follows that uε = vε in Ω, and,moreover, the boundary values satisfy Fu = Fv in Γε.

To prove that p-harmonious functions converge to the unique solution of theDirichlet problem for the p-Laplacian in Ω with fixed continuous boundaryvalues, we assume that Ω is bounded and satisfies the exterior cone condition.

Theorem 4.5. [12] Consider the unique viscosity solution u to

{div(|∇u|p−2∇u)(x) = 0, x ∈ Ω

u(x) = F (x), x ∈ ∂Ω,(13)

and let uε be the unique p-harmonious function with boundary values F . Then

uε → u uniformly in Ω as ε → 0.

The above limit only depends on the values of F on ∂Ω, and therefore anycontinuous extension of F |∂Ω to Γε0 gives the same limit.

The key to prove this theorem is to pass from the discrete setting of p-harmonious functions to the continuous case of p-harmonic functions. Thisis done by means of a characterization of p-harmonic functions in terms ofasymptotic mean value properties.

Theorem 4.6. [11] Let u ∈ C(Ω) such that for all x ∈ Ω we have

α

2

(supBε(x)

u+ infBε(x)

u

)+ β

∫Bε(x)

u = u(x) + o(ε2), as ε → 0.

Then u is p-harmonic in Ω. Here α and β are chosen as in (11).

Introduction to Random Tug-of-War Games and PDEs 147

The converse of this theorem holds if we weaken the asymptotic expansionto hold only in the viscosity sense. See [11] for details. Another approachto pass from the discrete to the continuous for fully-nonlinear equations hasbeen given by Kohn and Serfaty [6] by using a deterministic control theoryapproach.

5 Directed Trees

Let T be a directed tree with regular 3-branching as in Sect. 1. Let u : T �→ R

be a real valued function. The gradient of u at the vertex v is the vector in R3

∇u(v) = (u(v0)− u(v), u(v1)− u(v), u(v2)− u(v)).

The divergence of a vector X = (x, y, z) ∈ R3 is

div(X) = x+ y + z.

A function u is harmonic if it satisfies the Laplace equation

div(∇u) = 0. (14)

Observe that a function u is harmonic if and only if it satisfies the mean valueproperty

u(v) =1

3(u(v0) + u(v1) + u(v2))

Set X = T ∪∂T , X = T and Y = ∂T . The measure μ(v) is the normalizedcounting measure in S(v)

μ(v) =1

3(δv0 + δv1 + δv2) .

The pay-off function F : ∂T �→ R is defined on the unit interval [0, 1]. We areready to play games in T .

Think of a random walk started at the top vertex ∅ and move downwardby choosing successors at random with uniform probability. When you get at∂T at the branch point b determined by the random walk, you get paidf(b) dollars. Every time we run the game we get a sequence of verticesv1, v2, . . . , vk, . . . that determine a point on b the boundary ∂T . The set of allboundary points that start at a given vertex vr at level r is a ternary intervalof length 3−r that we denote by Ivr . Averaging out over all possible playsthat start at vr we obtain the value function

148 J.J. Manfredi

Evr [f(t)] = u(vr) =

1

|Ivr |∫Ivr

f(b) db, (15)

which is indeed harmonic in T . Therefore we have the well-known

Lemma 5.1. Dirichlet Problem in Trees (p=2): Given a continuous (indeedin L1([0, 1])) function f : [0, 1] �→ R the unique harmonic function u : T �→ R

such thatlimr→∞u(br) = f(b)

for every branch b = (br) ∈ ∂T is given by (15).

Let us now play a Tug-of-War game with noise. Choose α ≥ 0, β ≥ 0such that α+ β = 1. Start at ∅. With probability α the players play Tug-of-War. With probability β move downward by choosing successors at random.When you get at ∂T at the point b player II pays f(b) dollars to player I. Thevalue function u verifies the dynamic programming principle or mean valueproperty

u(v)=α

2

(max

i{u(vi)}+min

i{u(vi)}

)+β

(u(v0)+ u(v1)+ u(v2)

3

)(16)

that we can interpret as a PDE on the tree by using the following formulafor a generalized divergence depending on the parameters α and β.

Definition 5.2. Let X = (x, y, z) be a vector in R3. The (α, β)-divergence

of X is given by

divα,β(X) =α

2(max{x, y, z}+min{x, y, z}) + β

(x+ y + z

3

).

Theorem 5.3. [18] We have the equivalence

DPP ≈ MVP ≈ PDE

in the sense that the function u satisfies (16) in the tree T if and only if

divα,β (∇u) = 0 (17)

Some particular cases are:

(i) The Linear Case:

α = 0, β = 1 that corresponds to the linear case p = 2 of harmonic functions(14).

Introduction to Random Tug-of-War Games and PDEs 149

(ii) The Discrete ∞-Laplacian:

α = 1, β = 0 that corresponds to the case p = ∞. In this case the divergenceis

div∞(X) = div1,0(X) =1

2(max{x, y, z}+min{x, y, z})

and the equation is the discrete ∞-Laplacian div∞ (∇u) = 0.

(iii) The Discrete p-Laplacian:

For α �= 0 and β �= 0 we can select p as in (11), but the role of n is notintrinsically defined, to obtain the discrete p-Laplacian We remark that thisis the non-divergence form of the p-Laplacian (17). A discrete version of thep-Laplacian in divergence form can be found in [5].

While the formula (15) for the solution to the Dirichlet problem for p = 2is explicit, there are not such formulas to my knowledge for the case p �= 2.However, the game theoretic interpretation allows us to find explicit formulasin some special, but interesting cases.

Suppose that f is monotonically increasing. In this case the best strategyS�I for player I is always to move right and the best strategy S�

II for playerII always to move left. Starting at the vertex v at level k

v = 0.b1b2 . . . bk, bj ∈ {0, 1, 2}

we always move either left (adding a 0) or right (adding a 1). In this case Ivis the Cantor-like set Iv = {0.b1b2 . . . bkd1d2 . . .}, dj ∈ {0, 2}.Theorem 5.4. [18] The (α, β)-harmonic function with boundary values fin the tree T is given by

u(v) =

∫Iv

f(b)dPα,βv db,

where Pα,βv is a probability in [0, 1].

Moreover in the case α = 0, β = 1, which corresponds to p = 2 themeasure P

0,1v is just the Lebesgue measure, and in the case α = 1, β = 0,

which corresponds to the case p = ∞, the measure P1,0v is a Cantor measure

supported in Iv.

To see why this theorem is true observe that

u(v) = supSI

infSII

EvSI ,SII

[f(b)] = EvS�I ,S

�II[f(b)].

Since the strategies used are always the same, we are indeed in a linearsituation. All we need to do is to compute the probability P

vS�I ,S

�II.

150 J.J. Manfredi

6 Epilogue

We take the opportunity to state two open problems for p-harmonic functionsin R

n that have challenged the mathematical community for more than thirtyyears.

Open Problem 6.1 (Strong Comparison Principle): Suppose that uand v are p-harmonic functions in BR(x0) such that

{u(x) ≤ v(x) for x ∈ BR(x0), and

u(x0) = v(x0).(18)

Does it follow that u ≡ v in BR(x0)?

Open Problem 6.2 (Unique Continuation Property): Suppose that uis p-harmonic in B2R(x0) and that

u(x) = 0 for every x ∈ BR(x0).

Does it follow that u ≡ 0 in B2R(x0)?

Of course, the answer to this problems is clearly yes in the linear casep = 2. The answer to both problems is also yes in the planar case n = 2 sincecomplex methods are then available [1, 10].

It is natural to try to apply new techniques developed in Analysis tothese problems in the hope of improving our understanding of them. Whenthe viscosity theory was developed, it was first proved that the notions ofSobolev weak solution and viscosity solution agree [3] allowing us to studythe p-harmonic equation not only by variational methods for divergenceform equations but also using viscosity methods for non-divergence formequations. Progress in various problems followed: The ∞-eigenvalue problem[4], unexpected superposition principles [2] and [7], and various proofs wereextended and simplified by using sup-convolutions [8].

The non-linear potential theory on trees of Kauffman and Wu [5] provedessential to settle a long standing conjecture of Martio on the lack ofsubadditivity of p-harmonic measures even at the zero level [9].

In the opinion of this author, the connection between the p-harmonicequation and discrete stochastic games discovered in the case p = ∞ firstby Peres, Sheffield, Schramm, and Wilson [16] and for finite p by Peres andSheffield [17] (see also [15]) that opens the door to the use of game theoreticand control theoretic methods to the study of the p-Laplace equation, willhave substantial applications [15]. A distinguishing feature of the stochasticgames approach is that, since it is based on discrete stochastic processes,it provides good discrete approximations to p-harmonic functions, making adirect connection with the analysis in trees of [5].

Introduction to Random Tug-of-War Games and PDEs 151

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