on the number of distinct sites visited by a random walk

On the Number of Distinct Sites Visited by a Random Walk"

M. D. DONSKER AND S. R. S. VARADHAN

1. Introduction

Consider a spatially homogeneous random walk on Z d with one-step transition probability distribution T, i.e., to each point j E Z d is assigned a non-negative probability ri, x j a Z d T~ = 1. If we let X, , X , , X,, . . . be independent random variables taking values in Z d and having common distribution T, then Y,, = X, + X , + + X,, is the position of the random walk after n steps. Let D,, = { j € Z d : j = Y, for some r, l S r S n } , and let Dlf be the cardinality of D,,, i.e., Df is the number of distinct sites in Z d visited by the random walk in the first n steps.

For v > O , consider E[exp{-vDSf}], where E{ } denotes expectation over the random walks starting from the origin. In this paper we study the asymptotic behavior of E[exp {-vDlf}] as n -+ m. This behavior depends, of course, o n what we assume about T ; therefore we state now precisely what these assumptions are. It is convenient to make these hypotheses on the characteristic function of the distribution T. For t E R,, let &(t ) = x j e Z d Tiei('.') which is then periodic of period 27r in each of its d variables, t = ( t , , t,, - . . , t,) . About & ( f ) we assume the following:

( i ) & ( t ) = 1 if and only if r = 27r(n,, n , , * - , nd)r where n,, n2, - - * , nd are integers.

(ii) &(t)=l-fl(t)+o(lrlP) as It l- .O, where O < a S 2 and e-nc') is the characteristic function of a symmetric stable law of index a! in Rd which is non-degenerate, i.e.,

C aiitit, for some

positive definite matrix {ai j} , a ! = 2 , f l ( t ) =

*The research for this paper was done at the Courant Institute of Mathematical Sciences and supported by the National Science Foundation under NSF Grant No. MCS-77-02687. Repro- duction in whole or in part is permitted for any purpose of the United States Government.

Communications on Pure and Applied Mathematics, Vol. XXXII, 721-747 (1979) @ 1979 John Wiley & Sons, Inc. 0010-3640/79/0032-0721$01 .OO

722 M. D. DONSKER AND S. R. S. VARADHAN

In the expression for a(?), M is a symmetric measure on the unit ball

means that the support of M spans &. Let {rjl} be the n-step transition probabilities. It follows from hypotheses

(i) and (ii) above that for any j e Z d there exists an no such that .rrYu>O. Moreover, from hypothesis (ii) it follows that as n + 00

Sd-1 in ‘ R, (M(dy) = M(d(-y))), and the assumption of non-degeneracy

provided n/c t + k as n + 03. In other words, hypotheses (i) and (ii) imply that the random walk is irreducible and that the distribution 7~ belongs to the domain of normal attraction of a non-degenerate symmetric stable law of index a, O < a 12.

Let {x(s), O S s < ~ } be a symmetric stable process in Rd of index a, 0 < a 5 2, corresponding to a(t) and let Ln be its infinitesimal generator. Our main result is that under the hypotheses above we have

THEOREM 1. For v > 0,

where

d + a ahL d’ (d+rr ) k (u , L ) = k(v, L, d, a)= V ~ ’ ( ~ + ~ ) ( T ) ( T ) ,

and

A,= inf A(G), G:IGI- 1

the infimum being over all open sets G in R, of unit volume and A(G) the smallest eigenvalue of - Ln with Dirichler boundary conditions for G.

In [l] (Lemma 3.10), we showed that the constant AL appearing in Theorem 1 is always strictly positive. We note also that k(v, L) in Theorem 1 is the same constant that appears in the asymptotics for the symmetric stable sausage (Theorem 3 in [l]) which is obviously the continuous analogue of the problem treated here. Consider the special case of Theorem 1 where a = 2 and the infinitesimal generator Ln = +A. Then infG, lGl=l A(G) is attained for

THE NUMBER OF DISTINCT SITES VISITED BY A RANDOM WALK 723

the sphere of unit volume (Lemma 3.13) in [l]) so that in this special case

where yd is the lowest eigenvalue of -;A for the sphere of unit volume in d dimensions with zero boundary values. The prototype random walk in Z d , where transition is possible only to adjacent lattice points with equal probability, satisfies hypotheses (i) and (ii) for a = 2 and Theorem 1 applies with k(v, L) given in this special case by (1.2).

It is clear that in the asymptotic evaluation of E[exp {-Of}] as n .+ 0~ the random walks which contribute most to the integral are those for which Off are small so that what is required here are estimates on the probabilities of large deviations for Of.

2. Notation, Local Limit Theorem, and Upper Estimates

Certain estimates required for proving Theorem 1 are best made when the random walk is on a compact space; thus we find it convenient to map the walk on Z d into an appropriately selected lattice in the d-dimensional torus T,. In this mapping the periodicity of the walk must be taken into account.

Let p=g.c.d. {n: . r r : # O } be the period, and let Yo={?€ R, : Ii?(t)l= 1) so that Yo 2 (27r(n,, n,, . * , n d ) } where n, , n2, * . - , nd are all integers. If H c Z d is the annihilator of Yo, i.e.,

then H is a subgroup of Z d and Zd/H is cyclic with p elements. Moreover, if j is such that T, > 0, then H, H + j , * , H + (p - l ) j are the cyclically moving subclasses of the random walk.

Let A > 0 be given, and let {a,,} be a sequence of integers such that p I a,, for every n, and

as

Consider R,/Zd = T,, the d-dimensional torus. We can think of Td as the d-dimensional cube, Ixil 54, i = 1,2, * * * , d, with opposite sides identified. For


each n, let

i XETd: x=-forsome j E z d an

and, for x E @I;), let

+r,(x)= c T,. 1 :;/an =*(mod 1)

Thus, ii,(x) serves as the one-step transition probabilities for a random walk on the subgroup YF’c Td. Moreover, since p 1 a,, the random walk on 9:’ is also periodic of period p , the cyclically moving subclasses being the images of H, H + j, * * , H + (p - l ) j under the canonical mapping of Z d into 9“’). We shall denote these cyclically moving subclasses by I-i(d?‘ for r L 1 where a;;+, = H?‘ .

For any p>O, let

and let 6(p, x) = C ; & d @(p, x + j ) . For x E 2:), denote by ii!,(x) the /-step transition probability for the random walk on y:), i.e.,

From our assumptions, the following local limit theorem follows. The proof is omitted since it conforms to the usual pattern for local limit theorems.

THEOREM 2.

(2.2) I -- n-==

+ P r d )

Let XI, X 2 , * be independent random variables taking values in 9:) with common distribution +,,(x), and let ?, = XI + X2+ - - * + XI (addition on Td). Denote by B f ( l ) the number of distinct sites in $2) visited by


TI, g2, * . - , PI . We recall that

Dn={jEZd : j = Y, for some l s r s n n ) ,

so that if we let

0, = {X E Td : j/an = x (mod 1) for some j E 0,) ,

we observe that fin ~2:) and Dif 50:. What is important is to note that the distribution of 0: is the same as the distribution of fi if(n). We explain now why this is important for getting upper estimates on the expectation appearing in Theorem 1.

For A, a Bore1 set in T d , we consider

(2.3)

i.e., the proportion of visits to A made by the random walk in 2:) in the first 1 steps. We note that, for fixed w, I, and n, tl (w, .) is a probability measure on Td. Let & ( T d ) be the space of all probability measures on T d endowed with the weak topology. We shall subsequently consider the distribution of Ll(wl .) in & ( T d ) , i.e., let 6"' be a probability measure on &(Td) defined by

P ~ n ~ ( c ) = P { t l ( w l ~ ) E C},

where C is a set of measures in A ( T , ) and P is the probability measure on the random walks in 2:) starting from the origin.

For x E T d , define

(2.4)

where

(2.5) ( 0 otherwise

Let Ix : t i n ) ( w , x ) > 01 denote the Lebesgue measure of {x: LIn)(w, x)>O}.


We observe that

1 / x : Ly'(0, x) > 01 = -;i fi,Jf(l) .

For fixed a, I, and n, tln)(u, .) is a probability density on Td. Let LI(Td) be the space of all probability densities on Td endowed with the L, topology. Let a!") be the probability measure on LI(Td) defined as the distribution of Lin)(w;) on LI(Td). In other words, if B is a set in LI(Td) we define

(2.7) QI"'(B) = P[Li"'(o, *) E B] ,

where P is again the probability measure on the random walks in L$' starting from the origin. Clearly,

From the comments and definitions above,. we now see that

(2.8) E{exp{ -vD~}}~E{exp{ -v~~}}=EQ~"'{exp{ -vad , Ix : f ( x ) > O l } } .

Corresponding to the n(r) which appears in the hypotheses on T, there is a symmetric stable process {x(s), 0 5 s Sm} in & the infinitesimal generator of which acting on smooth functions on Rd is given by

a2 u f f = 2 ,

(2.9) L,u=

2

The process {x(s), 0 5 s < m} can be projected onto the torus Td. Denote it by {l?(s), 0 5 s <a}; then this is again a process with independent increments the generator t, of which acts on smooth functions on Td. The relation between L, and L, is very simple: Take a function u on T d , view it as a periodic function on Rd, apply L, to this periodic extension thereby getting a periodic function which is now L, applied to u.

Let % be the space of positive functions W E c ( T d ) each of which is strictly bounded away from zero. For p E d(Td), we define

(2.10)

and if p is given by a density f it will be convenient to denote I&) by It(f).

THE NUMBER OF DISTINn SITES VISITED BY A RANDOM WALK 727

For the A picked just before (2.1) we define

1

We shall prove below (Theorem 6) that if CcL1(Td) is closed, then

- 1 lim y-, log @“’( c> 5 -inf ZA (f) . n-re an f C C

(2.1 1)

Since Ix : f ( x ) > O l is a lower semicontinuous function on L1(Td), it follows from (2.11) (see corollary to Theorem 6 below) that for v > O

Recalling (2.1) and using (2.8) we then conclude that

1 - (2.13) n-m lim ~ nd,(d+a.) log E[exp {-vDf}]5 -Ad inf {vlx :

But (2.13) holds for all A > O so that

REL,(T~)

(2.14)

and at the end of Section 4 we shall show that

where k ( v , L ) is defined in the statement of Theorem 1. From (2.14) and (2.15) we have


In Section 5 we shall show

1 (2.17) nym lim - n d / ( d + a ) log E[exp{-uD~}]2 -k (u , L )

and thus will have proved Theorem 1.

3. Preliminary Results We first prove

THEOREM 3. lf C is closed in &(Td), then

Proof: Let U E % and for x € T d define the operator

Let

(3.2) 1 . ("") ( ",-" Vn(x)=l0g 2 (x)=log 1+"---

From elementary properties of Markov processes we deduce (c.f. (2.2) in [2]) that, for each W E % and all n,

(3.3)

With Lnu as defined in (2.9), we now assert that, for all UE%,

(3.4)

uniformly for x E Td as n .--* 0. To see (3.4), it suffices to verify it for functions u of the form ei ( r 'x ) on Td, where t is a multiple of 27r since linear combinations of such functions are dense in %. It is easy to check that, whether a = 2 or O < a <2, for these special functions u, LnU = -e'("%(t)


and finu = ei('*x)i?(?/an). Hence

and verifying (3.4) for these special functions u is equivalent to showing

(3.5)

But f l ( t ) has the property that Cl(r/a,) = (l/az)fl(t), and therefore (3.5) follows from hypothesis (ii) on # ( t ) in the introduction.

From (3.4), for any UE%,

and hence, as n - m ,

( "y) finy ++a) V,(x)=log l + L =-

uniformly for x E Td. Thus, for each u E %, there is a sequence of constants 6, -+ 0 such that

(3.6) Vn(x)S- -+an . a:: [? 3

From (3.3) and (3.6), for each ~ ~ " 1 1 ,

This implies that, for each ~ ~ " 1 1 ,

730 M. D. DONSKER AND S . R. S. VARADHAN

and, therefore, letting b, = nla; , we have

where we note from (2.1) that

(3.9)

Now (3.8) holds for all U E % and hence

Since C is closed in &(Td) and & ( T d ) is compact in the weak topology, we see that C is compact, and it follows (c.f. [4] just before (2.13)) that

(3.11) - 1 i u lim -logP',"'(C)Ssup inf I ( L ) ( x ) p ( d x ) . n- b, $&€CUP% T,, u

Thus, from (3.11) and the definition of Zc(p) we get

- 1 lim -log P ' , " ' (C)~sup ( - z i ( p ) n-- b, (rec (3.12)

= - inf Z,-(p) . ) I E C

From (3.9) and (2.1) we see that b, - at/A"+d which allows us to rewrite (3.12) as

- 1 1

= - inf IA(p) , * E C

which completes the proof.

THEOREM 4. Let C be closed in &(Td). Then

- 1 lim 7 log Q',"'(C) S - inf IA ( p ) . n- a, * E C


Proof: Let C be weakly closed in &(Td). With ,yn defined by (2.5), we observe that xn * p converges weakly to p as n -+ uniformly for p E &(Td). Thus, for any E > 0 there exists no such that, for n 2 no, x,, * p E C implies p € C , where c' is the set of measures in &(Td) within distance E

(Prokhorov distance) of C. Therefore, for n E no, Q!,'"(C) d P',"'(C"), and letting c" denote the weak closure of c' we have from Theorem 3,

(3.13)

- 1 - 1 lirn 7 log Q!,'"(C) 5 lirn 7 log P,(")(Ce) n-rn an n-- an

5 - inf_ IA(p). &€C'

This last holds for every E > 0, and since IA ( p ) is lower semicontinuous and &(Td) is compact in the weak topology, we obtain the theorem by letting E - + O in (3.13).

Let E > O be given and let &(x) be a c" function on T d with & ( x ) = $e (-x), t,bc (x) 2 0, JTd JI, (x) dx = 1, and JI, (x) + S(x) as E -+ 0. Let K, : LI(Td) + LI(Td) be defined by

(3.14)

We now prove

THEOREM 5. For every S > O ,

1 _ _ (3.15) lirn lim 4 log a:)[ f : I ((K,f)(x)-f(x)( dx 2 6

E - ~ o n-m a n T d

Proof: From the definition of measure on L,(Td) and P',"' measure on A(Td) , we have

(3.16)


Writing xn*g =x,g and using the symmetry of xn and CL,, we see that the last expression in (3.16) is the same as

(3.17)

Moreover, if we let Mn.s c C(Td) be the set of all functions V(x) of the form V=(K,-I)x.g, where g € c (Td) with llgll51, then (3.17) is the same as (see definition of Pi"' measure just after (2.3))

sup [V(P l )+V(P2)+ . . .+v (Y, ) l sna] , V E M ...

(3.18)

where, as before, p k = XI + X2 +. . . + r?k and Al , X2, . . * are independent random variables with values in 9:) having common distribution +,,(x). Thus

(3.19)

In Lemma 4.1 below we show that, given n, E, 6, there exist functions V,, V2; * a , VJ all in M,,E such that, for any function V E M ,,+,

inf sup I V(x) - Vi(x)l 5 4 8 , I S i S J ~€92~'"'

and, moreover, J = J ( n , E , 6) 5([8/6]+ 1)""". In other words, in Lemma 4.1 we show that M,, is totally bounded, and we estimate the number of spheres of radius i 6 it takes to cover Mn,c. Thus

(3.20)

THE NUMBER OF DISTINCT SITES VISlTED BY A RANDOM WALK 733

In Lemma 4.4 below we shall show that there exists an no such that n 2 no implies, for any 8>0, and any V of the type being considered here,

E[exp { e[ V( 9,) + V( p2) + - . . + V(p,,)n] (3.21) -= c exp {48( nu/(" +d) + p)} exp { n ~ , (ev)} ,

where c is a constant and

with

&(p) = - inf jTd log ( y ) ( x ) p ( d x ) , UC91,

'4L1 being the set of functions U E C ( T ~ ) such that u > O on Td. Thus, from (3.19), (3.20), and (3.21) we obtain, for n 2 no and any 8 >O,

In (3.22) choose 8 = ~ / n ~ ' ( ~ + ~ ) where r > 0; then (3.22) becomes

x l S i d J SUP exp { nAn( 6 vi) ] . Recalling that lVlS2 for all V under consideration, In(p)>O, and

7 34 M. D. DONSKER AND S. R. S. VARADHAN

we see that

Hence, in (3.24) nothing is changed if, in taking the supremum over all p,

we take it instead over all p such that j n ( p ) S y , 7 . i.e.,

But, it follows from Lemma 4.5 below that, for any of the V under consideration, i.e., V = (K - I)x,g where g E C( Td) with llgll5 1, and any t > 0,

where h(a)-,O as 0-0, A,(n)+O as n - , ~ for all r>0, and k , ( & ) - + O as E + O for all r>0. The functions h ( a ) , A,(n) and k , ( ~ ) are all independent of V.

Thus, from (3.25) and (3.26), we get

1 - -- a/(o + d ) SUP [7{2h(t~)+2A,(n)+ k , ( & ) } - u ] .

O C O S I

From (2.1), nd’(d’a’/at-+ l / A d as n -+ 33, and hence from (3.23) and (3.27) we obtain, for all r > O and all 7>0,

67 1 - (3.28) Slog([!] + 1 ) --+a lim sup [7{2h(ta)+2A1(n)+ k , ( ~ ) ) - - ( + ] 2hd A n - o c o s s

THE NUMBER OF DISTINCT SlTES VISITED BY A RANDOM WALK 735

From (3.28) we get, for all t > O , T > O ,

1 _- lim lim 7 log at)[ f : ITd ( (K , f ) (x ) - f ( x ) l dx 2 61 e-0 n-m an (3.29)

1 5- [ sup - A d o c o s s

In (3.29) let uf = U / T and the right side of inequality (3.29) becomes

where the last inequality follows from the fact that without loss of generality we could pick the function h to go to 0 monotonically.

From (3.29) and (3.30), letting t + 0 , we get

1 -- lim lim ~ l o g a',"' f I(K,f)(x) - f ( x ) l dx B u e-0 n-m an [ ' ITd (3.31)

5 -%+ log ([ P] + 1) , 2A

and finally, if we let T + in (3.31), we obtain (3.15).

4. Lemmas for Theorem 5 and Proof of Theorem 6

We now prove five lemmas used in the proof of Theorem 5 in the preceding section.

LEMMA 4.1. Let g E C(T,) with llgll5 1 and let M,,c c c(Td) be the set of all functions V ( x ) of the form V = ( K , - I ) x n g . For any S > O there exist functions V , , V , ; 9 , VJ in M",, such that, for any V E M,,,,

(4.1) inf sup I V ( x ) - V , ( x ) l ~ $ t i , I S i S J x s 9 ~ '

and, moreover, J = J(n, E , 8 ) s ([8/6]+ 1)"m".


Proof: We note that IV152 for any VEM,,, and also that 9;' has a: points. Subdivide the interval [-2,2] into subintervals of length so that there are at most [S /S]+l subintervals. Now [-2,2]"." consists of cubes of sides 48 and there are at most ( [ U S ] + 1)""" such cubes. But, any V E M,, , , as a function on 2':), can be thought of as an element of [-2,2]""". Thus, Mn,t. c [-2,2]".". Think of the subcubes constructed above as closed and, for a particular cube C, pick one of its elements if M,,= n C # 0. In this way we pick V,, V,; * , V,, where J is the number of cubes having non-empty intersection with M,,e. Thus J = J(n, E , a) 5 ([S/S]+ 1)""". Moreover, if V E M,,,, then V ~ [ - 2 , 2 p - " and must belong to some cube, i.e., the intersection of Mn,E with that cube is non-empty and hence we must have picked some element V, from that cube which means (4.1) holds.

LEMMA 4.2. Let 1 = "here exists an integer no and there exists a constant c such that, for n 2 no,

for all X E ~ ' $ " , y ~ S e l ; ' , Y - X E @ $ ; , where O S q S p - 1 .

Proof: Suppose first that x # @ z / ; then Y $ ! H : / + ~ so that i i ' , ( x ) = O and i i ! , + q ( y ) = O which means (4.2) holds. Next, suppose XE@'/ which implies y ~ H $ j + ~ . From Theorem 2 (with p = 1 and noting that 6 is strictly positive for X E Td) we can choose no such that n 2 no implies

and

Hence, for all x E H:/, we have

and, for all YE@$/+^, we have

THE NUMBER OF D I S I N C T SlTES VISITED BY A RANDOM WALK 737

Letting

we have the lemma.

LEMMA 4.3. Let x and y be any two points in 9:). There exists an integer no such that n 2 n, implies, for any 8 > 0 ,

where C,,, = c exp {48nu'(U+d)+ p}. The constant c is the constant of Lemma 4.2 and, indeed, no is the same as picked in Lemma 4.2. The notation Ex { } denotes expectation where the starting point is x.

Proof: Choose 1 and n, as in Lemma 4.2. Since x and y are in $:),

for some q, 0 5 q 5 p - 1. Using the fact that I VI 5 2, we have y - x E

(4.4)

Since (2 - x) - (2 - y ) = y - x E Hz:, we have from Lemma 4.2 and (4.4),

which is the lemma.

LEMMA 4.4. There exists an integer n, such that n B n , implies, for any e>o,

738 M. D. DONSKER A N D S . R. S. VARADHAN

Where C,,e is the constant in Lemma 4.3, and

with (c.f. [2])

I , (p) = - inf 1 log ( y ) ( x ) p ( d x ) , U € * I Td

(4.7)

and is the set of functions U E C(Td) such that u > O on Td.

Proof: From Lemma 4.3, there exists an integer no such that n Z n o implies, for any O>O,

E[exp {0[ V ( ?,) + - + V( ?,)]}I 5 C,,e id Ex[exp { e [ v ( ? ~ + * + v(?,,)D].

xeTd (4.8)

It follows from the proof of Theorem 1 in [2] that

inf Ex [exp { O[ V ( 9,) + * * - + V ( ?,,)]}] 5 exp { nA, (Ow} , xtTd

and hence we have (4.5).

LEMMA 4.5. Let p be a probability measure on 9:) such that f , ( p ) S u/n"'("+d', where u>O. Let V = ( K , - I ) x , g , where llgllSB. Then, for any t > O ,

(4.9)

where h(a)+O as a-0, A,(n)+O as n + m for all t>0, and k , ( E ) + O as E - + O for all t>o .

Proof: For any t > O , let n, = [fn'l'(d+ol)] and define

THE NUMBER OF DISTINa SITES VISITED BY A RANDOM WALK 739

Let

and let

Since llgllSB, we note that IV(x)lS2B on Td. Let piin,, be defined by: for X E 9:)

and write

The I-function has the property that if

and if m3=a7rl+(1-u)7r2, where O<a<l , then

(4.13) 1 3 ( p ) = - u p 9 1 inf I Td log ( ~ ) p ( d x ) S a l , ( p ) + ( l - n ) I , ( p ) .

Also, if np) indicates the n-step transition probabilities, then

(4.14) I y ) ( p ) = - u€*r inf I Td log ( $ ) p ( d x ) s n I l ( p ) .

Using (4.13) and (4.14) we see from (4.10) that -

(4.15) In,, ( P I 5

740 M . D. DONSKER AND S. R. S. VARADHAN

By Lemma 4.1 in [ 2 ] it follows from (4.15) and (4.11) that (in the variation norm)

(4.16) IlP*",, - PIIS h ( 4 ,

where h(u) + 0 as u + 0. Thus, from (4.16) and (4 .12) , we get

(4.17)

where

and

Since 6 is a density function on Td, we see that, for each t > 0, k,(&) + 0 as E + 0 .

To show that, for each t > O , A , ( n ) + O as n +m, it suffices to show, for each f > O ,

(4.18) -0 as n + m ,

i.e.,

THE NUMBER OF DISTINCX SITES VISITED BY A RANDOM WALK 74 1

Let A, = { X E Td : Jxil<1/2a,,, l s i s d } . O n A,,, the function x,, is equal to a t ; thus, if x ~ A , + z for some z ~ L f i ) , then 6n,r(x)=at+,,r(z). Observe also that meas (Td - U zE9p {A,, + z}) = 0. Since z E Lfi', we have z E a:' for some q, 0 5 q 5 p - 1. Consequently,

a t in,, (x) = - + y r ( z ) P

for some r, 0 9 r 5 p - 1, and z E Now, from Theorem 2,

which gives us (4.18) and the lemma.

THEOREM 6. Let CcL,(T,) be closed. Then

- 1 lim - log Q!,")(C) 9 -inf 1, (f) . (4.19) ,,+- a: fEC

Proof: Consider the mapping from A(TJ with its weak topology to LI(Td) with its strong topology given by p + K , p , where

with +be defined just before (3.14). This map is continuous and hence { p €&(T,) : K , p E C} is closed in &(T,). Thus, from Theorem 4 we get

- 1 lim 7 log Q!,"'[f : Kef E C] 5 - inf Z (f) . n-m a,, f : K.fp C

(4.20)

For any 6 > 0, let C, = {g E LI(Td) : llg -f"< 6 for some f E c} and let c, be the L, closure of C,. From (4.20), applied to this closed set c6 instead of C, we have


Consequently, for any set c closed in LI(Td), we have, for all 6 > 0 and all E > O ,

- 1 lim 7 log Q',n'(C) n-m a,,

To prove the theorem it suffices then to show that

5 - inf lA (f) . r E c

In view of Theorem 5, this means showing

(4.24) -- lirn lim inf IA (f) 2 inf IA ( f ) . 6-0 E-0 f : K . f ~ C a f E C

In fact, we prove even more, namely that

(4.25) 6-0

To see that (4.25) is true, let cf,} be a sequence in LI(Td) such that IA(fn) -+ I and KEfn E cam, where E, --P 0 and 6, --* 0; we must show that there exists f~ C such that ZA (f) S 1. If 1 = 00, there is noting to prove; thus assume 1 <w. But, the set of f in LL(Td) such that I,(f)Sl is compact in L1(Td), and the result follows from this compactness and the fact that IA(f) is lower semicontinuous. This completes the proof of Theorem 6.

From Theorem 6, using the arguments of Section 3 in [3], (c.f. remarks just after (2.11) above) we obtain

COROLLARY TO THEOREM 6. For any u > 0 ,

- 1 lirn 7 log EQ2"'[exp { - uad, Ix : f(x) > 0l)l (4.26)

S - inf {v Ix : g ( x ) > O l + I , ( g ) } . g EL i ( T d


As pointed out in (2.14), this implies

We shall now show that

as defined in the statement of Theorem 1. Thus, we will have shown that

1 (4.29) % log E[exp {- uDf}] S - k ( u, L ) .

n-m

To see that (4.28), holds we note first that, from the definition of IA(g ) ,

Let Ti be the torus in R, : lx,l S $ A , i = 1,2; - a , d, with opposite sides identified. Let

We can then rewrite the expression on the right of (4.30) as

(4.31) sup inf { u ] x : h ( x ) > O l + I t ( g ) } , A s 0 heLI(Td)

where Ie (h ) is the I-function for the symmetric stable process on the torus TA with generator L. It now follows from Lemmas 3.5, 3.6, and 3.9 of [l] that

(4.32) sup inf { v lh (x )>Ol+I t }Z k(u, L ) . A > O h E Ii(Td’)

Hence from (4.30), (4.31), and (4.32) we obtain (4.28).


To prove Theorem 1 it remains to show that

1 (4.33) lim ~ I o g E [ e x p { - u D { } ] Z - k ( u , L)

n-m

which is the object of the next section.

5. Tbe Lower Bound

Let G be a bounded open set in Rd containing the origin such that the Lebesgue measure of its boundary is zero. Consider the original random walk on Zd and recall the definition

D n = { j ~ Z d : j = Y, for some r, 1 s r s n n )

Let us pick an open set U with compact closure such that

and I VlS IGI + q, where q > 0 is a given arbitrarily small positive number. Let b, = r ~ ' ' ( ~ + ~ ) and define the set

b,U = {y E Rd : y = b,x for some x E U } ;

we have, clearly, by letting (b,U)# denote the number of lattice points in the set b,U,

E[exp {- uDt}] 2 E[exp {-ID:}; 0, c b, U ]

2exp{-u(b,U)#}P[D, c b,U]. (5.1)

In Lemma 5.1 below we shall show that

where A(G) is the smallest eigenvalue of -Ln with Dirichlet boundary conditions for G. Using (5.2) we obtain from (5.1)


since 9 > O was arbitrary we conclude that

1 lirn .d/(a+d) log E[exp {- vDlf}] 2 - vlGl- A ( G) . n-

(5.3)

Denoting by 3 the class of all bounded open sets that are continuity sets for Lebesgue measures we conclude from (5.3) that

1 lim;;;17inl;ijlogE[exp{-vDf}]b - inf [vIGI+A(G)I. n-- G s %

(5.4)

It follows from the proof of Lemma 3.9 in [l] that

(5 .5 ) inf [vlGl+A(G)]= k(v, L,), GEYl

as defined in the statement of Theorem 1. Thus (5.4) and (5.5) yield (4.33) and the proof of Theorem 1 is complete except for (5.2) which we state and prove as a lemma.

LEMMA 5.1.

Proof: For j E Z d consider

1 S k 5 1 n , - t G ] , Yf“ bn

where Pi[ ] denotes probabilities over the random walk starting from j . Let

(5.6) q(U,G,I, ,n)= inf P.

Here 1, is a sequence of integers tending to 33 such that l,,/na’(n+d) -+ A > 0, Nn is a sequence of integers such that Nn/nd’(a+d’+ 1 / A with l,,Nn 2 n for every n, and A > O is an arbitrary positive number. It follows from the Markov property that

746 M. D. DONSKER AND S . R. S . VARADHAN

From (5.7) we get

1 1 - lim ~ l o g I T D n c 6 , U ] b - lim logq(U,G,I , ,n) . ,--roo A “ZZ (5.8)

If {x(s), O S s < a } is the symmetric stable process in R,, of index a corresponding to a( r), then from (5.6) and the invariance principle we see that

(5.9) lim q( U, G, I , , n ) 2 inf Px[x(s) E U, 0 5 s 5 A , x(A) E GI. n-- x e G

Thus from (5.8) and (5.9), we have

inf P X [ x ( s ) ~ U , O 5 s ~ A , x ( A ) ~ G ]

The left-hand side being independent of A,

log P{ 0, = 6, V} 1 lim (5.11) n x

inf P , [x(s )~ U , O S s < A , x(A)E GI

Now

inf. Px [x(s) E U, 0 5 s i A , x( A ) E GI x e G

L inf_ P,[x(s) E V, 0 5 s S A - 13 inf_ Px [x(s) E U, 0 S s 5 1, x( 1) E GI , x s G X t v

where V is any open set such that

Since the second term is positive for any stable process and does not depend


on A, we conclude that

1 P , [ x ( s ) ~ U , 05 s SA , x(A)E GI

P, [x(s) E V, 0 d s 5 A - 111

It now follows from Theorem (8.1) of [5 ] that

and the proof of Lemma 5.1 is complete.

Bibliography

[l] Donsker, M. D., and Varadhan, S. R. S., Asymptotics for the Wiener sausage, Comm. Pure Appl. Math. 28, 1975 pp. 525-565.

[2] Donsker, M. D., and Varadhan, S. R. S., Asymptotic evaluation of certain Markoo process expectations for large time, I , Comm. Pure Appl. Math. 28, 1975, pp. 1-47.

[3] Varadhan, S. R. S., Asymptotic probabilities and differential equations, Comm. Pure Appl. Math. 19,1966, pp. 261-286.

[4] Donsker, M. D., and Varadhan, S. R. S., Asymptotic evaluation of certain Weiener inregrals for large time, Functional Integration and Its Applications, Proceedings of the International Conference, London; Clarendon Press, Oxford, 1975, pp. 15-33.

[5] Donsker, M. D., and Varadhan, S. R. S., Asymptotic evaluation of certain Markoo process expectations for large time, 111, Comm. Pure Appl. Math. 29. 1976, pp. 389-461.

Received November, 1978.

on the number of distinct sites visited by a random walk

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