AA (cid:1) course (cid:1) on (cid:1) Tug (cid:1) of (cid:1) War (cid:1) games (cid:1) with (cid:1) random (cid:1) noise
Introduction (cid:1) and (cid:1) basic (cid:1) constructions
Marta (cid:1)
Lewicka
This is a preprint of the following work: Marta Lewicka, A Course on Tug-of-War Games with Random Noise, 2020, Springer, Chapter 2.Reproduced with permission of Springer Nature Switzerland AG. The final authenticated version is available online at: https://doi.org/10.1007/978-3-030-46209-3 (cid:1) reface
The goal of these Course Notes is to present a systematic overview of the basicconstructions and results pertaining to the recently emerged field of Tug of Wargames, as seen from an analyst’s perspective. To a large extent, this book rep-resents the author’s own study itinerary, aiming at precision and completenessof a classroom text in an upper undergraduate to graduate level course.This book was originally planned as a joint project between Marta Lewicka(University of Pittsburgh) and Yuval Peres (then Microsoft Research). Due toan unforeseen turn of events, neither the collaboration nor the execution of theproject in the priorly conceived forms, could have been pursued.The author wishes to dedicate this book to all women in mathematics, withadmiration and encouragement. The publishing profit will be donated to theAssociation for Women in Mathematics. Marta Lewicka,Pittsburgh, October 2019.v
The linear case: random walk and harmonicfunctions
In this Chapter we present the basic relation between the linear potential the-ory and random walks . This fundamental connection, developed by Ito, Doob,L´evy and others, relies on the observation that harmonic functions and martin-gales share a common cancellation property, expressed via mean value proper-ties . It turns out that, with appropriate modifications, a similar observation andapproach can be applied also in the nonlinear case, which is of main interestin these Course Notes. Thus, the present Chapter serves as a stepping stonetowards gaining familiarity with more complex constructions of Chapters 3-6.After recalling the equivalent defining properties of harmonic functions inSection 2.1, in Section 2.2 we introduce the ball walk . This is an infinite hori-zon discrete process, in which at each step the particle, initially placed at somepoint x in the open, bounded domain D ⇢ R N , is randomly advanced to a newposition, uniformly distributed within the following open ball: centered at thecurrent placement, and with radius equal to the minimum of the parameter ✏ and the distance from the boundary @ D . With probability one, such process ac-cumulates on @ D and u ✏ ( x ) is then defined as the expected value of the givenboundary data F at the process limiting position. Each function u ✏ is harmonic,and we show in Sections 2.3 and 2.4, that if @ D is regular , then each u ✏ coin-cides with the unique harmonic extension of F in D . One su cient conditionfor regularity is the exterior cone condition , as proved in Section 2.5.Our discussion and proofs are elementary, requiring only a basic knowledgeof probabilistic concepts, such as: probability spaces, martingales and Doob’stheorem. For convenience of the reader, these are gathered in Appendix A.The slightly more advanced material which may be skipped at first reading,is based on the Potential Theoretic and the Brownian motion arguments from,respectively, Appendix C and Appendix B. Both approaches allow to deducethat functions in the family { u ✏ } ✏ (0 , are one and the same function, regardlessof the regularity of @ D . This fact is obtained first in Section 2.6* by proving8 .1 The Laplace equation and harmonic functions u ✏ coincide with the Perron solution of the Dirichlet problem for bound-ary data F . The same follows in Section 2.7* by checking that the ball walkconsists of discrete realisations along the Brownian motion trajectories, to thee ↵ ect that u ✏ equal the Brownian motion harmonic extension of F .Thus, the three classical approaches to finding the harmonic extension by:(i) evaluating the expectation of the values of the (discrete) ball walk at itslimiting infinite horison boundary position;(ii) taking infima / suprema of super- and sub-harmonic functions obeying com-parison with the boundary data;(iii) evaluating the expectation of the values of the (continuous) Brownian mo-tion at exiting the domain;are shown to naturally coincide when F is continuous. Among the most important of all PDEs is the
Laplace equation . In this Sectionwe briefly recall the relevant definitions and notation; for the proofs and areview of basic properties we refer to Section C.3 in Appendix C.Let D ⇢ R N be an open, bounded, connected set. The Euler-Lagrange equa-tion for critical points of the following quadratic energy functional: I ( u ) = Z D | r u ( x ) | d x is expressed by the second order partial di ↵ erential equation: u ⌘ N X i = @ u ( @ x i ) = D , whose solutions are called harmonic functions . The operator is defined inthe classical sense only for C functions u , however a remarkable property ofharmonicity is that it can be equivalently characterised via mean value prop-erties that do not require u to be even continuous. At the same time, harmonicfunctions are automatically smooth. More precisely, the following conditionsare equivalent (the proof will be recalled in Section C.3):(i) A locally bounded, Borel function u : D ! R satisfies the mean valueproperty on balls : u ( x ) = ? B r ( x ) u ( y ) d y for all ¯ B r ( x ) ⇢ D . The linear case: random walk and harmonic functions (ii) A locally bounded, Borel function u : D ! R satisfies for each x D and almost every r (0 , dist( x , @ D )) the mean value property on spheres : u ( x ) = ? @ B r ( x ) u ( y ) d N ( y ) . (iii) The function u is smooth: u C ( D ) , and there holds u = D .We also remark at this point that, Taylor expanding any function u C ( D )and averaging term by term on ¯ B ✏ ( x ) ⇢ D , leads to the mean value expansion ,also called in what follows the averaging principle : ? B ✏ ( x ) u ( y ) d y = u ( x ) + ✏ N + u ( x ) + o ( ✏ ) as ✏ ! + , (2.1)which in fact is consistent with interpreting u as the (second order) errorfrom harmonicity. This point of view is central to developing the probabilisticinterpretation of the general p -Laplace equations, which is the goal of theseCourse Notes. While we will not need (2.1) in order to construct the randomwalk and derive its connection to the Laplace equation in the linear setting p = p at any p (1 , ), and ultimately leading to the Tug of War games with random noise. In this Section we construct the discrete stochastic process whose value willbe shown to equal the harmonic function with prescribed boundary values.The probability space of the ball walk process is defined as follows. Con-sider ( ⌦ , F , P ), where ⌦ is the unit ball B (0) ⇢ R N , the -algebra F consists of Borel subsets of ⌦ , and P is the normalised Lebesgue measure: P ( D ) = | D || B (0) | for all D F , For any n N , we denote by ⌦ n = ( ⌦ ) n the Cartesian product of n copies of ⌦ , and by ( ⌦ n , F n , P n ) the corresponding product probability space. Further,the countable product ( ⌦ , F , P ) is defined as in Theorem A.12 on: ⌦ ⌘ ( ⌦ ) N = Y i = ⌦ = n ! = { w i } i = ; w i B (0) for all i N o . .2 The ball walk -algebra F n with the sub- -algebra of F consisting of setsof the form F ⇥ Q i = n + ⌦ for all F F n . Note that {F n } n = where F = { ; , ⌦ } ,is a filtration of F and that F is the smallest -algebra containing S n = F n . Definition 2.1
Let D ⇢ R N be an open, bounded, connected set. The ballwalk for ✏ (0 ,
1) and x D , is recursively defined (see Figure 2.1) as thefollowing sequence of random variables { X ✏ , x n : ⌦ ! D} n = : X ✏ , x ⌘ x , X ✏ , x n ( w , . . . , w n ) = X ✏ , x n ( w , . . . , w n ) + ✏ ^ dist( X ✏ , x n , @ D ) w n for all n w , . . . w n ) ⌦ n . (2.2) Figure 2.1 The ball walk and the process { X ✏ , x n } n = in (2.2). We will often write: x n = X ✏ , x n ( w , . . . , w n ). Intuitively, { x n } n = describe theconsecutive positions of a particle initially placed at x D , along a discretepath consisting of a succession of random steps of magnitude at most ✏ . Thesize of steps decreases as the particle approaches the boundary @ D . The po-sition x n D is obtained from x n by sampling uniformly on the open ball B ✏ ^ dist ( x n , @ D ) ( x n ). It is clear that each random variable X ✏ , x n : ⌦ ! R N is F n -measurable and that it depends only on the previous position x n , itsdistance from @ D and the current random outcome w n ⌦ . Lemma 2.2
In the above context, the sequence { X ✏ , x n } n = is a martingalerelative to the filtration {F n } n = , namely: E ( X ✏ , x n | F n ) = X ✏ , x n P a.s for all n . The linear case: random walk and harmonic functionsMoreover, there exists a random variable X ✏ , x : ⌦ ! @ D such that: lim n !1 X ✏ , x n = X ✏ , x P a.s. (2.3) Proof Since the sequence { X ✏ , x n } n = is bounded in view of boundednessof D , Theorem A.38 will yield convergence in (2.3) provided we check themartingale property. Indeed it follows that (see Lemma A.17): E ( X ✏ , x n | F n )( w , . . . , w n ) = Z ⌦ X ✏ , x n ( w , . . . , w n ) d P ( w n ) = x n + ✏ ^ dist( x n , @ D ) Z ⌦ w n d P ( w n ) = X ✏ , x n ( w , . . . , w n ) for P n -a.e. ( w , . . . , w n ) ⌦ n . It remains to prove that the limiting random variable X ✏ , x : ⌦ ! ¯ D satisfies P -a.s. the boundary accumulation property: X ✏ , x @ D . Observe that: lim n !1 X ✏ , x n = X ✏ , x \ { X ✏ , x D} ⇢ [ n N , (0 , ✏ ) \ Q A ( n , ) , (2.4)where A ( n , ) = dist( X ✏ , x i , @ D ) and | X ✏ , x i + X ✏ , x i | for all i n . Then: A ( n , ) ⇢ ! ⌦ ; | w i |
12 for all i > n . Indeed, if ! = { w i } i = A ( n , ) with < ✏ , it follows that: | X ✏ , x i + ( ! ) X ✏ , x i ( ! ) | = ✏ ^ dist( X ✏ , x i ( ! ) , @ D ) | w i + | ( ✏ ^ ) | w i + | = | w i + | , which implies that | w i + | for all i n . Concluding: P A ( n , ) lim i !1 P ( B (0)) i n = n N and all (0 , ✏ ) . Hence, the event in the left hand side of (2.4) has probability 0. ⇤ Given now a continuous function F : @ D ! R , define: u ✏ ( x ) ⌘ E ⇥ F X ✏ , x ⇤ = Z ⌦ F X ✏ , x d P . (2.5)Note that the above construction obeys the comparison principle . Namely, if F , ¯ F : @ D ! R are two continuous functions such that F ¯ F on @ D , then thecorresponding u ✏ and ¯ u ✏ satisfy: u ✏ ¯ u ✏ in D . .2 The ball walk Remark 2.3
It is useful to view the boundary function F as the restriction on @ D of some continuous F : ¯ D ! R , see Exercise 2.7 (i). Then we may write: u ✏ ( x ) = lim n !1 Z ⌦ F X ✏ , x n d P . (2.6)Since for each n F X ✏ , x n is jointly Borel-regular in thevariables x D and ! ⌦ n , it follows by Theorem A.11 that x E [ F X ✏ , x n ] is Borel-regular. Consequently, u ✏ : D ! R is also Borel.In what follows, we will denote the average A u of an integrable function u : D ! R on a ball B ( x ) ⇢ D by: A u ( x ) ⌘ ? B ( x ) u ( y ) d y . Directly from Definition 2.1 and (2.5) we conclude the satisfaction of the meanvalue property for each u ✏ on the sampling balls from (2.2): Theorem 2.4
Let D ⇢ R N be open, bounded, connected, and let F : @ D ! R be continuous. Then, the function u ✏ : D ! R defined in (2.5)and equivalently in (2.6), is continuous and satisfies:u ✏ ( x ) = A ✏ ^ dist ( x , @ D ) u ✏ ( x ) for all x D . Proof
Fix ✏ (0 ,
1) and x D . For each n ⌦ n , F n , P n ) asthe product of probability spaces ( ⌦ , F , P ) and ( ⌦ n , F n , P n ). ApplyingFubini’s Theorem (Theorem A.11), we get: E ⇥ F X ✏ , x n ⇤ = Z ⌦ Z ⌦ n F X ✏ , x n ( w , . . . , w n ) d P n ( w , . . . , w n ) d P ( w ) = Z ⌦ E ⇥ F X ✏ , X ✏ , x ( w ) n ⇤ d P ( w ) , where F : ¯ D ! R is some continuous extension of its given values on @ D , asin (2.6). Passing to the limit with n ! 1 and changing variables, we obtain: u ✏ ( x ) = Z ⌦ u ✏ X ✏ , x ( w ) d P ( w ) = Z ⌦ u ✏ x + ( ✏ ^ dist( x , @ D )) w d P ( w ) = ? B ✏ ^ dist( x , D )( x ) u ✏ ( y ) d y . The linear case: random walk and harmonic functions
Continuity of u ✏ follows directly from the averaging formula and we leave itas an exercise (see Exercise 2.7 (ii)). ⇤ The next two statements imply uniqueness of classical solutions to the bound-ary value problem for the Laplacian. The same property, in the basic analyticalsetting that we review in Section C.3, follows via the maximum principle.
Corollary 2.5
Let ✏ (0 , , x D and let { X ✏ , x n } n = be as in (2.2). In thesetting of Theorem 2.4, the sequence { u ✏ X ✏ , x n } n = is a martingale relative tothe filtration {F n } n = .Proof Indeed, Lemma A.17 yields for all n E u ✏ X ✏ , x n | F n ( w , . . . , w n ) = Z ⌦ ( u ✏ X ✏ , x n )( w , . . . , w n ) d P ( w n ) = Z ⌦ u ✏ ⇣ X ✏ , x n ( w , . . . , w n ) + ✏ ^ dist( x n , @ D ) w n ⌘ d P ( w n ) = ? B ✏ ^ dist( x n , D )( x n ) u ✏ ( y ) d y = ( u ✏ X x n )( w , . . . , w n ) , (2.7)valid for P n -a.e. ( w , . . . , w n ) ⌦ n . ⇤ Lemma 2.6
In the setting of Theorem 2.4, assume that u C ( ¯ D ) solves: u = in D , u = F on @ D . (2.8) Then u ✏ = u for all ✏ (0 , . In particular, (2.8) has at most one solution.Proof We first claim that given x D and ✏ (0 , { u X ✏ , x n } n = is a martingale relative to {F n } n = . This property follows exactly as in(2.7), where u ✏ is now replaced by u and where the mean value property forharmonic functions (C.8) is used instead of the single-radius averaging formulaof Theorem 2.4. Consequently, we get: u ( x ) = E [ u X ✏ , x ] = E [ u X ✏ , x n ] for all n . Since the right hand side above converges to u ✏ ( x ) with n ! 1 , it follows that u ( x ) = u ✏ ( x ). To prove the second claim, recall that u ✏ ( x ) depends only onthe boundary values u | @ D = F and not on their extension u on ¯ D . This yieldsuniqueness of the harmonic extension in (2.8). ⇤ We finally remark that the mean value property stated in Theorem 2.4 suf-fices to conclude that each u ✏ is harmonic (see Section C.3). One can also show .2 The ball walk { u ✏ } ✏ (0 , are the same, even in the absence ofthe classical harmonic extension u satisfying (2.8). This general result will begiven two independent proofs in Sections 2.6* and 2.7*. In the next Section,we provide an elementary proof in domains that are su ciently regular. An en-tirely similar strategy, based on showing the uniform convergence of { u ✏ } ✏ ! in¯ D and analyzing its limit, will be adopted in Chapters 3-6 for the p -harmoniccase, p (1 , ), in the context of Tug of War with noise. Exercise 2.7 (i) Let F : A ! R be a continuous function on a compact set A ⇢ R N . Verify that, setting: F ( x ) ⌘ min y A n F ( y ) + | x y | dist( x , A ) o for all x R N \ A , defines a continuous extension of F on R N . This construction is due toHausdor ↵ and it provides a proof of the Tietze extension theorem.(ii) Let u : R N ! R be a bounded, Borel function and let ✏ : R N ! (0 , ) becontinuous. Show that the function: x A ✏ ( x ) u ( x ) is continuous on R N . Exercise 2.8
Modify the construction of the ball walk to the sphere walk using the outline below.(i) Let ⌦ = @ B (0) ⇢ R N and let P = N be the normalised sphericalmeasure on the Borel -algebra F of subsets of ⌦ (see Example A.9).Define the induced probability spaces ( ⌦ , F , P ) and { ( ⌦ n , F n , P n ) } n = asin the case of the ball walk. For every ✏ (0 ,
1) and x D , let { X ✏ , x n : D ! R N } n = be the sequence of random variables in: X ✏ , x ⌘ x and for all n w , . . . , w n ) ⌦ n : X ✏ , x n ( w , . . . , w n ) = x n + ⇣ ✏ ^
12 dist( x n , @ D ) ⌘ w n where x n = X ✏ , x n ( w , . . . , w n ) . Prove that { X ✏ , x n } n = is a martingale relative to the filtration {F n } n = andthat (2.3) holds for some random variable X ✏ , x : ⌦ ! @ D .(ii) For a continuous function F : @ D ! R , define u ✏ : D ! R according to(2.5). Show that u ✏ is Borel-regular and that it satisfies: u ✏ ( x ) = ? @ B ✏ ^ dist( x , @ D )( x ) u ✏ ( y ) d N ( y ) for all x D . (iii) Deduce that if F has a harmonic extension u on ¯ D as in (2.8), then u ✏ = u for all ✏ (0 , The linear case: random walk and harmonic functions
The main result of this Section states that the uniform limits of values { u ✏ } ✏ ! of the ball walk that we introduced in Section 2.2, are automatically harmonic.The proof relies on checking that each limiting function u satisfies the meanvalue property on spheres. This is achieved by applying Doob’s theorem to u ✏ evaluated along its own walk process { X ✏ , x n } n = , and choosing to stop on exitingthe ball whose boundary coincides with the given sphere. Theorem 2.9
Let J ⇢ (0 , be a sequence decreasing to . Assume that { u ✏ } ✏ J defined in (2.5), converges locally uniformly in D , as ✏ ! , ✏ J,to some u C ( D ) . Then u must be harmonic.Proof In virtue of Theorem C.19, it su ces to prove that: u ( x ) = ? @ B r ( x ) u ( y ) d N ( y ) for all B r ( x ) ⇢ D . (2.9)Fix x D and r dist( x , @ D ), and for each ✏ J consider the followingrandom variable ⌧ ✏ : ⌦ ! N [ { + } : ⌧ ✏ = inf n X ✏ , x n < B r ( x ) , where { X ✏ , x n } n = is the usual sequence of the token positions (2.2) in the ✏ -ball walk started at x . Clearly, ⌧ ✏ is finite a.s. in view of convergence to theboundary in (2.3) and it is a stopping time relative to the filtration {F n } n = . ByCorollary 2.5, Doob’s theorem (Theorem A.31 (ii)) yields: u ✏ ( x ) = E ⇥ u ✏ X ✏ , x ⇤ = E ⇥ u ✏ X ✏ , x ⌧ ⇤ , while by passing to the limit with ✏ ! u ( x ) = lim ✏ ! , ✏ J E ⇥ u ✏ X ✏ , x ⌧ ⇤ = lim ✏ ! , ✏ J Z B r + ✏ ( x ) \ B r ( x ) u ( y ) d ✏ ( y ) . (2.10)The Borel probability measures { ✏ } ✏ (0 , r ) are here defined on ¯ B r ( x ) \ B r ( x )by the push-forward procedure, as in Exercise A.8: ✏ ( A ) ⌘ P X ✏ , x ⌧ A . We now identify the limit in the right hand side of (2.10). Observe that,by construction, the measures ✏ are rotationally invariant. Further, by Pro-horov’s theorem (Theorem A.10), each subsequence of { ✏ } ✏ ! , ✏ J has a fur-ther subsequence that converges (weakly- ⇤ ) to a Borel probability measure µ on ¯ B r ( x ) \ B r ( x ). Since each ✏ is supported in B r + ✏ ( x ) \ B r ( x ), the limit µ .3 The ball walk and harmonic functions @ B r ( x ). Also, µ is rotationally invariant in view of thesame property of each ✏ . Consequently, µ = N must be the uniquely de-fined, normalised spherical measure on @ B r ( x ) (see Exercises 2.10 and 2.11).As the limit does not depend on the chosen subsequence of J , we conclude:lim ✏ ! , ✏ J Z ¯ B r ( x ) \ B r ( x ) u ( y ) d ✏ ( y ) = ? @ B r ( x ) u ( y ) d N ( y ) . Together with (2.10), this establishes (2.9) as claimed. ⇤ Figure 2.2 The stopping position x ⌧ ✏ in the proof of Theorem 2.9. Exercise 2.10
Show that every (weak- ⇤ ) limit point of the family of proba-bility measures { ✏ } ✏ (0 , r ) defined in (2.10) must be rotationally invariant andsupported on @ B r ( x ). Exercise 2.11
Using the following outline, prove that the only Borel proba-bility measure µ on @ B (0) ⇢ R N that is rotationally invariant, is the normalizedspherical measure N .(i) Fix an open set U ⇢ @ B (0) and consider the sequence of Borel func-tions n x µ ( U \ B ( x , n )) µ ( B ( x , n )) o n = , where B ( x , r ) denotes the ( N @ B (0) centered at x and with radius r (0 , N ( U ) ⇣ lim inf n !1 N ( B ( x , n )) µ ( B ( x , n )) ⌘ · µ ( U ) , (2.11)where both quantities N ( B ( x , n )) and µ ( B ( x , n )) are independent of x @ B (0) because of the rotational invariance.8 The linear case: random walk and harmonic functions (ii) Exchange the roles of µ and N in the above argument and conclude:lim n !1 N ( B ( x , n )) µ ( B ( x , n )) = . Thus, µ ( U ) = N ( U ) for all open sets U , so there must be µ = N .This proof is due to Christensen (1970) and the statement above is a partic-ular case of Haar’s theorem on uniqueness of invariant measures on compacttopological groups. We now investigate conditions assuring the validity of the uniform convergenceassumption of Theorem 2.9. It turns out that such condition may be formulatedindependently of the boundary data F , only in terms of the behaviour of the ballwalk (2.2) close to @ D , which is further guaranteed by a geometrical su cientcondition in the next Section. In Theorem 2.14 we will show how the boundaryregularity of the process can be translated (via walk coupling) into the interiorregularity, resulting in the existence of a harmonic extension u of F on ¯ D , andultimately yielding u ✏ = u for all ✏ (0 , Definition 2.12
Consider the ball walk (2.2) on a domain D ⇢ R N .(a) We say that a boundary point y @ D is walk-regular if for every ⌘ , > (0 , ) and ˆ ✏ (0 ,
1) such that: P X ✏ , x B ( y ) ⌘ for all ✏ (0 , ˆ ✏ ) and all x B ˆ ( y ) \ D , where X ✏ , x is the limit in (2.3) of the ✏ -ball walk started at x .(b) We say that D is walk-regular if every y @ D is walk-regular. Lemma 2.13
Assume that the boundary point y @ D of a given open,bounded, connected domain D , is walk-regular. Then for every continuous F : @ D ! R , the family { u ✏ } ✏ ! defined in (2.5) satisfies the following. For every ⌘ > there is ˆ > and ˆ ✏ (0 , such that: | u ✏ ( x ) F ( y ) | ⌘ for all ✏ (0 , ˆ ✏ ) and all x B ˆ ( y ) \ D . (2.12) Proof
Given ⌘ >
0, let > | F ( y ) F ( y ) | ⌘ y @ D such that | y y | < . .4 Convergence at the boundary and walk-regularity Figure 2.3
Walk-regularity of a boundary point y @ D . By Definition 2.12, we choose ˆ ✏ and ˆ corresponding to ⌘ k F k + and . Then: | u ✏ ( x ) F ( y ) | Z ⌦ | F X ✏ , x F ( y ) | d P P X ✏ , x < B ( y ) · k F k + Z X ✏ , x B ( y ) | F X ✏ , x F ( y ) | d P ⌘ k F k + · k F k + ⌘ ⌘ , for all x B ˆ ( y ) \ D and all ✏ (0 , ˆ ✏ ). This completes the proof. ⇤ By Lemma 2.13 and Theorem 2.9 we achieve the main result of this Chapter:
Theorem 2.14
Let D be walk-regular. Then, for every continuous F : @ D ! R , the family { u ✏ } ✏ (0 , in (2.5) satisfies u ✏ = u, where u C ( ¯ D ) isthe unique solution of the boundary value problem: u = in D , u = F on @ D . Proof Let F : @ D ! R be a given continuous function. We will show that { u ✏ } ✏ ! is “asymptotically equicontinuous in D ”, i.e.: for every ⌘ > > ✏ (0 ,
1) such that: | u ✏ ( x ) u ✏ ( y ) | ⌘ for all ✏ (0 , ˆ ✏ )and all x , y D with | x y | . (2.13)Since { u ✏ } ✏ ! is equibounded (by k F k ), condition (2.13) imply that for everysequence J ⇢ (0 ,
1) converging to 0, one can extract a further subsequenceof { u ✏ } ✏ J that converges locally uniformly in ¯ D . Further, in view of (2.12) itfollows that u C ( ¯ D ) and u = F on @ D (see Exercise 2.17). By Theorem 2.9,we get that u is harmonic in D and the result follows in virtue of Lemma 2.6.0 The linear case: random walk and harmonic functions To show (2.13), fix ⌘ > > | F ( y ) F (¯ y ) | ⌘ for all y , ¯ y @ D with | y ¯ y | . By (2.12), for each y @ D there existsˆ ( y ) (0 , ¯ ) and ˆ ✏ ( y ) (0 ,
1) satisfying: | u ✏ ( x ) F ( y ) | ⌘ ✏ (0 , ˆ ✏ ( y )) and all x B ˆ ( y ) ( y ) \ D . The family of balls { B ˆ ( y ) ( y ) } y @ D is then a covering of the compact set @ D ; let { B ˆ ( y i ) ( y i ) } ni = be its finite sub-cover and set ˆ ✏ = min i = ... n ˆ ✏ ( y i ). Clearly: @ D + B (0) ⇢ n [ i = B ˆ ( y i ) ( y i )for some > < ¯ . This implies: | u ✏ ( x ) u ✏ ( y ) | ⌘ for all ✏ (0 , ˆ ✏ )and all x , y @ D + B (0) \ D with | x y | . (2.14) To conclude the proof of (2.13), fix ✏ (0 , ˆ ✏ ^ ) and let x , y D satisfydist( x , @ D ) , dist( y , @ D ) and | x y | < . Define ⌧ : ⌦ ! N [ { + } : ⌧ = min n
1; dist( x n , @ D ) < or dist( y n , @ D ) < , where { x n = X ✏ , x n } n = and { y n = X ✏ , y n } n = denote the consecutive positions inthe process (2.5) started at x and y , respectively. It is clear that ⌧ is finite P -a.s. in view of convergence to the boundary in (2.3), and it is a stopping timerelative to the filtration {F n } n = .By Corollary 2.5 and Doob’s theorem (Theorem A.31 (ii)) it follows that: u ✏ ( x ) = E ⇥ u ✏ X ✏ , x ⌧ ⇤ and u ✏ ( y ) = E ⇥ u ✏ X ✏ , y ⌧ ⇤ . Since | X ✏ , x ⌧ X ✏ , y ⌧ | = | x y | < and X ✏ , x ⌧ , X ✏ , y ⌧ @ D + B (0) \ D for a.e. ! ⌦ , we conclude by (2.14) that: | u ✏ ( x ) u ✏ ( y ) | Z ⌦ | u ✏ X ✏ , x ⌧ u ✏ X ✏ , y ⌧ | d P ⌘ . This ends the proof of (2.13) and of the Theorem. ⇤ Walk-regularity is, in fact, equivalent to convergence of u to the right bound-ary values. We have the following observation, converse to Lemma 2.13: Lemma 2.15
If y @ D is not walk-regular, then there exists a continuousfunction F : @ D ! R , such that for u ✏ in (2.5) there holds: lim sup x ! y , ✏ ! u ✏ ( x ) , F ( y ) . .5 A su cient condition for walk-regularity Proof
Define F ( y ) = | y y | for all y @ D . By assumption, there exists ⌘ , > { ✏ i } i = , { x j D} j = such that:lim j !1 ✏ j = , lim j !1 x j = y and P X ✏ j , x j < B ( y ) > ⌘ for all j , where each X ✏ j , x j above stands for the limiting random variable in (2.3) corre-sponding to the ✏ j - ball walk. By the nonnegativity of F , it follows that: u ✏ j ( x j ) F ( y ) = Z ⌦ F X ✏ j , x j d P Z { X ✏ j , xj < B ( y ) } F X ✏ j , x j d P > ⌘ > , proving the claim. ⇤ Exercise 2.16
Show that if D is walk-regular then ˆ and ˆ ✏ in Definition 2.12(a) can be chosen independently of y (i.e. ˆ and ˆ ✏ depend only on the param-eters ⌘ and ). Exercise 2.17
Let { u ✏ } ✏ J be an equibounded sequence of functions u ✏ : D ! R defined on an open, bounded set D ⇢ R N , and satisfying (2.12), (2.13) withsome continuous F : @ D ! R . Prove that { u ✏ } ✏ J must have a subsequence thatconverges uniformly, as ✏ ! ✏ J , to a continuous function u : ¯ D ! R . cient condition for walk-regularity In this Section we state a geometric condition (exterior cone condition) imply-ing the validity of the walk-regularity condition introduced in Definition 2.12.We remark that the exterior cone condition in Theorem 2.19 may be weakenedto the so-called exterior corkscrew condition, and that the analysis below isvalid not only in the presently studied linear case of p =
2, but in the nonlinearsetting of an arbitrary exponent p (1 , ) as well. This will be explained inChapter 6, with proofs conceptually based on what follows.We begin by observing a useful technical reformulation of the regularitycondition in Definition 2.4. Namely, at walk-regular boundary points y notonly the limiting position of the ball walk may be guaranteed to stay close to y with high probability, but the same local property may be, in fact, requestedfor the whole walk trajectory, with uniformly positive probability. Lemma 2.18
Let D ⇢ R N be an open, bounded, connected domain. For agiven boundary point y @ D , assume that there exists ✓ < such that forevery > there are ˆ (0 , ) and ˆ ✏ (0 , with the following property. Forall ✏ (0 , ˆ ✏ ) and all x B ˆ ( y ) \ D there holds: P n X ✏ , x n < B ( y ) ✓ , (2.15)2 The linear case: random walk and harmonic functionswhere { X ✏ , x n } n = is the ✏ -ball walk defined in (2.2). Then y is walk-regular.Proof Fix ⌘ , > m N be such that: ✓ m ⌘ . Define the tuples { ✏ k } mk = , { ˆ k } m k = and { k } mk = inductively, in: m = , ✏ m = k (0 , k ) , ✏ k (0 , ✏ k ) for all k = , . . . , m so that: P n X x n < B k ( y ) ✓ for all x B ˆ k ( y ) \ D and all ✏ (0 , ✏ k ) , k (0 , ˆ k ) for all k = , . . . , m . (2.16)We finally set: ˆ ✏ ⌘ ✏ ^ min k = ,..., m | ˆ k k | and ˆ ⌘ ˆ . Fix x B ˆ ( y ) \ D and ✏ (0 , ˆ ✏ ). We will show that: P n X ✏ , x n < B k ( y ) ✓ · P n X ✏ , x n < B k ( y ) for all k = , . . . , m . (2.17)Together with the inequality in (2.16) for k =
1, the above bounds will yield: P X ✏ , x < B ( y ) P n X ✏ , x n < B ( y ) ✓ m ⌘ . Since ⌘ and were arbitrary, the validity of the condition in Definition 2.12will thus be justified, proving the walk-regularity of y . Towards showing (2.17), we denote:˜ ⌦ = { n X ✏ , x n < B k ( y ) } ⇢ ⌦ . Without loss of generality, we may assume that P ( ˜ ⌦ ) >
0, because otherwise P n X ✏ , x n < B k ( y ) P n X ✏ , x n < B k ( y ) = ⌦ , ˜ F , ˜ P ) defined by:˜ F = { A \ ˜ ⌦ ; A F } and ˜ P ( A ) = P ( A ) P ( ˜ ⌦ ) for all A ˜ F . Also, let the measurable space ( ⌦ f in , F f in ) be given by: ⌦ f in = S n = ⌦ n andby taking F f in to be the smallest -algebra containing S n = F n . Then the fol-lowing random variable ⌧ k : ˜ ⌦ ! N : ⌧ k ⌘ min n X ✏ , x n < B k ( y ) .5 A su cient condition for walk-regularity ⌦ with respect to the induced filtration { ˜ F n = { A \ ˜ ⌦ ; A F n }} n = . We consider two further random variables below: Y : ˜ ⌦ ! ⌦ f in Y { w i } i = ⌘ { w i } ⌧ k i = Y : ˜ ⌦ ! ⌦ Y { w i } i = ⌘ { w i } i = ⌧ k + and observe that they are independent, namely:˜ P Y A ) · ˜ P Y A ) = ˜ P { Y A } \ { Y A } for all A F f in , A F . We now apply Lemma A.21 to Y , Y and to the indicator function: Z { w i } si = , { w i } i = s + ⌘ n X ✏ , x n ( { w i } i = ) < B k ( y ) that is a random variable on the measurable space ⌦ f in ⇥ ⌦ , equipped with theproduct -algebra of F f in and F . It follows that: P n X ✏ , x n < B k ( y ) = Z ˜ ⌦ Z ( Y , Y ) d ˜ P = Z ˜ ⌦ f ( ! ) d ˜ P ( ! ) , where for each ! = { w i } i = ˜ ⌦ we have: f ( ! ) = P ⇣ { ¯ w i } i = ⌦ ; n X ✏ , x n { w i } ⌧ k i = , { ¯ w i } i = ⌧ k + < B k ( y ) ⌘ = P ( ˜ ⌦ ) · P ⇣ n X ✏ , x ⌧ k n < B k ( y ) ⌘ P ( ˜ ⌦ ) · ✓ , in view of x ⌧ k B ˆ k ( y ) and the construction assumption (2.16). This ends theproof of (2.17) and of the lemma. ⇤ The main result of this Section is a geometric su cient condition for walk-regularity. When combined with Theorem 2.14, it implies that every continu-ous boundary data F admits the unique harmonic extension to any Lipschitzdomain D . This extension automatically coincides with all process values u ✏ ,regardless of the choice of the upper bound sampling radius ✏ (0 , Theorem 2.19
Let D ⇢ R N be open, bounded, connected and assumethat y @ D satisfies the exterior cone condition , i.e. there exists a finitecone C ⇢ R N \ D with the tip at y . Then y is walk-regular.Proof The exterior cone condition assures the existence of a constant R > ciently small ⇢ > z C satisfying: | z y | = ⇢ (1 + R ) and B R ⇢ ( z ) ⇢ C ⇢ R N \ D . (2.18)4 The linear case: random walk and harmonic functions
Let > ciently small and define z R N as in (2.18) with ⇢ = ˆ , where we set ˆ = + R . We will show that condition(2.15) holds for all ✏ (0 , Figure 2.4
The concentric balls in the proof of Theorem 2.19.
Fix x B ˆ ( y ) \ D and consider the profile function v : (0 , ) ! R in: v ( t ) = ( sgn( N t N for N , , log t for N = . By Exercise C.27, the radial function x v ( | x z | ) is harmonic in R N \ { z } ,so in view of Lemma 2.6 the sequence of random variables { v | X ✏ , x n z |} n = is a martingale with respect to the filtration {F n } n = . Further, define the randomvariable ⌧ : ⌦ ! N [ { + } by: ⌧ ⌘ inf n X ✏ , x n < B ( y ) , where we suppress the dependence on ✏ in the above notation. Applying Doob’stheorem (Theorem A.31 (ii)) we obtain: v | x z | = E ⇥ v | X ✏ , x z | ⇤ = E ⇥ v | X ✏ , x ⌧ ^ n z | ⇤ for all n , because for every n
0, the a.s. finite random variable ⌧ ^ n is a stopping time.Passing to the limit with n ! 1 and recalling the definition (2.3), now yields: v | x z | = Z { ⌧ < + } v | X ✏ , x ⌧ z | d P + Z { ⌧ =+ } v | X ✏ , x z | d P . .6* The ball walk values and Perron solutions v is a decreasing function, this results in: v (2 + R )ˆ v | x z | P ( ⌧ < + ) · v (3 + R )ˆ + P ( ⌧ = + ) · v ( R ˆ ) = P ( ⌧ < + ) · ⇣ v (3 + R )ˆ v ( R ˆ ) ⌘ + v ( R ˆ ) , in view of the following bounds: | x z | | x y | + | y z | < (2 + r )ˆ , | X ✏ , x ⌧ z | | X ✏ , x ⌧ y | | y z | = (1 + R )ˆ = (3 + R )ˆ , | X ✏ , x z | R ˆ . Finally, noting that v ((3 + R )ˆ ) v ( R ˆ ) < P ( ⌧ < + ) v ( R ˆ ) v ((2 + R )ˆ ) v ( R ˆ ) v (3 + R )ˆ = v ( R ) v (2 + R ) v ( R ) v (3 + R ) . This establishes (2.15) with the constant ✓ = v ( R ) v (2 + R ) v ( R ) v (3 + R ) <
1, that depends onlyon the dimension N and the cone C . By Lemma 2.18, the proof is done. ⇤ Remark 2.20
An alternative su cient condition for walk-regularity is thesimple-connectedness of D ⇢ R . The proof follows through the identificationof { X ✏ , x n } n = as the discrete realisation of the Brownian path in Section 2.7* andapplying the same reasoning as in the proof of Theorem 6.21. Indeed, in Chap-ter 6 we will give su cient conditions for the so-called game-regularity, in thecontext of the Dirichlet problem for p -Laplacian, p (1 , ), encompassingand extending the classical discussion in the present Chapter. In this Section we prove that all functions in the family { u ✏ } ✏ (0 , defined in(2.5) for a continuous F : @ D ! R , are always one and the same function,coinciding with the so-called Perron solution of the Dirichlet problem: u = D , u = F on @ D . (2.19)This material may be skipped at first reading, as it requires familiarity withmore advanced PDE notions of Perron’s method and Wiener’s resolutivity. Therelated presentation in the general nonlinear case of p , p (1 , ), can befound in Section C.7 of Appendix C. Below we recall this classical approachin the linear setting p =
2; for proofs we refer to the textbook by Helms (2014).6
The linear case: random walk and harmonic functions
Definition 2.21 (i) A function v C ( D ) is called superharmonic in D ,provided that for every ¯ B r ( x ) ⇢ D and every h C ( ¯ B r ( x )) that is harmonicin B r ( x ) and satisfies h v on @ B r ( x ), there holds: h v in B r ( x ).(ii) A function v C ( D ) is subharmonic in D , when ( v ) is superharmonic.(iii) Given a continuous boundary data function F : @ D ! R , we define the upper and lower Perron solutions to (2.19):¯ h F = inf n v C ( ¯ D ) superharmonic, such that F v on @ D o , h F = sup n v C ( ¯ D ) subharmonic, such that v F on @ D o . The usual maximum principle argument implies that if v , v C ( ¯ D ) are,respectively, subharmonic and superharmonic, and if v v on @ D , then v v in D . In this comparison result, the conclusion may be in fact strengthenedto: v < v or v ⌘ v in D . It follows that ¯ h F and h F are well defined functions,and also: h F ¯ h F . One may further prove, by means of the harmonic lifting,that ¯ h F and h F are harmonic in D . The celebrated Wiener resolutivity theorem in Wiener (1925) states the uniqueness of this construction:
Theorem 2.22
Let D ⇢ R N be open, bounded and connected. Every bound-ary data F C ( @ D ) is resolutive , i.e. the two functions ¯ h F and h F coincide in D . The resulting harmonic function is called the Perron solution to (2.19):h F = ¯ h F = h F . (2.20)We remark that h F does not have to attain the prescribed boundary value F ( x ) at each x @ D ; it necessarily does so, however, for all points outside ofa set whose 2-capacity is zero (see Section C.7).By identifying the super- / subharmonic functions via mean value inequali-ties and comparing u ✏ with ¯ h F and h F , we obtain the main result of this Section: Theorem 2.23
Let D ⇢ R N be open, bounded, connected and let F C ( @ D ) . For each ✏ (0 , , functions u ✏ in (2.5) satisfy: u ✏ = h F in D .Proof Let v C ( ¯ D ) be superharmonic and satisfy F v on @ D . Observe firstthat for any ball ¯ B r ( x ) ⇢ D we may apply Definition 2.21 to compare v and theharmonic extension u of v | @ B r ( x ) on B r ( x ) (see Exercise C.21) and get: ? @ B r ( x ) v ( y ) d N ( y ) = ? @ B r ( x ) u ( y ) d N ( y ) = u ( x ) v ( x ) . .7* The ball walk and Brownian trajectories ? B r ( x ) v ( y ) d y = | B r ( x ) | Z r Z @ B s ( x ) v ( y ) d N ( y ) d s | B r ( x ) | Z r Z @ B s ( x ) | @ B s ( x ) | d s · v ( x ) = v ( x ) . Fix ✏ (0 ,
1) and x D . The sequence of random variables { v X ✏ , x n } n = along the ball walk { X ✏ , x n } n = defined in (2.2), is then a supermartingale withrespect to the filtration {F n } n = , because: E v X ✏ , x n | F n = ? ✏ ^ dist ( X n , @ D ) v ( y ) d y v X ✏ , x n a . s . Consequently: E [ v X ✏ , x n ] E [ v X ] = v ( x ). Passing to the limit with n ! 1 and recalling the boundary comparison assumption, finally yields: v ( x ) E ⇥ v X ✏ , x ⇤ E ⇥ F X ✏ , x ⇤ = u ✏ ( x ) . We conclude that ¯ h F u ✏ by taking the infimum over all v as above. Since bya symmetric argument: h F u ✏ , the result follows in virtue of (2.20). ⇤ In this Section we show that the ball walk, introduced in Section 2.2, can beseen as a discrete realisation of the Brownian motion. In particular, we willdeduce the same result as in Section 2.6*, namely that all functions in thefamily { u ✏ } ✏ (0 , in (2.5) are always one and the same function. This materialmay be skipped at first reading; it is slightly more advanced and necessitatesfamiliarity with the construction of Brownian motion in Appendix B.We start with some elementary technical observations. Denote ( ⌦ B , F B , P B )the probability space on which the standard N -dimensional Brownian motion {B Nt } t is defined. We consider the product probability space ( ¯ ⌦ , ¯ F , ¯ P ) = ( ⌦ B , F B , P B ) ⇥ ( ⌦ , F , P ) with the space ( ⌦ , F , P ) in Section 2.2, and denoteits elements by ( ! B , ! ) with ! = { w i } i = B (0) N . Clearly, {B Nt } t is alsoa standard Brownian motion on ( ¯ ⌦ , ¯ F , ¯ P ). We further denote the product -algebras ¯ F t = F t ⇥ F , so that ¯ F s ⇢ ¯ F t ⇢ ¯ F for all 0 s t ; for every s [0 , t ]the random variable B Ns is ¯ F t -measurable.We call ¯ ⌧ : ¯ ⌦ ! [0 , ] a stopping time on ( ¯ ⌦ , ¯ F , ¯ P ) provided that { ¯ ⌧ t } ¯ F t for all t P (¯ ⌧ = + ) =
0. Then, the random variable B N ¯ ⌧ is¯ F ¯ ⌧ -measurable, namely: {B N ¯ ⌧ A } \ { ¯ ⌧ t } ¯ F t for all Borel A ⇢ R N and all t
0, which can be proved as in Lemma B.21.8
The linear case: random walk and harmonic functions
Let now D ⇢ R N be open, bounded, connected and fix x D , ✏ (0 , ⌧ k : ¯ ⌦ ! [0 , ] in:¯ ⌧ = , ¯ ⌧ k + ! B , { w i } i = = min n t ¯ ⌧ k ; B Nt ( ! B ) B N ¯ ⌧ k ( ! B , ! ) ( ! B ) = ✏ ^ dist( x + B N ¯ ⌧ k ( ! B , ! ) ( ! B ) , @ D ) | w k + | o , (2.21)and also: ¯ ⌧ ( ! B , ! ) = min t x + B Nt ( ! B ) @ D . (2.22)¯ ¯ P ¯ Lemma (cid:1) (cid:1)
Each (cid:1) ⌧ ¯ k (cid:1) in (cid:1) (2.21) (cid:1) and (cid:1) ⌧ ¯ (cid:1) in (cid:1) (2.22) (cid:1) is (cid:1) a (cid:1) stopping (cid:1) time (cid:1) on (cid:1) ( ⌦ , (cid:1) F ¯ (cid:1) , (cid:1) ) . (cid:1) Moreover, (cid:1) ⌧ ¯ k (cid:1) converge (cid:1) to (cid:1) ⌧ ¯ (cid:1) as (cid:1) k (cid:1) ! 1 , (cid:1) a.s. (cid:1) in (cid:1) ⌦ . Given a continuous boundary function F : @ D ! R , recall that: u ( x ) = Z ¯ ⌦ N ¯ ⌧ F x + B d ¯ P (2.23)defines a harmonic function u : D ! R , in virtue of Corollary B.29 that buildson the classical construction and discussion of Brownian motion presented inAppendix B. As in Remark 2.3, we view F as a restriction of some F C ( ¯ D ).Then, by Lemma 2.24 we also have: u ( x ) = lim k !1 Z ¯ ⌦ N ¯ ⌧ ¯ F x + B k d P . On the other hand, we recall that in (2.6) we defined: u ✏ ( x ) = ⌦ F X ✏ , x d P = k lim !1 Z Z ⌦ F X k ✏ , x d P . Theorem 2.25
For all ✏ (0 , and all x D there holds: u ✏ ( x ) = u ( x ) . In fact, we have: N ¯ ⌧ P B x + B A = P X ✏ , x A for all Borel A ⇢ R N . (2.24) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) ⇤ (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) ⇤ Exercise 2.26
Modify the arguments in this Section to the setting of thesphere walk introduced in Exercise 2.8. Follow the outline below:(i) Given x D and ✏ (0 , ⌦ B , F B , P B ): = ⌧ n t ⌧ k ; |B tN B N ⌧ k ⌧ = , k + min | = ✏ ^
12 dist( x + B ⌧ N k , @ D ) o , that converge a.s. as k ! 1 , to the exit time: ⌧ = min { t B tN @ D x } . The linear case: random walk and harmonic functions (ii) Let ( ⌦ , F , P ) and { X ✏ , x n } n = be as in Exercise 2.8 (i), and define u ✏ : D ! R according to (2.5) and (2.6). Prove that the push-forward of P on D via X ✏ , x k , coincides with the push-forward of P B via x + B N ⌧ k , for every k
0. Consequently, u ✏ ( x ) = R ⌦ F ( x + B N ⌧ ) d P , which is the harmonicextension of a given F C ( @ D ), independent of ✏ (0 , All constructions, statements of results and proofs in this Chapter have theircontinuous random process counterparts through Brownian motion, see M¨ortersand Peres (2010). The ball walk can be seen as a modification of the spherewalk in Exercise 2.8, which in turn is one of the most commonly used methodsfor sampling from harmonic measure, proposed in Muller (1956).The definition of the walk-regularity of a boundary point y , which in thecontext of Section 2.7* can be rephrased as: ⌘ , > ˆ (0 , ) x B ˆ ( y ) \ D P x + B N ⌧ B ( y ) ⌘ , is equivalent to the classical definition given in Doob (1984): P B ⇣ inf t > y + B Nt R N \ D = ⌘ = y in Definition C.46.Its equivalence with the Wiener regularity criterion, stating that R N \ D is 2-thick at y (compare Definition C.47 (ii)) can be proved directly, see M¨ortersand Peres (2010) for a modern exposition. In working out the proofs of thisChapter and the analysis in Section 2.7*, the author has largely benefited fromthe aforementioned book and from personal communications with Y. Peres.Various averaging principles and related random walks in the Heisenberggroup were discussed in Lewicka et al. (2019). In papers by Lewicka and Peres(2019a,b), Laplace’s equation augmented by the Robin boundary conditionshas been studied from the viewpoint of the related averaging principles in C , -regular domains. There, the asymptotic H¨older regularity of the values of the ✏ -walk has been proved, for any H¨older exponent ↵ (0 ,
1) and up to theboundary of D , together with the interior asymptotic Lipschitz equicontinuity.The “ellipsoid walk” linked to the elliptic problem: Trace A ( x ) r u ( x ) = cients matrix A satisfying det A =
1, and uniformly elliptic with theelliptic distortion ratio that is close to 1 in D , this lead to proving the localasymptotic uniform H¨older continuity of the associated process values u ✏✏