aa r X i v : . [ m a t h . A P ] M a y GRADIENT WALK AND p -HARMONIC FUNCTIONS HANNES LUIRO AND MIKKO PARVIAINEN
Abstract.
We consider a class of stochastic processes and establish itsconnection to p -harmonic functions. In particular, we obtain stochasticapproximations that converge uniformly to a p -harmonic function, with anexplicit convergence rate, and also obtain a precise diffusion representationin continuous time. The main difficulty is how to deal with the zero set ofthe gradient of the underlying function. Introduction
A connection between a stochastic game called a tug-of-war with noise and p -Laplace equation div( |∇ u | p − ∇ u ) = 0 , was first discovered by Peres and Sheffield in [PS08], see also [PSSW09]. Inone of the key results they show that if the gradient of a p -harmonic payofffunction is nonvanishing and a player follows the gradient strategy, then thegame value is close to the p -harmonic function.However, the zeros of the gradient pose deep problems both in the theoryof PDEs and tug-of-war games. This is the main difficulty we encounter inthis paper. To be more precise, we study a gradient walk i.e. we fix gradientstrategies for both the players, but we do not assume that the gradient isnonvanishing. A motivation for such an approach stems partly from desireto understand a local behavior of stochastic processes related to p -harmonicfunctions as well as to obtain a diffusion representation in continuous time for p -harmonic functions.The main results show that the expectation under the gradient walk ap-proximates a p -harmonic function, Theorems 3.1 (uniform convergence withrespect to the step size) and 3.3 (explicit rate with respect to the step size),without any assumptions on the zero set of the gradient of the underlying p -harmonic function. Naturally, the set where the gradient is small requires aspecial attention in terms of how we define the process.If the zero set of the gradient is known to be a finite set of points, we developa technique which is more flexible with respect to how we define the process atthe zero set of the gradient. A natural choice at those points is to choose the Date : October, 2015.2010
Mathematics Subject Classification.
Key words and phrases.
Diffusion representation, Feynman-Kac formula, Markov chain, p -Laplacian, tug-of-war with noise. next point at random according to a uniform probability distribution. Sucha process has the approximation property as shown in Theorem 4.2, and thecontinuous time version gives an exact diffusion presentation of a p -harmonicfunction, Theorem 4.1. Such stochastic approximations seem to be deeplyconnected to the structure of the zero set of the gradient for a p -harmonicfunction. In the plane, the zero set of the gradient is known to be discrete[BI87], but in higher dimensions, understanding structures of the zero set is adifficult open problem.1.1. Background.
Taking the average over the usual Taylor expansion u ( x + y ) = u ( x ) + ∇ u ( x ) · y + 12 D u ( x ) y · y + O ( | y | ) , over B (0 , ε ), we obtain u ( x ) − Z B (0 ,ε ) u ( x + y ) dy = − ε n + 2) ∆ u ( x ) + O ( ε ) , (1.1)when u is smooth. Above we used the notation Z B (0 ,ε ) u ( x + y ) dy = 1 | B (0 , ε ) | Z B (0 ,ε ) u ( x + y ) dy. On the other hand, by assuming ∇ u ( x ) = 0, evaluating the Taylor expansionwith y = ± ε ∇ u ( x ) |∇ u ( x ) | , and summing up we get u ( x ) − (cid:26) u (cid:18) x + ε ∇ u ( x ) |∇ u ( x ) | (cid:19) + u (cid:18) x − ε ∇ u ( x ) |∇ u ( x ) | (cid:19)(cid:27) = − ε N ∞ u ( x ) + O ( ε ) , (1.2)where ∆ N ∞ u = |∇ u | − P ni,j =1 u ij u i u j is the normalized infinity Laplace opera-tor, and u i , u ij denote the first and second derivatives respectively. Next if wemultiply (1.1) and (1.2) by β = (2 + n ) / ( p + n ) , ≤ p < ∞ , and 1 − β , andadd up the formulas, we get the normalized p -Laplace operator on the righthand side i.e. u ( x ) = 1 − β (cid:26) u (cid:18) x + ε ∇ u ( x ) |∇ u ( x ) | (cid:19) + u (cid:18) x − ε ∇ u ( x ) |∇ u ( x ) | (cid:19)(cid:27) + β Z B (0 ,ε ) u ( x + y ) dy + Cε ∆ Np u ( x ) + O ( ε )as ε →
0, where ∆ Np u = (∆ u +( p − N ∞ u ) denotes the normalized p -Laplacian.The equation ∆ Np u = 0 gives the same solutions as the usual p -Laplace oper-ator, see [JLM01, KMP12, JJ12]. If ∆ Np u = 0, then the above formula yields u ( x ) = 1 − β (cid:26) u (cid:18) x + ε ∇ u ( x ) |∇ u ( x ) | (cid:19) + u (cid:18) x − ε ∇ u ( x ) |∇ u ( x ) | (cid:19)(cid:27) + β Z B (0 ,ε ) u ( x + y ) dy + O ( ε ) , (1.3) RADIENT WALK AND p -HARMONIC FUNCTIONS 3 as ε →
0. A variant of this formula can then be used to characterize the p -harmonic functions [MPR10].The above formula suggests the following Markov chain, which we call thegradient walk: when at x step to x + ε ∇ u ( x ) |∇ u ( x ) | or to x − ε ∇ u ( x ) |∇ u ( x ) | with probability(1 − β ) /
2, respectively, or to a point chosen according to a uniform probabilitydistribution on B ( x, ε ) with probability β . Similarly, this defines one stepprobability measures at every point, which again induce a probability measureon the space of sequences according to the Kolmogorov construction. Take u ε ( x ) to be the expectation with respect to this probability measure whenstarting at x , stopping when exiting a domain (stopping time τ ), and takingthe boundary values from a smooth p -harmonic function u with ∇ u = 0 i.e. u ε ( x ) := E x (cid:2) u ( x τ ) (cid:3) . (1.4)For more details of the stochastic background, see for example [LPS14].If we drop the error term in (1.3) and replace u by u ε , then the resultingformula holds by the Markov property, see [MT09] Section 3.4.2, and tells us,how to compute the expectation at the point x : this is done by summing upthe three possible outcomes with the corresponding probabilities. This is thekey formula needed in establishing a connection with the expected value andthe corresponding p -harmonic function in the case ∇ u = 0.The formula (1.3) also suggest a version of a dynamic programming principleand a version of a tug-of-war with noise with good symmetry properties, seefor example in [MPR12, LPS13, LPS14].2. Preliminaries
We consider a domain B (0 , ⊂ R n , n ≥ , and a p -harmonic function u : B (0 , γ ) → R , γ >
0, throughout the paper. Further, let x ∈ B (0 , ε, η > |∇ u | > η , β = (2 + n ) / ( p + n ) and set µ x, = β L B ( x,ε ) + 1 − β (cid:0) δ x + ε ∇ u ( x ) |∇ u ( x ) | + δ x − ε ∇ u ( x ) |∇ u ( x ) | (cid:1) , (2.5)where L B ( x,ε ) denotes the uniform distribution in B ( x, ε ) ⊂ R n and δ x theDirac measure at x . In this notation, the equation (1.3) can be written as (cid:12)(cid:12)(cid:12)(cid:12) Z R n u ( y ) dµ x, ( y ) − u ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε . (2.6)Another technical tool we use repeatedly is the fact that the expected valueof the distance to a fixed point increases at every step in the gradient walk ata certain rate. Proposition 2.1.
For all x ∈ R n , ν ∈ S n − = ∂B (0 , , ε > and β ∈ (0 , ,it holds that − β (cid:0) | x + εν | + | x − εν | (cid:1) + β Z B ( x,ε ) | y | dy ≥ | x | + C ( n ) β ε | x | + ε ) , (2.7) where C ( n ) → as n → ∞ . LUIRO AND PARVIAINEN
Proof.
Observe first that | x + εν | + | x − εν | ≥ | x | by the triangle inequality.Assume for now x = 0, and denote by h x i ⊥ the space orthogonal to x , by y ⊥ the projection of y ∈ R n onto h x i ⊥ and by P ( y ) the reflection of y with respectto h x i ⊥ . Then P is a linear isometry, thus we can write Z B ( x,ε ) | y | dy = Z B (0 ,ε ) | x + y | + | x + P ( y ) | dy ≥ Z B ( x,ε ) | x + y + x + P ( y ) | dy = Z B (0 ,ε ) | x + y ⊥ | dy , where we used the triangle inequality and the fact y + P ( y ) = 2 y ⊥ . ByPythagoras’ theorem, the following estimate holds for all a, b ∈ R n , a · b = 0 : | a + b | = ( | a | + | b | ) = | a | + Z | a | + | b | | a | √ t dt ≥ | a | + | b | | a | + | b | ) = | a | + | b | | a + b | . This estimate implies that Z B (0 ,ε ) | x + y ⊥ | dy ≥ | x | + Z B (0 ,ε ) | y ⊥ | | x + y ⊥ | dy ≥ | x | + 12( | x | + ε ) Z B (0 ,ε ) | y ⊥ | dy . Finally, it is easy to see that for 0 < c < Z B (0 ,ε ) | y ⊥ | dy = n − n Z B (0 ,ε ) | y | dy ≥ n − n | B (0 , ε ) \ B (0 , cε ) || B (0 , ε ) | | cε | = n − n (1 − c n ) | cε | . By choosing c suitably depending on n yields the claim by combining the aboveestimates.If x = 0, take nonzero approximating sequence x i →
0, and use the aboveresult. (cid:3) Immediate evaluation of error
Above we considered the domain B (0 , p -harmonic function u : B (0 , γ ) → R , γ >
0, under the assumption of the non vanishing gradient.However, as already pointed out the difficulty lies in the zero set of the gradient,and now we drop the assumption ∇ u = 0. Define the one step probabilitymeasures µ x, = δ x + ε x | x | if x = 0 and µ , = δ ε | e | , RADIENT WALK AND p -HARMONIC FUNCTIONS 5 and µ x = ( µ x, , if |∇ u ( x ) | ≥ η , and µ x, , if |∇ u ( x ) | < η , where µ x, was defined in (2.5). The first version of a gradient walk which is aMarkov chain { x , x , . . . } ⊂ R n starting at x is determined by the transitionprobabilities µ x . Moreover, for x ∈ B (0 , u ε ( x ) := u ε,η,u ( x ) := E x [ u ( x τ )] (3.8)where τ is the first exit time from B (0 , u ε := u outside B (0 , ∂B (0 ,
1) at least by a step comparableto ε , it follows that τ is finite almost surely, and the same holds for otherstopping times defined in this paper as well.The name of the section reflects the fact that in the bad set {|∇ u ( x ) | < η } we obtain comparison result implying Theorem 3.1 in one step, cf. (3.13) below.In contrast, in Section 4 we define the bad set slightly differently and wait untilwe exit the bad set. The immediate evaluation of error uses properties of thestochastic process in the bad set in a subtle way, but on the other hand doesnot require any assumptions on the structure of the zero set of the gradient.It also allows us to obtain explicit convergence rate.Our first main theorem states that the value of the gradient walk convergesuniformly to the underlying p -harmonic function. Theorem 3.1.
Let u : B (0 , γ ) → R be a p -harmonic function, ≤ p < ∞ .Let η > and let u ε be the value of the gradient walk given by (3.8). Then forany C > , there is ε > such that || u ε − u || L ∞ ( B (0 , ≤ Cη for all < ε < ε .Proof. Let us denote g ( x ) := | u ε ( x ) − u ( x ) | . Choose an auxiliary comparisonfunction f ( x ) = cη ( c − | x | ), where c >
1. We will establish that g ≤ f , whichimplies the claim. To this end, assume the opposite, so that M := sup x ∈ B (0 , ( g ( x ) − f ( x )) > . (3.9)Suppose that M above is achieved at x ∈ B (0 ,
1) up to an arbitrary smallerror term κ > g ( x ) − f ( x ) ≥ M − κ . (3.10) LUIRO AND PARVIAINEN
First we consider the case |∇ u ( x ) | ≥ η . By the definition of u ε and (2.6) itfollows that g ( x ) − f ( x )= | u ε ( x ) − u ( x ) | − f ( x ) = (cid:12)(cid:12) Z R n u ε ( y ) dµ x , − u ( x ) (cid:12)(cid:12) − f ( x ) ≤ Z R n | u ε ( y ) − u ( y ) | dµ x , ( y ) + Cε − f ( x )= Z R n g ( y ) − f ( y ) dµ x , ( y ) + Z R n f ( y ) dµ x , ( y ) − f ( x ) + Cε ≤ M + Z R n f ( y ) dµ x , ( y ) − f ( x ) + Cε . (3.11)Combining this with (3.10) implies f ( x ) ≤ Z R n f ( y ) dµ x , ( y ) + Cε + κ . (3.12)However, Proposition 2.1 implies that this can not be true for any x ∈ B (0 , f if ε and κ are small enough.Consider then the case |∇ u ( x ) | < η (and x = 0, the case x = 0 is similar).In this case it follows that g ( x ) − f ( x ) = | u ε ( x ) − u ( x ) | − f ( x )= (cid:12)(cid:12) Z R n u ε ( y ) dµ x , − u ( x ) (cid:12)(cid:12) − f ( x )= | u ε ( x + ε x | x | ) − u ( x ) | − f ( x ) ≤ | u ε ( x + ε x | x | ) − u ( x + ε x | x | ) | + | u ( x + ε x | x | ) − u ( x ) | − f ( x )= g ( x + ε x | x | ) − f ( x + ε x | x | ) + f ( x + ε x | x | ) − f ( x )+ | u ( x + ε x | x | ) − u ( x ) |≤ M − cηε + | u ( x + ε x | x | ) − u ( x ) | , implying by (3.10) that | u ( x + ε x | x | ) − u ( x ) | ≥ cηε − κ . (3.13)However by using the fact that c > |∇ u ( x ) | < η , and u is C , it followsthat (3.13) can not be true for ε ≤ ε . Summing up, we have shown that (3.9)cannot be valid, and thus || u ε − u || L ∞ ( B (0 , ≤ c η . (cid:3) We could also have used stochastic approach in the proof above, consider theexpectation over a single step, estimate the accumulation of the error, and touse optional stopping theorem, cf. [PS08, Theorem 2.4] or [MPR12, Theorem
RADIENT WALK AND p -HARMONIC FUNCTIONS 7 C regularity of thelimit u as well as the drift property (2.7). The fact that u is p -harmonic isonly utilized through these properties.3.1. Convergence rate.
In this section, we aim at obtaining an explicit con-vergence rate of the gradient walk with respect to ε . First recall the notationwith multi-index σ and α ∈ (0 , || u || C ,α ( B ) = X | σ |≤ || D σ u || C α ( B ) , || u || C α ( B ) = sup B | u | + | u | C α ( B ) , | u | C α ( B ) = sup x,y ∈ B,x = y | u ( x ) − u ( y ) || x − y | α . We will also use || u || C ,α below and the definition is analogous to the one above.Let u : B (0 , γ ) → R , γ > p -harmonic function. Then there is α = α ( n, p ) ∈ (0 ,
1) and C = C ( n, p, γ, || u || L ∞ ( B (0 , γ )) ) such that || u || C ,α ( B (0 , ≤ C, (3.14)see for example [Ura68, Uhl77, Eva82, DiB83, Tol84]. Note that p -Laplaciandegenerates when the gradient vanishes, and C ,α -regularity is optimal. Wewill also use another standard estimate. Theorem 3.2.
Let u be p -harmonic function in B (0 , such that ≤ |∇ u ( x ) | in B (0 , . Then there is α = α ( n, p ) ∈ (0 , and C = C ( n, p, || u || L ∞ ( B (0 , ) such that u ∈ C ,α ( B (0 , )) and || u || C ,α ( B (0 , )) ≤ C. The theorem is well-known. It is based for example on using Schauderestimates (see for example [GT01]) combined with the estimate (3.14). To bemore precise, consider the normalized p -Laplacian a ij ( ∇ u ) u ij = 0 using theEinstein summation convention where u ij denotes the second derivatives and a ij ( q ) = δ ij + ( p − q i q j | q | , and δ ij = 1 if i = j and zero otherwise. Observe that a ij ( · ) is smooth when q us bounded away from zero. Then we have a ij ( ∇ u ) ∈ C α by (3.14), and u ∈ C ,α by the Schauder theory and ≤ |∇ u | . Then the heuristic idea is dif-ferentiating the equation and denoting by w = D ν u the directional derivativeto the direction ν , we get a ij,k ( ∇ u ) w k u ij + a ij ( ∇ u ) w ij = 0 , where a ij,k denotes the derivative of a ij with respect to the k th variable. The C ,α -estimate for w depends on the C α -norm for the coefficients, that is, on the LUIRO AND PARVIAINEN C ,α -estimate for u . Thus the estimate in Theorem 3.2 holds with a uniformcoefficient C = C ( n, p, || u || L ∞ ( B (0 , ). An alternative approach can be builton the divergence form equation and the weak formulation.Now we consider the gradient walk defined by the one step probability mea-sure µ x, = δ x + ε x | x | if x = 0 and µ , = δ ε | e | , (3.15)and µ x = ( µ x, , if |∇ u ( x ) | ≥ ε α ′ ,µ x, , if |∇ u ( x ) | < ε α ′ , (3.16)where α ′ < α/ α is given by (3.14), and µ x, by (2.5). As before, weset u ε ( x ) := E x [ u ( x τ )] . Theorem 3.3.
Let u : B (0 , γ ) → R be a p -harmonic function, ≤ p < ∞ ,and let u ε be the value of the gradient walk given above. Then for any C > there is ε > such that || u ε − u || L ∞ ( B (0 , ≤ Cε α ′ (3.17) for all < ε < ε .Proof. Let λ > r := ( λ/ (2 C ′ )) /α < γ , where C ′ isthe one in the estimate (3.14). Suppose that x ∈ B (0 ,
1) such that |∇ u ( x ) | = λ .It follows from (3.14) that B ( x, r ) ⊂ { z ∈ B (0 , γ ) : λ ≤ |∇ u ( z ) | ≤ λ } .Then set v ( y ) = u ( yr + x ) − u ( x ) λr so that ≤ |∇ v ( y ) | ≤ B (0 , v (0) = 0. It follows that || v || L ∞ ( B (0 , ≤
2. We are in the position of using the estimate from The-orem 3.2 for v with a constant C ( n, p ). In particular, let y ∈ B (0 , ), ˜ ε = ε/r and v (cid:18) y + ∇ v ( y ) |∇ v ( y ) | ˜ ε (cid:19) = u (cid:16) y ′ + ∇ u ( y ′ ) |∇ u ( y ′ ) | εr r (cid:17) − u ( x ) λr , where y ′ = x + yr . Then we have C ˜ ε ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β Z B (0 , ˜ ε ) v dy + 1 − β ( v (cid:18) y + ∇ v ( y ) |∇ v ( y ) | ˜ ε (cid:19) + v (cid:18) y − ∇ v ( y ) |∇ v ( y ) | ˜ ε (cid:19) ) − v ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) βλr Z B ( x,ε ) u dy ′ + 1 − β λr ( u (cid:18) y ′ + ∇ u ( y ′ ) |∇ u ( y ′ ) | ε (cid:19) + u (cid:18) y ′ − ∇ u ( y ′ ) |∇ u ( y ′ ) | ε (cid:19) ) − u ( y ′ ) (cid:12)(cid:12)(cid:12)(cid:12) , RADIENT WALK AND p -HARMONIC FUNCTIONS 9 and the estimate holds for y ′ ∈ B ( x, r ). Multiplying the both sides by λr , wesee that the desired formula holds with the error C ( ε/r ) λr = Cε λr − . If wefix λ = ε α ′ , the error term reads as λ − α ε = ε α ′ (1 − α ) ε up to a constant. On the other hand, if we have λ > ε α ′ , then the aboveargument also works and with a smaller error term. To sum up, a counterpartof formula (2.6) holds in the form (cid:12)(cid:12)(cid:12)(cid:12) Z R n u ( y ) dµ x, ( y ) − u ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε δ , (3.18)where δ = 1 + α ′ (1 − α ) > c > f ( x ) = cε α ′ ( c − | x | ) . Further, denote g ( x ) := | u ε ( x ) − u ( x ) | , let κ > x ∈ B (0 ,
1) such that g ( x ) − f ( x ) + κ ≥ sup x ∈ B (0 , ( g ( x ) − f ( x )) > . Suppose first that |∇ u ( x ) | ≥ ε α ′ . By calculation (3.11) using (3.18), we seethat f ( x ) ≤ Z R n f ( y ) dµ x , ( y ) + Cε δ + κ. However, by Proposition 2.1 f ( x ) − Cε α ′ ≥ Z R n f ( y ) dµ x , ( y ) . The contradiction follows if α ′ < α/ |∇ u ( x ) | < η = ε α ′ . By (3.14), it also holds thatsup y ∈ B ( x,ε ) | u ( y ) − u ( x ) − ∇ u ( x ) · ( y − x ) | ≤ C ′ r α implyingsup y ∈ B ( x,ε ) | u ( y ) − u ( x ) | ≤ C ′ ε α + ε |∇ u ( x ) | < C ′ ε α + ε α ′ ≤ ( C ′ ε α − α ′ + 1) ε α ′ . (3.19)Next recall f and observe that there is ε > cε α ′ for every 0 < ε < ε . Using this, we concludesimilarly as in the proof of Theorem 3.1. (cid:3) Delayed evaluation of error
In this section, we study a slightly different technique of showing that thegradient walk approximates a p -harmonic function with gradient vanishing ina finite set of points. In contrast to the previous sections, we take a ballneighborhood of a zero set of the gradient and then in that set we delay theevaluation of the error until we exit from the set. The good point in this method is that there is a lot of freedom to choose the probability measureas long as the resulting stochastic process exits the ball neighborhood almostsurely. In a sense, a very natural choice is to use the random walk in the zerosof the gradient and otherwise to utilize the gradient directions. This is thechoice we use below. A counterpart of this choice in continuous time allows usto establish a diffusion representation of p -harmonic functions.4.1. Diffusion representation of p -harmonic functions. Next we studya continuous time diffusion process related to a p -harmonic function. Thismay be combared to the Feynman-Kac formula in the classical setting. Acontinuous time tug-of-war game has been previously studied by Atar andBudhiraja in [AB10] and in the context of option pricing in [NP].Let u : B (0 , γ ) → R , γ > , be a p -harmonic function such that |∇ u ( x ) | > x = x , where x ∈ B (0 , A ( x ), an n × n matrix, whoseentries are given by a ij ( y ) = 12 ( δ ij + ( p − |∇ u ( y ) | − u i ( y ) u j ( y ) , when y = x δ ij , when y = x , where u i denotes the i th partial derivative, and δ ij = 1 if i = j and zerootherwise. In B (0 , γ ) \ { x } , the p -harmonic function u is real analyticand it holds that P ni,j =1 a ij u ij = 0. Also observe that min( p − , | ξ | ≤ P ni,j =1 a ij ξ i ξ j ≤ max( p − , | ξ | so that A is uniformly elliptic. Thus thereexists a strong Markov family ( X ( t ) , P x ) of solutions to the martingale prob-lem, see [Bas98] Chapter 6, Stroock and Varadhan [SV06] Chapter 6 and 12as well as [Kry73]. Define a stopping time τ as a first exit time from B (0 , ϕ ( X ( t ∧ τ )) − ϕ ( x ) − Z t ∧ τ n X i,j =1 a ij ( X ( s )) ϕ ij ( X ( s )) ds, (4.20)where t ∧ τ = min( t, τ ), is a P x -martingale for all φ ∈ C ( B (0 , δ )). Moreover, τ as well as the stopping times defined in the proof below are finite a.s., seefor example Exercise 40.1 in [Bas11].We may define v ( x ) = E x [ u ( X ( τ ))] . Theorem 4.1.
Let u : B (0 , γ ) → R be a p -harmonic function, < p < ∞ ,such that |∇ u ( x ) | > whenever x = x , where x ∈ B (0 , . Further, let v beas above. Then v = u in B (0 , .Proof. Let η > f ( x ) := η (1 − | x − x | ) and z := x − x . It holdswhenever x = x that ∇ f ( x ) = − η z | z | , D f ( x ) = η | z | (cid:18) z ⊗ z | z | − I (cid:19) , RADIENT WALK AND p -HARMONIC FUNCTIONS 11 and n X i,j =1 a ij ( x ) f ij ( x )= tr (cid:18) I + ( p − ∇ u ( x ) ⊗ ∇ u ( x ) |∇ u ( x ) | (cid:19) η | z | (cid:18) z ⊗ z | z | − I (cid:19) ! = η | z | tr z ⊗ z | z | + ( p − ∇ u ( x ) |∇ u ( x ) | ( ∇ u ( x ) |∇ u ( x ) | ) T z | z | ( z | z | ) T − I − ( p − ∇ u ( x ) ⊗ ∇ u ( x ) |∇ u ( x ) | ! = η | z | (cid:16) p − ∇ u ( x ) |∇ u ( x ) | · z | z | ) − n − ( p − (cid:17) ≤ η | z | (1 − n ) < . (4.21)Denote g ( x ) := | v ( x ) − u ( x ) | . We aim at showing g ( x ) − f ( x ) ≤
0. Thrivingfor a contradiction, let κ ∈ (0 , η/
2) and suppose that there is
M > x ∈ B (0 ,
1) such that g ( x ) − f ( x ) + κ ≥ M := sup B (0 , ( g ( x ) − f ( x )) . (4.22)Then we fix r and again split the argument into two cases: Now if x ∈ B (0 , \ B ( x , r ), take a ball B ( x , δ ) ⊂ B (0 , \ B ( x , r ), and stopping time τ ∗ as the first exit time from B ( x , δ ). Then by the strong Markov property,we have v ( x ) = E x [ v ( X ( τ ∗ ))] . (4.23)Moreover, u is a smooth p -harmonic function at the vicinity of B ( x , δ ) so that u ( X ( t ∧ τ ∗ )) − u ( x ) − Z t ∧ τ ∗ n X i,j =1 a ij ( X ( s )) u ij ( X ( s )) ds = u ( X ( t ∧ τ ∗ )) − u ( x )is a martingale. By the optional stopping theorem u ( x ) = E x [ u ( X ( τ ∗ ))] . (4.24)Similarly, f ( X ( t ∧ τ ∗ )) − f ( x ) − Z t ∧ τ ∗ n X i,j =1 a ij ( X ( s )) f ij ( X ( s )) ds is a martingale, and the optional stopping theorem combined with (4.21) gives − κ + f ( x ) > E x [ f ( X ( τ ∗ ))] , (4.25) for all small enough κ . Thus by (4.22), (4.23), (4.24) and (4.25) M − κ ≤ g ( x ) − f ( x ) = | v ( x ) − u ( x ) | − f ( x ) ≤ E x [ | v ( X ( τ ∗ )) − u ( X ( τ ∗ )) | − f ( X ( τ ∗ ))] + E x [ f ( X ( τ ∗ ))] − f ( x )= E x [ g ( X ( τ ∗ )) − f ( X ( τ ∗ ))] + E x [ f ( X ( τ ∗ ))] − f ( x ) < M − κ, a contradiction.If x ∈ B ( x , r ), we define a stopping time τ as the first exit time from B ( x , r ). First we observe that E x [ | u ( X ( τ ) − u ( x ) | ] ≤ E x [ | u ( X ( τ ) − u ( x ) | + | u ( x ) − u ( x ) | ] ≤ Cr α . Then we estimate M − κ ≤ g ( x ) − f ( x ) = | v ( x ) − u ( x ) | − f ( x )= | E x [ v ( X ( τ ))] − u ( x ) | − f ( x ) ≤ E x [ | v ( X ( τ )) − u ( X ( τ )) | ] − f ( x ) + Cr α ≤ E x [ | v ( X ( τ )) − u ( X ( τ )) | − f ( X ( τ ))] + E x [ f ( X ( τ ))] − f ( x ) + Cr α ≤ M + E x [ f ( X ( τ ))] − f ( x ) + Cr α , where the second step holds by the Markov property for v , and third step by C ,α regularity of u and ∇ u ( x ) = 0. This implies that − κ + f ( x ) ≤ E x [ f ( X ( τ ))] + Cr α which is a contradiction by the form of f when κ and r are small enough.Since η > (cid:3) Gradient vanishing in a finite set of points.
Next we return to thediscrete time setting. If the gradient vanishes in a finite set of points E ,then the above method can easily be modified to prove that the gradient walkapproximates the original p -harmonic function by only changing the processon E . More precisely, using the notation from Section 2, let us define for agiven p -harmonic function u : B (0 , γ ) → R , γ > x ∈ B (0 , ε > µ x = ( µ x, , if |∇ u ( x ) | > , and L B ( x,ε ) , if |∇ u ( x ) | = 0 . Then set similarly as before u ε ( x ) := E x [ u ( x τ )] . Theorem 4.2.
Let u : B (0 , γ ) → R be a p -harmonic function such that |∇ u ( x ) | > outside a finite set of points E ⊂ B (0 , . Then | u ε − u | → uniformly in B (0 , as ε → . Proof.
Let us denote E = { x , x , . . . , x N } . Since u is C , there is strictlyincreasing φ : [0 , ∞ ) → [0 , ∞ ), which is continuous and φ (0) = 0, such that | u ( x ) − u ( x i ) | ≤ φ ( | x − x i | ) | x − x i | . (4.26) RADIENT WALK AND p -HARMONIC FUNCTIONS 13 Let us choose η > f ( x ) := 4 φ (2 η ) N X i =1 (2 − | x − x i | ) =: N X i =1 f i ( x ) . We again denote g ( x ) := | u ε ( x ) − u ( x ) | , and show that for ε small enoughit holds that g ≤ f . Assume that the claim is not true, so that for any κ > x ∈ B (0 , g ( x ) − f ( x ) + κ ≥ M := sup x ∈ B (0 , ( g ( x ) − f ( x )) > . In the case dist( x , E ) > η we have |∇ u ( x ) | ≥ c >
0, and the desiredcontradiction follows for small enough ε in the same way as in the proof ofTheorem 3.1.Consider then the case dist( x , E ) ≤ η . We may assume that | x − x | ≤ η and | x − x i | ≥ η for all 2 ≤ i ≤ N . In this case, let τ ∗ < τ be the first exittime from B ( x , η ). Then, using the strong Markov property of a Markovchain, [MT09] Proposition 3.4.6, and the estimate (4.26), we have M − κ ≤ g ( x ) − f ( x ) = | u ε ( x ) − u ( x ) | − f ( x )= | E x [ u ε ( x τ ∗ )] − u ( x ) | − f ( x ) ≤ E x (cid:2) | u ε ( x τ ∗ ) − u ( x τ ∗ ) | (cid:3) + E x (cid:2) | u ( x τ ∗ ) − u ( x ) | (cid:3) − f ( x ) ≤ E x (cid:2) | u ε ( x τ ∗ ) − u ( x τ ∗ ) | (cid:3) + 3 φ (2 η ) η − f ( x )= E x (cid:2) | u ε ( x τ ∗ ) − u ( x τ ∗ ) | − f ( x τ ∗ ) (cid:3) + E x (cid:2) f ( x τ ∗ ) − f ( x ) (cid:3) + 3 φ (2 η ) η ≤ M + E x (cid:2) f ( x τ ∗ ) − f ( x ) (cid:3) + 3 φ (2 η ) η . Above we estimated E x (cid:2) | u ( x τ ∗ ) − u ( x ) | (cid:3) ≤ | u ( x τ ∗ ) − u ( x ) | + | u ( x ) − u ( x ) | ≤ ηφ (2 η ) + ηφ ( η ) ≤ ηφ (2 η ) by (4.26). Since κ can be chosen to be arbitrarysmall, the desired contradiction follows, if we can show that E x (cid:2) f ( x τ ∗ ) − f ( x ) (cid:3) + 3 φ (2 η ) η < . For this, we compute that E x (cid:2) f ( x τ ∗ ) − f ( x ) (cid:3) = E x (cid:2) N X i =1 f i ( x τ ∗ ) − f i ( x ) (cid:3) = E x (cid:2) f ( x τ ∗ ) − f ( x ) (cid:3) + E x (cid:2) N X i =2 f i ( x τ ∗ ) − f i ( x ) (cid:3) =4 φ (2 η ) (cid:20) E x (cid:2) | x − x | − | x τ ∗ − x | (cid:3) + E x (cid:2) N X i =2 | x − x i | − | x τ ∗ − x i | (cid:3)(cid:21) ≤ φ (2 η ) (cid:20) − η + N X i =2 (cid:16) | x − x i | − E x (cid:2) | x τ ∗ − x i | (cid:3)(cid:17)(cid:21) , implying that the desired inequality follows if | x − x i | ≤ E x (cid:2) | x τ ∗ − x i | (cid:3) (4.27)for all 2 ≤ i ≤ N .Now, suppose that (4.27) is not true, so that for any κ >
0, there is z ∈ B ( x , η ) such that0 < M e := sup z ∈ B ( x , η ) (cid:0) | z − x i | − e ( z ) (cid:1) ≤ | z − x i | − e ( z ) + κ , where we denoted e ( z ) := E z (cid:2) | z τ ∗ − x i | (cid:3) . It holds that e ( z ) = Z R n e ( y ) dµ z ( y ) . We proceed with the similar reasoning as before M e ≤| z − x i | − e ( z ) + κ = | z − x i | − Z R n e ( y ) dµ z ( y ) + κ = | z − x i | − Z R n | y − x i | dµ z ( y ) + Z R n (cid:0) | y − x i | − e ( y ) (cid:1) dµ z ( y ) + κ ≤| z − x i | − Z R n | y − x i | dµ z ( y ) + M e + κ . Observe above that if y is outside B ( x , η ), then | y − x i | − e ( y ) = 0 < M e .By substracting M e from the inequality, the desired contradiction follows byProposition 2.1, which guarantees that | z − x i | < Z R n | y − x i | dµ z ( y ) − κ for all z ∈ B ( x , η ) . (cid:3) Acknowledgements.
The second author was supported by the Academy ofFinland project
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