The hitting distribution of a line segment for two dimensional random walks
aa r X i v : . [ m a t h . P R ] J u l The hitting distribution of a line segmentfor two dimensional random walks
Kˆohei UCHIYAMADepartment of Mathematics, Tokyo Institute of TechnologyOh-okayama, Meguro Tokyo 152-8551e-mail: [email protected]
Abstract
Asymptotic estimates of the hitting distribution of a long segment on the realaxis for two dimensional random walks on Z of zero mean and finite variances are obtained:some are general and exhibit its apparent similarity to the corresponding Brownian density,while others are so detailed as to involve certain characteristics of the random walk. Let S n = z + ξ + · · · + ξ n be a two dimensional random walk of i.i.d. increments ξ , ξ , . . . andinitial position S = z moving on the square lattice Z , which we suppose to be embeddedin the complex plane C . Let n be a positive integer and denote by H I ( n ) z ( s ) the probabilitythat the first visit (after time 0) to the interval {− n + 1 , . . . , n − } of the random walk S · starting at z takes place at s . For the later use it is convenient to define a positive number n ∗ and an interval I ( n ) by I ( n ) = ( − n ∗ , n ∗ ) = { u ∈ R : | u | < n ∗ } , n ∗ = n − / . Then H I ( n ) z ( s ), s ∈ I ( n ), is written as H I ( n ) z ( s ) = P z [ ∃ j ≥ , S j = s and S k / ∈ I ( n ) for 1 ≤ k < j ] , where P z stands for the probability of the walk starting at z ∈ Z + i Z . An explicit expressionof the corresponding distribution for Brownian motion is readily derived from the Poissonkernel for the unit disc in view of the conformal invariance of harmonic measures. Let h I ( n ) x denote the Brownian analogue of H I ( n ) z , namely the density of hitting distribution of theinterval I ( n ) for the two dimensional standard Brownian motion starting at z . Then, for x ∈ R \ [ − n ∗ , n ∗ ] h I ( n ) x ( s ) = q x − n ∗ π | x − s | · q n ∗ − s ( s ∈ I ( n )) (1)(see Appendix (A), in which we compute h I ( n ) z ( s ) for general z ). From the Donsker’s in-variance principle it is expected that H I ( n ) z ( s ) behaves similarly to h I ( n ) z ( s ) if the covariancematrix of ξ is isotropic, but it is not clear at all in what sense they are similar. In the presentpaper we compute exact asymptotic forms of H I ( n ) x ( s ) for all x ∈ Z as | x − s | ∧ n → ∞ : thefirst of them exhibits its apparent similarity to h I ( n ) x ( s ) and the others give finer estimatesthat involve certain characteristics of the random walk. The case of non-real initial siteswill be briefly discussed in Appendix (D). key words : harmonic measure in a slit plane, line segment, asymptotic formula, random walk of zeromean and finite variances. AMS Subject classification (2010) : Primary 60G50, Secondary 60J45. S startsat a point x ∈ Z ⊂ C and X = ( X n ) is its trace on the real axis, namely X is a one-dimensional random walk on Z imbedded in ( S n ) with X n being the position of the n -threturn of S to the real axis, then H ( n ) x ( · ) equals the hitting distribution of I ( n ) for X n . It isremarked that the increment distribution of X is almost Cauchy in the sense that its tailsare asymptotically C/ | x | in both directions [15].For the symmetric simple random walk H. Kesten has obtained the upper bound H I ( n ) ∞ ( s ) := lim | z |→∞ H I ( n ) z ( s ) ≤ C [ n ( n − s )] − / (0 ≤ s < n ) (2)in [4] (the limit on the left-hand side exists ([11]:Theorem 14.1, p.141)) and applied it toa study of the DLA model in [5] (cf. also [6]; a unified exposition is found in [8]). For arectangle with a side on the real axis Lawler and Limic [9] give an explicit expression forthe hitting distribution of its boundary for a simple random walk started inside it and, bytaking limits, derive from it the corresponding ones for a half-infinite strip and a quadrant.For a quadrant of the plane, one half of it split along its diagonal line and the complementsof these regions as well Fukai [2] obtains very detailed evaluations of the hitting distributionsby exploiting the properties special to simple random walk.Throughout this paper we suppose that the walk S n is irreducible, E [ S ] = 0 and E [ | S | δ ] < ∞ either for δ = 0 or for some δ > /
2; (3)we make an explicit reference to δ in the latter case, while no reference is understood tomean the case δ = 0. Theorem 1
Let δ > / in (3). Then uniformly for integers s ∈ I ( n ) and x, | x | ≥ n , as n → ∞ H I ( n ) x ( s ) = h I ( n ) x ( s ) " O (cid:18) q ( | x | − n ∗ ) ∧ ( n − | s | ) (cid:19) . (4)From Theorem 1 it follows that H I ( n ) ∞ ( s ) = 1 π · q n ∗ − s " O (cid:18) q n − | s | (cid:19) ( s ∈ I ( n )); (5)(2) is thus refined. Indeed, the probability that the walk starting at z hits the real line inthe interval I ( N ) tends to zero for any N > n as | z | → ∞ so that H I ( n ) ∞ ( s ) is representedas the limit of a convex combination of H I ( n ) x ( s ), | x | > N , which with, e.g., N = 2 n showsthe relation above in view of (4).When either | x | − n or n − | s | remains bounded, (4) does not determine the asymptoticform of H I ( n ) . The next theorem improves the estimate in this respect in the case δ = 0; inparticular it determines a precise asymptotic form of H I ( n ) ∞ ( s ) valid uniformly for s ∈ I ( n ),which is not provided by (5). (See Section 4 (Theorems 9, 10) for the case δ > / µ ( y ) and ν ( y ), y ≥
0, associatedwith the imbedded random walk ( X n ) — the trace of S — on the real line mentioned above.They are characterized as positive solutions of the Wiener-Hopf equations µ ( y ) = E − y [ µ ( − X ); X ≤
0] and ν ( y ) = E y [ ν ( X ); X ≥ µ (0) ν (0) = πσ − e P ∞ k =1 k − P [ X k =0] , except for determi-nation of µ (0) > ν (0)) that is in our disposal: we are to single out µ (0) appropriatelyfor the present purpose. Here σ is the square root of the determinant of the covariancematrix Q of the i.i.d. increments ξ k : σ := (det Q ) / : the quadratic form of Q is given by E [( S · θ ) ]. The equations above plainly say that µ and ν are harmonic for, respectively,the walks − X and X killed on hitting the negative half-line. We extend µ and ν to y < µ ( y ) and ν ( y ) , y ∈ Z are (strictly) increasing. Wecan and do choose µ (0) so that µ ( y ) √ y −→ σ and µ ( − y ) √ y −→ y → ∞ , (6)which entails the same property for ν in place of µ ([13]: Theorem 1.1). (For more detailssee Appendix (C).) Theorem 2 (i)
Uniformly for ≤ s < n and x ≥ n , as n → ∞ and x − s → ∞ H I ( n ) x ( s ) = σ π · ν ( x − n ) µ ( − n + s ) x − s · s x + nn + s (cid:16) o (1) (cid:17) . (ii) Uniformly for − n < s ≤ and x ≥ n , as n → ∞ H I ( n ) x ( s ) = σ π · ν ( x − n ) ν ( − n − s ) x − s · s x + nn − s (cid:16) o (1) (cid:17) . Theorem 2 describes the asymptotic behavior of H I ( n ) x ( s ) in the case x ≥ n ; by symmetrywe have a similar result in the case x ≤ − n , actually a translation of Theorem 2 in view ofthe duality of µ and ν . Theorem ′ (i) Uniformly for − n < s ≤ and x ≤ − n , as n → ∞ and s − x → ∞ H I ( n ) x ( s ) = σ π · µ ( − x − n ) ν ( − n − s ) s − x · s − x + nn − s (cid:16) o (1) (cid:17) . (ii) Uniformly for ≤ s < n and x ≤ − n , as n → ∞ H I ( n ) x ( s ) = σ π · µ ( − x − n ) µ ( − n + s ) s − x · s − x + nn + s (cid:16) o (1) (cid:17) . By using the asymptotic form of the hitting distribution of the real line we can readilydeduce asymptotic forms of H I ( n ) z ( s ) for z / ∈ R from Theorems 2 and 2 ′ (see Appendix (D)).Here we record only the case when z = ∞ . Since 1 / √ n + s may be replaced by µ ( − n + s )as n → ∞ we obtain the following Corollary 3
Uniformly for s ∈ I ( n ) , as n → ∞ H I ( n ) ∞ ( s ) = π − µ ( − n + s ) ν ( − n − s )(1 + o (1)) . From Theorems 2 and 2 ′ we obtain the second corollary:3 orollary 4 Uniformly for integers n ≥ , s ∈ I ( n ) and x ∈ Z \ I ( n ) , H I ( n ) x ( s ) ≍ h I ( n ) x ( s ) , namely there exists a positive constant C independent of n, s and x such that C − h I ( n ) x ( s ) ≤ H I ( n ) x ( s ) ≤ Ch I ( n ) x ( s ) . The next theorem provides the asymptotic form of H I ( n ) x ( s ) when x ∈ I ( n ). In view ofthe corresponding result for the first visit of the real axis, that may reads P z [ ∃ j ≥ , S j = s and S k / ∈ R for 1 ≤ k < j ] ∼ σ lim y → | y | h x + iy ( s )with h z ( s ) = | y | /π ( y + ( x − s ) ) (see (32)), we extend h I ( n ) x ( s ) to the variables x ∈ I ( n ), x = s by h I ( n ) x ( s ) = lim y → | y | h I ( n ) x + iy ( s ) . In Appendix (B) we compute this limit and find that h I ( n ) x ( s ) = n ∗ − xsπ ( x − s ) q ( n ∗ − x )( n ∗ − s ) x, s ∈ I ( n ) , x = s. (See (42); also the identities (41) and (31) for an underlying idea.) Let S (1)1 and S (2)1 bethe real and imaginary parts of S , respectively and let Y be the component of S (1) that isperpendicular to S (2)1 under P , namely Y = S (1)1 − ωS (2)1 where ω = E [ S (1)1 S (2)1 ] /E [( S (2)1 ) ]. Theorem 5
Let Y be as above and suppose the moment condition E h | Y | log | Y | i < ∞ . (7) Let x, s ∈ I ( n ) . Then (i) as ( n − | s | ) ∧ ( n − | x | ) ∧ | x − s | → ∞ H I ( n ) x ( s ) = σ h I ( n ) x ( s )(1 + o (1));(ii) if s < x , as ( n − x ) / ( n − s ) → H I ( n ) x ( s ) = σ π · ν ( − n + x ) ν ( − n − s ) √ n √ x − s ) / (cid:16) o (1) (cid:17) ;(ii ′ ) if s > x , as ( n − s ) / ( n − x ) → H I ( n ) x ( s ) = σ π · µ ( − n + s ) µ ( − n − x ) √ n √ s − x ) / (cid:16) o (1) (cid:17) . Observe first that the condition ( n − x ) / ( n − s ) → x − s → ∞ , xn → , x − sn − s → n − xsn ( n − s ) → , and then that the formula of (ii) implies (and is actually finer than) the formula of (i).Under the condition | x − s | → ∞ the cases (ii) and (ii ′ ) together exhaust the case when( x ∨ s ) /n →
1. We have an obvious analogue for (ii ′ ), which is a dual statement of (ii). Alsoobserve that h I ( n ) x ( s ) as well as H I ( n ) x ( s ) is bounded away from zero and infinity whenever | x − s | is bounded above by any constant. These observations lead to the following corollary.4 orollary 6 Suppose E [ | Y | log | Y | ] < ∞ . Then, for x, s ∈ I ( n ) , C − h I ( n ) x ( s ) ≤ H I ( n ) x ( s ) ≤ Ch I ( n ) x ( s ) . In particular if x is kept within any bounded distance from n and so is s from − n , then H I ( n ) x ( s ) ≍ /n . Remark 1.
Under the same supposition as in Theorem 5 the formulae obtained abovecan be extended to the general starting positions x + iy as in [13] but with the resultingformula somewhat complicated (see (39)).Denote by H + z ( s ) the probability that the first visit (after time 0) to the positive realaxis of the walk starting at z ∈ C takes place at s ∈ { , , , . . . } : H + z ( s ) = P z [ ∃ j ≥ , S j = s and S k / ∈ { , , , . . . } for 1 ≤ k < j ] . Similarly let H − z ( s ) denote the distribution of the first visiting sites (after time 0) of the set {− , − , − , . . . } . The proofs of Theorems 1 and 2 rest on the results on H ± x ( s ) obtained in[13] (Theorem 1.1; see also [14] for (9)) as given in the following theorem (and also in (23),(24) and (25) later). Theorem ([13], [14])
Let s < . Then for x ≥ , as x ∨ ( − s ) → ∞ H − x ( s ) = σ π · ν ( x ) µ ( s ) | x − s | (cid:16) o (1) (cid:17) . (8) If E [ | Y | log | Y | ] < ∞ in addition, then as | x − s | → ∞ under x < , s < , H − x ( s ) = σ π · | x + s | ν ( x ) µ ( s ) | x − s | (cid:16) o (1) (cid:17) . (9)It is warned that in [13] the condition E [ | S (1)1 | log | S (1)1 | ] < ∞ is errorneously assumedwhere it should be (7) as in Theorem above.Comparing the formulae given in Theorem with those in Theorems 2(i) and 5 we findthe quite reasonable conclusion that H I ( n ) x ( s ) /H − x − n ( s − n ) → x/n → s/n → x is larger than n or not).For the symmetric simple random walk (i.e., P [ S = x ] = 1 / x ∈ {± , ± i } ) we canimprove the error estimate in [13] for H ± x ( s ) and accordingly that in Theorem 1 for H I ( n ) x ( s ). Proposition 7
Let S n be the symmetric simple random walk. Then H − x ( s ) = 1 π · x − s s x ∨ − s × " O (cid:18) − s (cid:19) + O (cid:18) x ∨ (cid:19) ( x ≥ , s <
0) (10) and as n → ∞ H I ( n ) x ( s ) = h I ( n ) x ( s ) " O (cid:18) | x | − n ∗ ) ∧ ( n − | s | ) (cid:19) ( | x | ≥ n, s ∈ I ( n )) . (11) Here the error estimates are uniform for integers x, s subject to the respective constraintsindicated in parentheses.
Theorem 1 is proved in Section 2 by taking for granted certain several results whoseproofs are given in Section 3. Proof of Proposition 7 is given at the end of Section 3. InSection 4 we make detailed estimation of H I ( n ) x ( s ) when x or s are near the edges of I ( n )under the moment condition (3); in particular Theorem 2 is proved. Theorem 5 is provedin Section 5. 5 Proof of Theorem 1
The proof of Theorem 1 primarily rests on the asymptotic estimates obtained in [13] ofhitting distributions for half-real-lines. The first visit of I ( n ) occurs after the consecutiveovershoots in the first k alternating entrances of the walk X into the half-lines ( −∞ , − n )and ( n, ∞ ) for some k = 0 , , , . . . and we simply sums up the relevant probabilities over k . Motivated by this we bring in the sequence of probability kernels in below.Throughout this section we pick and fix a (large) positive integer n , which we shall notdesignate in the notation introduced in this section even though it depends on n . Let ( S )stand for the indicator of a statement S : ( S ) = 1 or 0 according as S is true or not. Definefor integers x ≥ n and y > − n , Q ( x, y ) = − n X s = −∞ H − x − n ( s − n ) H + s + n ( y + n ) ,Q I ( x, y ) = Q ( x, y ) ( − n < y < n ) ,K I ( x, y ) = H − x − n ( y − n ) ( − n < y < n ) , and Q = (the identity matrix), Q = Q and inductively Q k ( x, y ) = ∞ X u = n Q k − ( x, u ) Q ( u, y ) ( k = 1 , , . . . ) , (12)and finally Λ( x, y ) = ∞ X k =1 Q k ( x, y ) ( y ≥ n ) . Then for x ≥ n, − n < s < n , H I ( n ) x ( s ) = (1 + Λ)( Q I + K I )( x, s )= ∞ X y = n [ ( y = x ) + Λ( x, y )][ Q I ( y, s ) + K I ( y, s )] . (13)The kernels Q, K I , Q I and Λ are probabilities with self-evident meaning. We are tocompare them with the corresponding ones, denoted by q, k I , q I and λ , for the standardtwo dimensional Brownian motion B ( t ). In doing this it is recalled that the interval I ( n )is defined to be ( − n + 1 / , n − /
2) instead of [ − n + 1 , n − L ± = { t ∈ R : ± t > } and τ L ± = inf { t > B ( t ) ∈ L ± } , and define h ± x ( s ) = P BMx [ B ( τ L ± ) ∈ ds ] /ds ( x ∈ L ∓ , ± s > , where P BMz denotes the law of B ( t ) starting at z . Then for real x > n ∗ , y > − n ∗ , q ( x, y ) = Z − n ∗ −∞ h − x − n ∗ ( s − n ∗ ) h + s + n ∗ ( y + n ∗ ) ds,q I ( x, y ) = q ( x, y ) ( − n ∗ < y < n ∗ ) ,k I ( x, y ) = h − x − n ∗ ( y − n ∗ ) ( − n ∗ < y < n ∗ )6nd q k and λ are given in analogous ways; in particular q = q and λ ( x, y ) = ∞ X k =1 q k ( x, y ) ( y > n ∗ ) . We know that h − x ( s ) = √ xπ ( x − s ) · √− s ( x > , s < , (14) h I ( n ) x ( s ) = q x − n ∗ π ( x − s ) · q n ∗ − s ( x > n ∗ , − n ∗ < s < n ∗ ) . The function Q is extended to that of reals by Q ( u, v ) = Q ( x, y ) for ( u, v ) ∈ ( x − , x + ] × ( y − , y + ] , and similarly for Λ and K I . (The summation in (12) can be then replaced by the integrationover y > n ∗ .) With Q thus extended put η = Q − q. We shall prove the relations (I) through (VII) given below. The symbol f ≍ g meansthat C − g ≤ f ≤ Cg . Here and in what follows C denotes a positive constant which maydepend on the law P [ S = · ] but is independent of any variables x, n, y, s contained thereinexplicitly or inexplicitly and may change from line to line. The products of two functions(of two variables) are understood to be that of integral operators in an analogous way to(12): e.g., ηq ( x, y ) = Z ∞ n ∗ η ( x, u ) q ( u, y ) du (15)(the range of the integration is always the half-line u > n ∗ where q ( u, · ) is defined).Let x > n ∗ , y > − n ∗ and − n ∗ < s < n ∗ ; x, y, s are real numbers in (I) through (III).(I) q ( x, y ) ≍ √ x − n ∗ √ n ∗ + y · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − y log x + 2 ny + 2 n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The function t − log(1 + t ) is understood to be continuously extended to t = 0. On usingthe inequalities 1 /b < ( b − a ) − log( b/a ) < /a (0 < a < b ) we infer that ( x − y ) − log[( x +2 n ) / ( y + 2 n )] ≍ x − if | x − y | ≤ n , which combined with (I) yields the bound of q I givenin the next item where we also display the explicit form of k I for convenience.(I ′ ) q I ( x, s ) ≍ √ x − n ∗ √ n ∗ + s · x (cid:16) xn ∗ (cid:17) ; k I ( x, s ) = 1 π √ x − n ∗ ( x − s ) √ n ∗ − s . (II) | η ( x, y ) | q ( x, y ) = o (1) as ( x − n ) ∧ ( n + y ) → ∞ if δ = 0 , ≤ C q ( x − n ∗ ) ∧ ( n ∗ + y ) if δ > .7III) | η | ( + λ )( q I + k I )( x, s )= " q n ∗ − s ∧ h I ( n ) x ( s ) × o (1) as n → ∞ if δ = 0 , ≤ C x/n ∗ ) √ x q n ∗ − s if δ > .Here | η | in (III) stands for the integral operator of kernel | η ( x, y ) | as in (15), , also in (III),for the identity operator, and δ in (II) and (III) for the constant in (3).For integers x ≥ n, − n < s < n ,(IV) ∞ X y = n Λ( x, y ) ≤ C s x − n ∗ x , (V) ∞ X y = n Λ( x, y ) 1 y − s ≤ C n s x − n ∗ x · log 3 nn − s , (VI) n + N X y = n Λ( x, y ) ≤ C s x − n ∗ x · Nn ( N = 1 , , . . . ) . There exist functions ε j ( t ), j = 1 ,
2, of (a single variable) such that as t → ∞ , ε j ( t ) = o (1) or O (1 / √ t ) according as δ = 0 or δ > / | k I − K I | ( x, s ) ≤ k I ( x, s )[ ε ( x − n ∗ ) + ε ( n − s )] . The proofs of these results are postponed to the next section. In the rest of this sectionwe prove Theorem 1 taking them for granted.By symmetry we may suppose x ≥ n . From the identity H I ( n ) x = ( + Λ)( K I + Q I )( x, · )and a similar one for h I ( n ) x it follows that H I ( n ) x − h I ( n ) x = ( + Λ)( K I − k I + Q I − q I ) + (Λ − λ )( k I + q I )( x, · ) . Writing Q = Λ − Λ Q and q = λ − qλ (valid on [ n ∗ , ∞ ) ) one finds the identity Λ q − Qλ = Λ ηλ ,which yields Λ − λ = η + Λ η + ηλ + Λ ηλ = ( + Λ) η ( + λ ) . (16)Let q I + k I act on the both sides from the right. Let x and s be integers such that x ≥ n and − n < s < n . Using (III), (IV) and the simple inequality x − α log x/n ≤ ( eα ) − n − α ( α > , first observe that Λ | η | ( + λ )( q I + k I )( x, s ) ≤ s x − n ∗ x · ε ( n ) √ n − s , ε ( t ) is a function of the same meaning as ε j ( t ) in (VII), and then, further using (16)and (III), that | Λ − λ | ( q I + k I )( x, s ) ≤ " ε ( n ) √ x − n ∗ + C (cid:18) xn (cid:19) q x ( n − s ) ≤ C ′ ε ( n ) + 1 √ x − n ∗ ! h I ( n ) x ( s ) . (17)The last inequality in particular impliesΛ( k I + q I )( x, s ) ≤ Ch I ( n ) x ( s ) . (18)Let δ > / | k I − K I | ( x, s ) ≤ C n s x − n ∗ x log 3 nn − s + h I ( n ) x ( s ) ! √ n − s , but we have n − log[3 n/ ( n − s )] ≤ / q n ( n − s ) so thatΛ | k I − K I | ( x, s ) ≤ C s x − n ∗ nx ( x − s ) + h I ( n ) x ( s ) ! √ n − s ≤ C ′ h I ( n ) x ( s ) 1 √ n − s . (19)On the other hand by (II) and (I ′ ) we have | q I − Q I | ( y, s ) ≤ C √ y − n + 1 √ n + s ! · q I ( y, s ) ≤ C y/n √ y · n + s (20)and Λ | q I − Q I | ≤ C √ x − n ∗ / √ xn ( n + s ) ≤ C ′ h I ( n ) x ( s ) / √ n + s , which in conjunction with(19) gives ( + Λ) (cid:16) | q I − Q I | + | k I − K I | (cid:17) ( x, s ) ≤ Ch I ( n ) x ( s ) √ n − s + 1 √ n + s ! . (21)The bounds (VII), (17), (20) and (21) together yield the formula of Theorem 1 in the case δ > / Proof of (I). Let x ≥ n ∗ and y > − n ∗ . It follows from (14) that q ( x, y ) = 1 π Z − n ∗ −∞ √ x − n ∗ ( x − u ) √− u + n ∗ √− u − n ∗ ( y − u ) √ n ∗ + y du = 1 π s x − n ∗ y + n ∗ J n ( x + n ∗ , y + n ∗ ) , (22)where J n ( a, b ) = Z ∞ √ t dt √ t + 2 n ∗ ( t + a )( t + b ) ( a = y + n ∗ > , b = x + n ∗ > n ∗ ) . n ∗ ≤ a < b , then J n ( a, b ) = 1 a Z ∞ √ t dt q t + 2 n ∗ /a ( t + 1)( t + b/a ) ≍ b + 1 a Z ∞ dt ( t + 1)( t + b/a )= 1 b + 1 b − a log a + b a ≍ b − a log ba . This shows (I) in the case y ≥ n ∗ .In the case − n ∗ < y < n ∗ , a similar computation gives J n ( a, b ) = 12 n ∗ Z ∞ √ t dt √ t + 1 ( t + a/ n ∗ )( t + b/ n ∗ ) ≍ b + 1 b − a log 2 n ∗ + b n ∗ + a . Thus (I) has been proved.
Proof of (II). First suppose that δ > / C such that for x ≥ n and s < n , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H − x − n ( s − n ) − π · x − s s x − n ∗ n − s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ch − x − n ∗ ( s − n ∗ ) √ n ∗ − s + 1 √ x − n ∗ ! . (23)In making application of this and its obvious analogue for H + there arise four terms to beestimated for computation of the difference Q − q = H − H + − h − h + = ( H − − h − ) H + + h − ( H + − h + ) (the right side of (23) is counted two terms), which are equal to those obtainedby inserting the factors1 √ x − n ∗ + 1 √− u + n ∗ and 1 √ n ∗ + y + 1 √− u − n ∗ under the integral symbol of the integral of (22). Among them only two terms requirecomputation, which we are to show to be not larger than the sum of the other two. To thisend, we make the same change of variables that led to the second equality of (22) and findthat it suffices in view of (I) to verify the following inequalities Z ∞ √ t dt ( t + 2 n ∗ )( t + a )( t + b ) ≤ Z ∞ dt √ t + 2 n ∗ ( t + a )( t + b ) ≤ π ( a ∨ b ) √ a ∧ b ( a = y + n ∗ > , b = x + n ∗ as before). The first one is trivial. The second one is verified byletting ( a ∨ b ) − R ∞ [ √ t ( t + a ∧ b )] − dt dominate the integral in the middle. This completesthe proof in the case δ > / δ = 0 is similarly dealt with based on the corresponding result for H − x ( s )(Theorem 1 of [13]). Proof of (III). Consider the case δ > /
2. Set A ( y ) = 1 √ x − n ∗ q ( x, y ) h I ( n ) y ( s ) q n ∗ − s , ( y ) = q ( x, y ) 1 √ y + n ∗ h I ( n ) y ( s ) q n ∗ − s and I A = Z ∞ ( x − n ) ∨ n A ( y ) dy and I B = Z ( x − n ) ∨ nn ∗ B ( y ) dy. Notice that 1 / √ x − n ∗ ≤ / √ y + n ∗ if and only if y ≤ x − n ∗ and that according to (II) | η | h I ( n ) ( x, s ) ≤ C ( I A + I B ) / q n ∗ − s . By (I) A ( y ) ≍ √ y − n ∗ y − s · x − y log x + 2 ny + 2 n . A simple computation shows that Z ( x − n ) ∨ (2 n )( x − n ) ∨ n A ( y ) dy = O (1 / √ x )and Z ∞ ( x − n ) ∨ (2 n ) A ( y ) dy ≤ C Z ∞ ( x − n ) ∨ (2 n ) y / yx ! dy = O (1 / √ x ) . Thus I A = O (1 / √ x ) (uniformly in s < n ∗ ).As for I B first we observe B ( y ) ≍ √ x − n ∗ √ y + n ∗ √ y − n ∗ y − s · x − y log x + 2 ny + 2 n . Let x ≥ n . It is easy to see that R nn B ( y ) dy = const (1 / √ x ) log( x/n ∗ ) , while Z x n B ( y ) dy ≤ C √ x Z x n y · x − y log x + 2 ny + 2 n dy ≤ C √ xx + 2 n Z n/ ( x +2 n ) u (1 − u ) log 1 u du and the last member is dominated by a constant multiple of (1 / √ x )(log x/n ) owing to theequality R a u − log(1 /u ) du = | log a | (0 < a < δ > /
2. Forthe case δ = 0 the same argument as above leads to the upper bound o (1) / q n ∗ − s ; theidentity qh I · = h I · − ( k I + q I ) ( ≤ h I · ) gives the other bound h I · × o (1). The proof of (III) iscomplete. Proof of (IV). Put p n = sup x,y ≥ n Q ( x, y ). Then p n = O (1 √ n ) and X y ≥ n Q k ( x, y ) ≤ p k − n X y ≥ n Q ( x, y ) ≤ p k − n X y ≥− n H − x − n ( y ) ≤ p k − n C s x − n ∗ x , hence P ∞ y = n Λ( x, y ) ≤ P k P y ≥ n Q k ( x, y ) ≤ C ′ q ( x − n ∗ ) /x .11 emma 8 Uniformly for integers x ≥ n, − n < s < n , ∞ X y = n Q ( x, y ) 1 y − s ≍ √ x − n ∗ x √ n xn ! log 3 nn − s . Proof . By employing (I) and the bound ( x − y ) − log x +2 ny +2 n ≍ x − [1 + log( x/y )] valid for n ≤ y ≤ x + n one sees that n X y = n Q ( x, y ) 1 y − s ≍ Z nn √ x − n ∗ ( y − s ) √ n + y · x − y log x + 2 ny + 2 n dy ≍ √ x − n ∗ x √ n xn ! log 2 n − sn − s as well as ∞ X y =2 n +1 Q ( x, y ) 1 y − s ≤ C √ x − n ∗ x √ n xn ! (break the summation according as y is larger than x ∨ (2 n ) or not and consider the cases x ≤ n and x > n separately). These together yield the estimate of the lemma. ✷ Proof of (V). Letting 2 q x/n dominate 1 + log( x/n ) in the right-hand side of the asymptoticformula of Lemma 8 and employing (IV), we have ∞ X y = n Λ( x, y ) 1 y − s = ∞ X y = n ( Q + Λ Q )( x, y ) 1 y − s ≤ C s x − n ∗ x log[3 n/ ( n − s )] n . Thus (V) is proved.
Proof of (VI). As in the proof of Lemma 8 we have n + N X y = n Q ( x, y ) ≤ C q ( x − n ∗ ) /x N/n ;the estimate of (VI) is then follows from (IV) as in the preceding proof. Proof of (VII): This follows immediately from (23) if δ > /
2. In the case δ = 0 use (8). Proof of Proposition 7 . Both formulae (10) and (11) of Proposition 7 are proved in a similarway as Theorem 1 and in their proofs given below we omit details. We first show thededuction of (11) from (10). For the symmetric simple random walk the right-hand side of(23) can be replaced by Ch − x ( s )[( − s ) − + ( x ∨ − ] (cf. [13]) and accordingly we deducethat ( | k I − K I | + | q I − Q I | )( y, s ) ≤ Ck I ( y, s )[( y − n ∗ ) ∧ ( n − s ) ∧ ( n + s )] , | η | ( x, y ) ≤ Cq ( x.y )( x − n ∗ ) ∧ ( n ∗ + y ) , ∞ X y = n Q ( x, y )( y − s ) √ y − n ∗ ≍ √ x − n ∗ x √ n xn ! √ n − s , ∞ X y = n Λ( x, y )( y − s ) √ y − n ∗ ≤ C √ x − n ∗ n √ x √ n − s ∞ X y = n Λ( x, y ) k I ( y, s ) y − n ∗ ≤ C √ x − n ∗ √ x n ( n − s ) , and with these bounds we can proceed as above to obtain (11). Proof of (10) : The proof is based on an asymptotic expansion of the potential functionof the random walk (cf. [3], [12], [7]) from which an application of the reflection principleimmediately yields H im ( s ) = | m | π ( | s | + m ) + O | s | + | m | ! ( m = 0) , where H im ( s ) stands for the probability that the first visit to the real axis of the simplerandom walk starting at im ∈ i Z takes place at s ∈ Z . We proceed as in Section 2. Bearingsymmetry of the walk in mind, this time we define for y ∈ Z and x ≥ Q ( x, y ) = ∞ X m = −∞ H ix ( m ) H im ( y ) if x > H ( y ) if x = 0and inductively Q k ( x, y ) = P ∞ u =0 Q k − ( x, u ) Q ( u, y ) ( k = 1 , , . . . ). We have the correspond-ing quantities h , h − , q and q k for the standard Brownian motion. Then for s < x ≥ H − x ( s ) = Λ( x, s ) := ∞ X k =1 Q k ( x, s ) , h − x ( s ) = λ ( x, s ) := ∞ X k =1 q k ( x, s ) . We know that C − λ ≤ Λ ≤ Cλ for some constant C > Q to the real variables and put η = Q − q as before. An elementary computation then givesin turn η ( x, y ) = O ( | xy | − ∧ | x y | − ) , | η | λ ( x, y ) = O ( | xy | − / ∧ | x y | − ) , Λ | η | ( x, y ) = O ( | x / y | − )and Λ | η | λ ( x, s ) ≤ C | x / s / | + ≍ x − s s x ∨ − s × " − s + 1 x ∨ ( s ≤ − , x ≥ , where | a | + = | a | ∨
1. Thus (10) follows because of the identityΛ − λ = (1 + Λ) η (1 + λ ) . The proof of Proposition 7 is complete. ✷ H I ( n ) x ( s ) near the edges We continue the arguments of the preceding section to estimate H I ( n ) x ( s ) mainly in the casewhen δ > / n − s or n + s is small in comparison with x − n . The case when δ = 0 or x − n is not large can be similarly dealt with and is only briefly discussed at theend of this section. 13heorems 9 and 10 given below are based on the following result from [13] ((10) ofTheorem 1.3 and its dual): if δ > / x ≥ s > H − x ( − s ) = √ xπ ( x + s ) µ − ( s ) + O (cid:16) x (cid:17) , (24) H + − x ( s ) = √ xπ ( x + s ) ν − ( s ) + O (cid:16) x (cid:17) , (25)where µ − ( s ) = µ ( − s ) and ν − ( s ) = ν ( − s ). The following theorem concerns particularly tothe case when ( x − n ) / ( n − s ) → ∞ so that h I ( n ) x ( s ) >> ( x − s ) − . Theorem 9 If δ > / in (3), then uniformly for integers n > , ≤ s < n and x ≥ n , H I ( n ) x ( s ) = √ n ∗ − s µ ( − n + s ) h I ( n ) x ( s ) + O log nn + 1 x − s ! . Proof . Make decomposition H I ( n ) x = K I + Q I + Λ( K I + Q I ) and infer from (24) that K I = √ n ∗ − s µ − ( n − s ) k I + O (1 / ( x − s )) , (26)Λ K I ( x, s ) = ∞ X y = n Λ( x, y ) " √ n ∗ − s µ − ( n − s ) k I ( y, s ) + O (cid:18) y − s (cid:19) . In view of (I) we have sup s ≥ ,x>n ∗ q I ( x, s ) ≤ C/n, which in particular shows that λq I ( x, s ) = O (1 /n ) uniformly for 0 ≤ s < n ∗ , x > n ∗ . Thus, on employing (I ′ ), for s > q I + λq I = O (1 /n ) and Q I + Λ Q I ≤ C ( q I + λq I ) = O (1 /n ) . (27)By (V) of the preceding section we have ∞ X y = n Λ( x, y ) 1 y − s ≤ C s x − n ∗ x log nn , (28)so that Λ K I ( x, s ) = √ n ∗ − s µ − ( n − s )Λ k I ( x, s ) + O ( n − log n ) . Here the factor q ( x − n ∗ ) /x on the right side of (28) is replaced by 1: the loss of accuracyto the estimate of H I ( n ) x caused by this replacement is small in comparison with the errorterm O (1 / ( x − s )) in (26). By (17) Λ k I = λk I + (Λ − λ ) k I = λk I + O (1 /n ); henceΛ K I = √ n ∗ − s µ − ( n − s ) λk I + O ( n − log n ) , which together with (26), (27) yields the assertion of the theorem. ✷ heorem 10 If δ > / in (3), then uniformly for integers n > , − n < s < and x ≥ n , H I ( n ) x ( s ) = √ n ∗ + s ν ( − n − s ) h I ( n ) x ( s ) × O s s + n ∗ n · log n ! + O s xn ( x − n ∗ ) ! . Proof . We make decomposition H I ( n ) x ( s ) = K I ( x, s ) + − n X y = −∞ H − x − n ( y − n ) H I ( n ) y ( s )= K I ( x, s ) + − n X y = −∞ ( H − x − n − h − x − n )( y − n ) H I ( n ) y ( s )+ − n X y = −∞ h − x − n ( y − n ) H I ( n ) y ( s ) . (29)For evaluation of the second sum of the last line we substitute the estimate of Theorem 9for H I ( n ) x ( s ) (with S n replaced by − S n , hence √ n ∗ − s µ − ( n − s ) by √ n ∗ + s ν − ( n + s )), usethe expression of h I ( n ) x ( s ) analogous to the first expression in (29) of H I ( n ) x ( s ) and observethat sup − n Proof of (ii). The Wiener-Hopf equation for ν may be written as ∞ X x = n H x ( x ) ν ( x − n ) = ν ( − n + x ) (36)(see (44)). We claim that as ( n − x ) / ( n − s ) → J n,s,x := X x ≥ n H x ( x ) H I ( n ) x ( s ) = " ν ( − n − s ) √ n − s · σ ν ( − n + x ) √ n π ( n − s ) (cid:16) o (1) (cid:17) . (37)16ut ξ="n" − x and observe that owing to (6) the summation in (36) may be restricted to x ≤ n + Kξ by choosing K large enough . Then, on looking at (34), the claim (37) followsif we show that for each ε> K > X x ≥ n + Kξ H x ( x ) H I ( n ) x ( s ) ≤ εJ n,s,x . However, by simple consideration this reduces to Z ∞ Kξ √ n + u ( u + ξ ) ( u + n − s ) √ u du ≤ ε √ n ( n − s ) √ ξ , which is certainly true if K is large enough. Thus the claim is verified. One can easily checkthat X x ≤− n H x ( x ) H I ( n ) x ( s ) = O n + s ) ∧ √ n ] n ! = o ( J n,s,x ) . Finally notice that ( n − s ) / ( x − s ) → ✷ Proof of (i). We may suppose that ( n − x ) / ( n − s ) is bounded away from zero, the case( n − x ) / ( n − s ) → X n ≤| x |≤ n + K H x ( x ) H I ( n ) x ( s ) = o (cid:18) s − x ) (cid:19) (indeed this is valid if ( n − x ) / ( n − s ) → ∞ ; the contribution of − n − K ≤ x ≤ − n is easyto estimate), namely the sum above is negligible if compared with H x ( s ). Hence one canreplace the ratio appearing last in (35) by 1. Also ν ( − n − s ) may be replaced by 1 / √ n + s and, substituting the resulting expression into (31) and applying Lemma 11 of Appendix(B), we conclude the formula of (i). ✷ Remark 2.
We could have employed the identity H I ( n ) x ( s ) = H − x − n ( s − n ) + X x ≥− n H − x − n ( x − n ) H I ( n ) x ( s ) s ∈ I ( n ) , (38)instead of (31). This way is simpler owing to (9) except for a tedious computation for exactevaluation of the definite integral of a certain rational function. (A) Let D be the complement of the line segment with edges at ± D = C \ { s : − ≤ s ≤ } and denote the Poisson kernel (density of harmonic measure) for D by h D ( z, s ± i h [ − , z ( s ) = h D ( z, s + i
0) + h D ( z, s − i h I ( n ) z ( s ) = h [ − , z/n ∗ ( s/n ∗ ) /n ∗ .
17e compute h D ( z, s ± i
0) by using the conformal invariance of harmonic measures. Thefunction z = ( w + w − ) univalently maps the exterior of the unit circle onto D . Denote by f ( z ) its inverse map, which may be represented by f ( z ) = z + √ z − , z ∈ D with the standard choice of a branch of the square root (so that f ( ± s ) = ± s ± √ s − s > f ( s ± i
0) = s ± i √ − s for − < s < w = f ( z ) moves on a circlecentered at the origin counter-clockwise starting at a point R > z describes the ellipse[2 x/ ( R + R − )] + [2 y/ ( R − R − )] = 1 (which surrounds the segment − ≤ s ≤ R ↓
1) rotating also counter-clockwise and starting at the point f ( R ) = ( R + R − ) ∈ (1 , R ). (Cf. [1]:p.94 or [10]:p.269). The Poisson kernel for the exterior of theunit circle is given by K ( Re iθ , e iθ ′ ) = R − π ( R − R cos( θ − θ ′ ) + 1) , R > . Put θ ( z ) = arg f ( z ) . Then, for − ≤ s ≤ θ ( s ± i
0) = ± arccos s ∈ ( − π, π ), so that | dθ ( s ± i | = ds/ √ − s ;thus the conformal invariance shows that h D ( z, s ± i π · | f ( z ) | − | f ( z ) | − | f ( z ) | cos [ θ ( z ) − θ ( s ± i · √ − s (39)= 12 π · | f ( z ) | − | f ( z ) | − s ℜ f ( z ) ± √ − s ℑ f ( z )] + 1 · √ − s , which, for z = x ∈ R \ [ − , h [ − , x ( s ) = 2 h D ( x, s + i
0) = √ x − π | x − s | · √ − s . Let Q = ( σ ij ) be a 2 × h D ( z, s ± i
0) the corresponding hitting density for the process Q / B t , a two-dimensional Brownianmotion of mean zero and the covariance matrix tQ . Then for z ∈ D ,˜ h D ( z, s ± i
0) = h D (˜ z, s ± i , ˜ z = ( x − ωy ) + iλy, (40)where ω = σ /σ and λ = σ /σ = q σ /σ − ω . If z is real, the identity above followsimmediately from the rotation invariance of the standard Brownian motion. In view of thestrong Markov property of S its full validity is deduced from the identity thus restricted inconjunction of the corresponding identity for the Poisson kernel of the upper half plane (see(E) below). (B) In Section 5 (at the end of it) we have used the following lemma.
Lemma 11
For x, s ∈ ( − n ∗ , n ∗ ) with s = x , s − x ) + Z | ξ |≥ n ∗ ξ − x ) h I ( n ) ξ ( s ) dξ = n ∗ − xs ( s − x ) q ( n ∗ − x )( n ∗ − s ) . (41)18 roof. By the scaling property we may suppose the interval I ( n ) to be [ − , h D be asin (A) and h z ( s ) be the Poisson kernel on the upper half plane: h z ( s ) = y/π ( y + ( s − x ) ).Then h [ − , z ( s ) = h D ( z, s + i
0) + h D ( z, s − i
0) = h z ( s ) + Z | ξ | > h z ( ξ ) h [ − , ξ ( s ) dξ, which shows lim y ↓ πy − h [ − , x + iy ( s ) equals L.H.S. of (41). The lemma therefore follows if weverify that if | x | < | s | < y ↓ π [ h D ( x + iy, s + i
0) + πh D ( x + iy, s − i y = R.H.S. of (41) . (42)If w = − (1 − x + y ) + i xy and φ = π − arg w ∈ ( − π/ , π/ | f ( z ) | = ( x + | w | / sin φ ) + ( y + | w | / cos φ ) and we see that y − ( | f ( z ) | − → / √ − x . In view of (39) this shows thatlim y ↓ πh D ( x + iy, s ± i y = 12(1 − cos( θ x ∓ θ s )) q (1 − x )(1 − s )where cos θ t = t with θ t ∈ (0 , π ) for − < t <
1. Now (42) follows from the identity11 − cos( θ x − θ s ) + 11 − cos( θ x + θ s ) = 2 − θ x cos θ s (cos θ x − cos θ s ) = 2(1 − xs )( x − s ) . ✷ (C) Let ( X n ) be the imbedded walk on the real axis mentioned in Section 1. In otherwords ( X n ) is the one-dimensional random walk with the transition probability p X ( x, y ) = H ( y − x ), where H ( x ) be the hitting distribution of the real line for our random walkstarting at the origin as being introduced in (31). Let µ ( x ), x ≥ X n ) and ν its dual; they are normalizedso as to satisfy lim x →∞ µ ( x ) /ν ( x ) = 1 and given by µ ( x ) = √ πe θ + σ ( v + · · · + v x ) , ν ( x ) = √ πe − θ + σ ( u + · · · + u x )for x = 0 , , , . . . ([11]:p. 212), where if c = exp (cid:16) − P ∞ k =1 1 k P [ X k = 0] (cid:17) , θ + = ∞ X k =1 k (cid:16) − P [ X k > (cid:17) + 12 log c = 12 ∞ X k =1 k (cid:16) P [ X k < − P [ X k > (cid:17) , (43) v = u = 1 / √ c and √ cv k (resp. √ cu k ) equals the probability that the ascending (resp.descending) ladder-height process visits k (resp. − k ) ([11]:p.202, 203). Then µ and ν arepositive solutions of the Wiener-Hopf integral equations associated with the kernels p X ( x, y )and p X ( y, x ) ( x, y ≥ x = 0 , , , . . . , µ ( x ) = ∞ X k =0 µ ( k ) H ( x − k ) and ν ( x ) = ∞ X k =0 ν ( k ) H ( − x + k ) (44)19see [11]:p.332 for uniqueness of positive solutions). (D) Suppose that E [ | Y | log | Y | ] < ∞ (as in Theorem 5). Put σ j = E [( S ( j )1 ) ] ( j = 1 , σ = E [ S (1)1 S (2)1 ]. Then putting for s, x, y ∈ Z H z ( s ) = P z [ ∃ j, S n = s and S k / ∈ R for 1 ≤ k < j ] ( z = x + iy )it is shown in [15] (Theorem 1.2) that as | x − s | + | y | → ∞ H z ( s ) = 1 ∨ σ a ( y ) π k s − z k (1 + o (1)) . (45)(Note that σ a ( y ) ≥ | y | [11]:P28.8, P31.1.) Here k z k = σ − ( σ x − σ xy + σ y )and a ( y ) , y = 0 is the potential function of the one-dimensional walk S (2) n . Denote thePoisson kernel on the upper half plane by h z ( s ) as in (B). Putting λ = σ /σ and ω = σ /σ we have k z k = [( x + ωy ) + ( λy ) ] /λ , and for y = 0, (45) is written as H z ( s ) = 1 ∨ σ a ( y ) | y | h x − ωy + iλy ( s )(1 + o (1)) . (46)Using this we can readily deduce from Theorems 2 and 2 ′ that as | z | ∧ | z − s | → ∞ H I ( n ) z ( s ) = σ a ( y ) | y | µ ( − n + s ) ν ( − n + s ) q n ∗ − s h I ( n ) x − ωy + iλy ( s )(1 + o (1)) (47)for y = 0. (E) In view of Donsker’s invariance principle the relation (46) (resp. (47)) incidentallyshows that h x − ωy + iλy ( s ) (resp. h [ − , x − ωy + iλy ( s )) is the density of the hitting distribution of thereal line (resp. the interval I ( n )) for the process Q / B t .We give a direct algebraic verification. We may replace B t by U B t with any orthogonalmatrix U and choose U so that the matrix Q / U sends the real line to itself. A simplealgebraic manipulation leads to( Q / U ) − = σ σ − ω λ ! . Let z = x + iy , ˜ z = x − ωy + iλy and c = σ /σ . Noting that c ˜ z corresponds to z , we thenfind, with the notation analogous to ˜ h D ( z, s ) in (40), that˜ h z ( s ) = ch c ˜ z ( cs ) , of which the right-hand side equals h ˜ z ( s ), yielding the analogue of (40) as desired. References [1] L. V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill, 1979202] Y. Fukai, Hitting distribution to a quadrant of two-dimensional random walk, KodaiMath. Jour. (2000) 35-80.[3] Y. Fukai and K. Uchiyama, Potential kernel for two-dimensional random walk, Ann.Probab. (1992) 1979-1992.[4] H. Kesten, Hitting probabilities of random walks on Z d , Stoch. Proc. Appl. (1987)165-184.[5] H. Kesten, How long are the arms of DLA, J. Phys. A (1987) L29-L33.[6] H. Kesten, Some caricatures of the multiple contact diffusion-limitted aggregation and η -model, pp.179-227 in Stochastic Analysis, eds. M.T. Barlow and N.H. Bingham, Cam-bridge Univ. Press, 1991.[7] G. Kozma and E. Schreiber, An asymptotic expansion for the discrete harmonic po-tential, Electronic J.P. (2004) 1-17.[8] G.F. Lawler, Intersections of random walks, Birkh¨auser, 1991.[9] G. F. Lawler and V. Limic, Random walks: A modern introduction, Cambridge Univ.Press, 2010.[10] Z. Nehari, Conformal mapping, McGraw-Hill, 1952[11] F. Spitzer, Principles of Random Walks, Van Nostrand, Princeton, 1964.[12] K. Uchiyama, Green’s functions for random walks on Z d , Proc. London Math. Soc. (1998), 215-240.[13] K. Uchiyama, The hitting distribution of the half real line for two dimensional randomwalks. Arkiv f¨or Matematik (2010), 371-393.[14] K. Uchiyama, Erratum to: The hitting distribution of the half real line for two dimen-sional random walks. Arkiv f¨or Matematik50