One technique for studying the approach to equilibrium of a continuous time Markov process is to consider the restriction to the L^2 space of an invariant distribution. When the process is reversible with respect to this distribution, the generator is a selfadjoint operator. We study the L^2 spectrum of the generator for certain random walks on Z^d, where the reversible invariant distribution is concentrated near the origin and decays rapidly with distance to the origin. For the related diffusions on R^d we find that the generators are unitarily equivalent to Schrödinger operators

Sullivan, W. G.

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DI AS STP -84 6L2 Convergence of CertaIn FatidomVJaiks on Za and related DiffusionsOne technique for studying the approach to equilibrium of a continuous timeMarkov process is to consider sne restriction to the Er space of an invariant distribizion. When the process is reversible with respect to this distribution, thegenerator is a seifadjoint operator. We study the £2 spectrum of the generatorfor certain random walks on Zd, where the reversible invariant distribution isconcentrated near the origin and decays rapidly with distance to the origin. Forthe related dinsions on Rd we find that the generators are unitariiy equivalentto Schrödinger operators.Wayne C. SullivanDepartment of MathematicsUniversity CollegeDubiin, 1rnandandSchool of Theoretical Physicsoblin Iust:tute br Advanced Studtesthen convergence to equhbr:nni is expcuentiaily fast in L2 (;I. Such convergence has cen shown for special :iyiianic spin system having distinct phasesand is conjectured for dynamic two-Qlmcnsional ising models in the two ohaseregion (see 4], 51).Here we use the spectral approach to study the convergence of some simplerandom wall: models. The original motivation is in the worh of i3aEet-Puld-deSmedt on a boson system coupled to a heat bath ]. Arising in then su’lieswas a Markov process on the positive reals for which an estimate of die rat: ofconvergence was desired. ifl 61 we gave estimates of convergence for processesof this type and related Markcv chains on the positive integers, in the presentwork we extend the analysis to certain random walks on Zd and lock brieflyat the diusion limit which yields Schrbdinger operators.§ Formula tom The £2 approach is applicable to rather general continuoustime Markov processes. We shall formulate here in lerrus of a countable discretestate space i and later specialize to Z5. To each I2 we associate a inircsubset of fl denoted z called the nesghbours of . We require matI) ‘S ‘SOy.For Zd we define§1 introductiorn One standard approach to continuous time Markov processes is to start with a generator D, construct a transition function Pt(, d)which in an appropriate sense satisfies= exp(tD) (1)dz={z±e:i=1.”d} (3)where {e1} are the standard unit vectors.We assume that we are given a strictly positive probability distributionon . We shall construct the generator D from a matrix Q which satisSes thereversibility conditionand then study the properties of P, e.g. invariant measures. For certainphysical systems it is more natural to start from a probability measure p anddevelop processes which leave p invariant. Of particular interest are generatorswhich satisfy the detailed balance condition with respect to p. In terms of pand lb this yields the relationwhenever i E ôz. We definer(z)Q(z, y) = r(y)Q, )= :; Q(z,z,)y Eö(4)tqdz)P(z,dy) = p(dy)Pt(y,dz) (2)which is the condition of time reversibility for the Markov process with initialdistribution p and transition function P,The assumption of time reversibility introduces additional techniques forthe study of the process. The aspect we consider here is convergence in theLa sense. When p and lb satisfy (2), lb gives rise to a seladjoint courracrim semigrou in L2(p). The generator D acts as a negative semi-definiteselfadjiat operator ir Lçp) with elgenvalue zero corresponding to the invariant distribution. if the spectrum of D is otherwise bounded away from zero,and for the real yalued function f equal to zero except o a finite subset of fI(Qf)(z)= Q()yEO(Df)(z) = (Qf)(z) — q(z)j(;).1) is densely defined and dissinative(Df,J),. (f,f)(5)(6)1. 2.in £2(r) where Theorem 1. Let ir be a strictly positive probability distribution on Z. If(f, g) f(z)g(fr(z).We now introduce some particular generators in terms of Q, I = 0, 1, 2, 3.We need only define Q(x, y) for y E ox and take q, Q and D as above;for 1=0,1,2.Qo(z,y) = 7r(y)/[r.(z) +ir(y)]Q (x, y) = min{1, r(v)/r(z)}Q2(z,y) =Q (z, y) = max{1, 7r(y)/r(z)}Proposition 1. The closure of 132 is the generator of a positive contractionsenhigroup in £‘ (7r) provided that the cardinality of O is uniformly boundedfor x E fl.Conclusions about the spectral properties of 13 presume that the closureof 13 is the generator of a positive contraction semigroup on £‘(7r). When oxis uniformly bounded, this obtains for D and D as bounded operators andfor D2 by Proposition 1. Related methods can prove that this property holdsfor D in the case of r of the form in Theorem 2.When the closure of D is the generator of a positive contraction semigroupon £(z), the associated P is a Feller semigroup which extends to £, I pcc. The action of the generator in £2(7r), which we also denote by D, is anegative semidefinite selfadjoint operator. We use the following notation:= i{A E £2(r) spectrum of — 13)A(D) = inf{the [0, A] spectral projection of — 13 is infinite dimensionai}The value of A is significant because it determines the rate of exponentialconvergence to equilibrium in £2 (r). In [6] we give direct estimates for A1, butfor Zd, d> 1, the methods seem more suited for estimating ). We note thatif A > 0, then ? > 0. From (7) we hay.forj = 1,oo, I = 0,1,2.For Z’ we have the following)(D) Aj(D;+1) (s)liPç (x)/x(n)<oolirn sup x(-)/(-72) <Gothen we have )(D) > 0, 1 0, 1, 2,3. Conversely, if either limit superior isinfinite, then A(D) = 0,1 = 0,1,2.In general the condition is not sufficient for A (D) = 0, but this is the casefor ir of the type in Theorem 2.For D > 1, the results are for a specific family of examples. We considerprobability distributions on Zd of the form= K exp —cr[’c,cr>0with K the normalizing constant,Theorem 2. For r as given above we have0 < a <1 ?(D) = 0, 1 = 0,1,2,3 (9)a=1=0<Ac(Dj)<oo, 1=0,1,2,3a>i=.O<A(D)<oo, i=0,1a > I ) (Di) = o, I = 2, 3 (12)§3 The Diffusion Limit. Before proving the above results we shall takean elementary look at the diffusion limit of operators like those above. Thelimiting diffusion operators on L2(R’, rdx) when symraetrized toL2(Rd) takethe formwith denoting the Laplace operator and V a multiplicative operator. Thus,apart from irrelevant constants, the limiting operators are Scbrödinger operators (see [3]).We shall consider limits of operators based on D2. The reader may verifythat D and D lead to these same limiting operators, while D yields half theselimits. Let x : R’ — R be a strictly positive twice continuously differentiablefunction such that< 00All of these are reversible for r and we have(D+Lf) (Dgf,f),,. (7)(10)(11)3. 4.for each positive integer n, with the sum over E Zd. Define §4 Remaining proof’s. For Proposition 1: if 8z is uniformly bounded in2, then Q2 acts a.s a bounded linear operator in £2(r). MuItiplicaion bye(z —logr(z) —q is the generator of a contraction semigroup in £2(). By Theorem 13.2.1 of2 Rile-Phillips [2], the closure of D2 generates a contraction sernigroup inThe generator D is defined by the matrix Q. where Then for A> 0 and any ,f E £2fr) there exists q E £2fr) withôx={±ej/n:j=1”.d} (A—D2)g = f________Now £r) C t’(r) and is dense. We verify that the above holds in t’fr) andy) = n y)/(z) the result follows from the Hilie-Yosida Theorem.Proposition 2. Let f : Rd R be twice continuously differentiable, Then Proof of Theorem 1. If we have for all Ti 0 and some a > 000 00Jim Df=Lf —2Ve’Vf (13) —.2r(—z)<!— r(n) a 2r(—n) — apointwise. Th operator acting on f E L2(Rd, dz) is formally unitarily equ&- an adaptation of the proof of Theorem I of 6] shows thatalent to— ([Ve]2—i..c) g (14) A1(D3) a2/4acting on g EL2(R’). A1(D2) ‘./r/(i —Proof. To get (13), expand j’ to second order in a Taylor series and take limits. A1(D3) [a/(1 —The mapping taking f E L2(Rd, 7rdz) to g E L2(Rd) is given by A1(D0) ag(z) = f(x) (15) As the uulI space of D is one dimensionalExpression (14) follows from a straightforward calculation. A00 > A1 > 0Thus we obtain Schr6dinger operators which can be expressed with V of in each easethe fom For the case where one of the inferior limits is infinite— for definitenessV = — e. we assumeFor example, e(z) = z12 corresponds to the harmonic oscillator potential. Jim inf irfr)/ E ir() = 0, (IG)The following shows that quite a broad class of Schrddinger operators can beexpressed in this way.consider the functionsProposition 3. Let the Schrödinger operator — + V possess a strictlypositive twice continuously differentiable eigenfunction ‘: f (z) = 0 for x1 < n, = I for a1 n (17)+ = As d 1, a1 = a, but the above definition will be used again below. We havefor the real constant A. Then V can he expressed in the formfr)—/(Jn ft),r = \/(n — 1)7r(n)/V = A ± Ve]2 —It is not difficult to deduce from (16) and the above thatwith c = — lega.Proof. Calculation liminf(—D2f7;,Jn),r/(fn, fn) = 0,6.so (D2) 0 by Lemma 1 below. The result for D0 and D1 follows from (8). Thenlim(D3fn,f,)/(fn,fn) 0Lemma 1. If there exists a constant a and a sequence of functions {f} E n—Do£2(r) such thatand Lemma 1 together with (8) gives conclusion (9).= 0 for jzj <For D2 and a 1 we employ Lemma 2 with p(z) 1. for this case we1iminf(—Dfn,fn)/(fn,fn), a (is) haven—oo a1. 1then —s(z)+q(z) = —2d+ [r(z+ej) +r(z —el)31/r(x) (19)Prool If b < A00, then the [0, b] spectral projection of —D is finite dimensional,so the contribution of this part of the spectrum to (18) goes to zero as . For a > 1 the right hand side of (19) approaches +00 as — co. Hencehence b a. we have (12). For a = 1 the right hand side of (19) can for large Izi beLemma 2 If there exists a constant a and a positive function o such th approximated arbitrarily closely by a sum of d terms of the form r + 1/f — 2with with at least one value of r satisfying r exp[—c/(2dj1. Also for a = 1,Q(z, y) = Q(z, y)/ir(z)p(y)/(p(z)ir(y))Q2 cosh(c/2) QoQ(z,y)yEas so we have conclusion (10).Now we consider D1 for a > I and takewe have thatliminf q(z)—8(z) a,(z)=exp[—cz+2z/3then A00 a.with > 0 constant. Then q(z) — s(z) can be represented as the sum of 2dProof. Assume that for all IzI r’ terms of the following two types. With y = z ± e1 for 1 < Jz the term is ofthe form 1 — exp [1uI — IzU, while for ui > izi the term is of the formq(z) —s(x) b.(expcUzia— iyM)(1 — exp/3[izI —1i1) (20)For f such that f(z) = 0 for ivl < n and {z: f(z) 0} is finite, with u(z) =/(x)/p(z)f(z) we haveIt is not difficult to deduce that terms of the form (20) go to zero as —+ 00(ff)1 = (g,g)p so that(—Df,f) = ([q — s]f,f) + (sg,g)p + (—Qg,g). liminfq(z) — s(z) 1— exp(—/d.By an adaptation of Lemma 1 of [61, the last two terms together are nonnegativeSince > 0 is arbitrary, A00(D1) 1. Also 2Qo Q and both D1 and D0soare bounded so we have (11).(—DJ, f),. b(j, f),..Tiic uallty extends to all fin the £2fr) domain of D which satisfy 1(z) = 0 Concluding remarks. The random walks on we considered showed thefor [z[ < ri, which implies A00 b. following behavior. If the reversible invariant distribution ir decays as 00Proof of Theorem 2. For 0 < a < 1, if we apply D3 to f given by (17) for at a rate less than exponential, thea 0 is an accumulation point of the spectrumof the generator. If the rate of decay is exponential or greater, then 0 is an(—D3f,fn)ir/(fn, f,.,) {zi = n}hr{zi } isolated point of the spectrum.We have given a very elementary approach to the diffusion limit. WithFor fixed integer k we hay. a bit more effort it is possible to relate the spectral properties of the limitto those of the discrete approximations. For this to be useful we need betterlim r{z1 = = n + k} = 1. estimates on the spectra of the discrete models. Since Schrödinger operators7. 8.have received such considerable study, it might be more useful to be able tomake assertions about the spectra of the discrete models based on the propertiesof the diffusion limit.References1. Buffet.E., Pulé,i, de Smedt,P.: On the dynamics of Bose-Einstein condensation. Ann. Inst. Henri Poincar Analyse non-linéaire 1 413-451(1984)2. Hille,E. and Phillips,R.S.: Functional Analysis and Semi-groups, 2nd ed.Providence: AMS (1957)3. Reed,M. and Simon,B.: Methods of Modern Mathematical Physics, Vol. 4.New York: Academic Press (1978)4. Suflivan,W.G.: Mean square relaxation times for evolution of random fields.Commun. Math. Phys. 40 249-258 (1975)5. Sullivan,W.C.: Exponential convergence in dynamic Ising models with distinct phases. Phys. Let. 53A 441-2 (1975)6. Sullivan,W.G.: The L2 spectral gap of certain positive recurrent Markovchains and jump processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete (inpress)9.

L^2 Convergence of Certain Random Walks on Z^d and related Diffusions

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L^2 Convergence of Certain Random Walks on Z^d and related Diffusions

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