AN ESTIMATE ON THE DISTRIBUTION AND MOMENTS OF THE LAST EXIT TIME OF AN ELLIPTIC DIFFUSION PROCESS

Available online at www.sciencedirect.com.MalhemiutCcHientiasCIENCEdoiREct.Acta Mathematica Scientia 2006,26B(4):639- 645数学物理学报www.wipm.ac.cn/ publish/AN ESTIMATE ON THE DISTRIBUTION ANDMOMENTS OF THE LAST EXIT TIME OF ANELLIPTIC DIFFUSION PROCESS*LiBo(李波)School of Mathematics and Statistics, Huazhong Normal University, Wuhan 430079, ChinaSchool of Mathematics and Statistics, Wuhan University, Wuhan 430072, ChinaLiu Luqin(刘禄勤)Abstract Let LB be the last exit time from a compact set B of an lliptic difusionprocess X. A moderate estimate for the distribution of LB is obtained, and the sufficientand necessary condition for E* (L旨) <∞is proved.Key words Last exit time, moment, transience, distribution, difusion process2000 MR Subject Classifcation 60.J25, 60J601 IntroductionLet X = (S2,F,Ft,Xt,0t,P*) be a Hunt process with state space (E,E). We will usethe notations and terminology adopted in Blumenthal and Getoor [2] except explicitly statedotherwise. For any B C E define LB = sup{t≥0: Xt∈B}, and call LB the last exit timefrom B of X.The investigation of last exit time of Markov processes has a long history. In 1967 Takeuchi[17] studied the existence of moments for last exit time of stable processes. Chung [5] (1973) .established a relationship between last exit distribution and equilibrium measure for Markovprocesses (see also Chung [4] (1982), Chapter 5). See Li et al [10] (1993) for more results alongthis line. Pitman [12] (1975) and Getoor [8] (1979) obtained the precise distribution density ofLE(0,r) for standard Brownian motion in Rd (d≥3). Chen [3] (1985) studied the distributiondensity of last exit time for Bessel processes. Wang [18] (1980), [19](1995), [20] (1996) and Wu[21] (1984) investigated the last exit distributions for several classes of Markov processes. Forgeneral Markov processes, it is very dificult to obtain the precise expression of the last exitdistributions.It is natural to use last exit time to define the transience of Markov process X. Recall thatif LB<∞a.s. for all compact set B, then X is called transient (see Chung [4] for details). X* Received August 14, 2004. Research supported in part by Tianyuan Fund ofr Mathematics of NSFC(10526021) and A Grant from Ministry of Education中国煤化工MHCNM HG640ACTA MATHEMATICA SCIENTIAVol.26 Ser.Bis called strongly transient if E* (Lp) <∞for all compact set B. Sato [15] (1995) proved acriterion for weak and strong transience of Levy processes. A further problem is to study thefiniteness of higher moments of the last exit time, that is, when does E*(L管) <∞(k≥1, Bcompact)? Takeuchi [17] (1967) showed that if X is a symmetric stable Levy process of order ain Rd, then the last exit time of X from a ball has the finite k-moment if and onlyif (k+1)a < d.Hawkes [9] (1977) generalized this result to symmetric Levy processes. For Brownian motion inRd (d≥3), Wang [18] (1980) obtained the formula for E0 (LB), and Yin Chuancun [22] (2000)gave the precise expressionof E* (L管) (Vx∈Rd).The elliptic diffusion is an important class of Markov processes. Aronson [1] obtained thelower and upper bounds of the transition density of the elliptic diffusion (See also Davies [6]).Schiling [14] (1996) proved the lower and upper bounds for the Hausdorff dimension of thesample paths of this process. Liu and Xiao [11] (1998) investigated the escape rates for llipticdifusion. Recently, Gao and Liu [7] (1999) studied the large deviation for this kind of diffusionprocesses.In this article, we will obtain a suitable estimate of the last exit distribution, and then willuse it to study the existence of moments of last exit time for the elliptic diffusion processes.2 PreliminariesFor any given λ∈(0,1], let A(入) denote the class of all measurable, symmetric matrix-valued functions a: Rd←+ Rd 8 Rd that satisfy the elliptic conditionλI1β≤E a(x)$s≤1 forall x,∈∈R'.For each a∈A(入), let L三V●(aV) be the corresponding second order partial differentialoperator. By Theorem II.3.1 of Strook [16], we know that L is the infinitesimal generator ofa d-dimensional symmetric diffusion process X = (92,F, Ft,Xt,0t,P") with state space Rdwhich is a Hunt process in the sense of Blumenthal and Getoor [2] and has continuous samplepaths. Moreover, its transition density function p(t,x,y) ∈C((0,∞)x Rd x Rd) satisfies thefollowing inequalityexpM|y-c )≤p(t,x,y):M。(2.1)Mt李牙expItfor all (t,x,y) ∈(0,∞)x Rd x Rd, where M = M(a,d) is a constant. The estimate (2.1) isdue to Aronson [1]. We call X an elliptic diffusion process.In general, the process X is bounded by two (nonstandard) Brownian motions X(1) ={X{'),t≥0} and X(2)= {x{2),t ≥0}-with generators本△and￥△, respectively, runningat different speeds.ForanyBorelsetBinRd,letLp=sup{t≥0:Xt∈B},thelastexittimefromB;Tp=inf{t>0:Xt∈B},thefirsthittingtimeofB(sup(0=0,inf0=∞).LetP(t,x,A)=P*(Xt∈A) be the transition function of X, and let Pf(x)= fpP (t,x, dy) f(y)be the transition operator; Uf(x) = E' (f0∞f(Xt)dt) denotes the potential of nonnegativefunction f. Define PAf(x) = E* (f(Xr,);TA <∞).中国煤化工MHCNM HGNo.4Li & Liu: AN ESTIMATE ON THE DISTRIBUTION AND MOMENTSWe recall some definitions. X is called transient ifLB<∞P*-a.e. Vx∈ Rd，V compact B ;X is called strongly transient ifE(LB)<∞，Vx∈ Rd，V compact BNote that X is transient if and only if .p* (limIXI|=∞)=1for every x∈RdIt is well known that Brownian motion in Rd is transient if and only ifd≥3. By a result ofTakeuchi [17], it is strongly transient if and only ifd≥5. It is clear from (2.1) that the llipticdifusion X is transient if and only ifd≥3. We will prove in the next section that X is stronglytransient if and only ifd≥5.We will use K, K1,... to denote positive constants whose precise values are not importantand may be different in different cases.3 Main ResultsLet X = (2, F,Ft, Xt,Ot, Pr) be the lliptic diffusion process in Rd(d≥3) specified inSection 2. It is known that X is transient and symmetric. As in the preceding section, p(t,x, y)denotes its transition density which satisfies (2.1). Letu(x,y)=p(t,x, y)dtbe its potential density. From (2.1) it is easy to see that (put u(x,x)厂1 = 0)u(x,y)=∞→x=yVx∈R',y- + u(x, y)~ is finite continuous.Thus from Theorem 1 on page 212 of Chung [4] we have the following Lemma.Lemma 3.1 For any compact set B, there exists a unique finite measure μB withsupp(μB) C 8B, such that for any x∈Rd we havePel(x)=P*(LB>0)= I.u(x, y)μuB(duy).(3.1)The measure μB is called the equilibrium measure of B, and C(B) = μB(B) is called thecapacity of B under X. It is known from Chung [4] that Hunt's Hypothesis (B) holds, and foreach compact set B,C(B)= sup{v(B): v finite, supp(v)C B,Uv≤1 on B}.Lemma 3.2 For any ball B(0,r)M fr(告-1) °2≤(B(:,n)≤MItr(告-)”,(3.2)中国煤化工MHCNM HG642ACTA MATHEMATICA SCIENTIAVol.26 Ser.BProof Let X(1) be the (nonstandard) Brownian motion whose transition density is givenbypi(t,y-x)= 717 expM|y-x| .and let X(2) be the (nonstandard) Brownian motion whose transition density is given by[p2(t,y-x)= 727 exp(一),where71 = M1+tπ生,72= M-1- tπ~ t are two finite positive constants. Letul)(x,y)=。pr(,y -x)dt,u(x,y)三. p2(t,y- x)dtbe the potential density of X(1) and X(2), respectively. For any measure μ, defineUμ(x)= Ia u(x, y)u(dy),U)μ(x)= ld u(x()u(du),U(2)u(x)= loa u(2)(x,y)u(dy).By (2.1) we can see thatrU(1)μ(x)≤Uμ(x)≤72 -1U(2)μ(x)Vx, Vμ.(3.3)Hence, using C(1)(B) and C(2)(B) to denote the capacities of B under X(1) and X(2), respec-tively, for any compact set B we have72C(2)(B)= 72sup{u(B): μ finite, supp(2)C B, U(2)μ≤1 on B}= sup{v(B): v finite, supp(v)C B, U(2)v≤γ2 on B}≤sup{v(B): v finite, supp(v)C B, Uv≤1, on B }=C(B)≤sup{v(B): v finite, supp(w)C B,U(1)v≤γ1, on B}= ηnC(1)(B).(3.4)Let u(0)(x, y), C(0)(B) denote the transition density and capacity of standard Brownian motion,respectively. Then it is easy to see thatu(2)(x,y) =立u(x,y)and henceC(2)(B)= sup{u(B): U(2)μ≤1 on B, μ finite, supp(u)C B}= sup{μ(B):六U(0)μ≤1 on B, μ finite， supp(u)C B}: Msup{v(B): U(O)v≤1 on B, v finite, supp(v)C B}=￥C(0)(B).(3.5)中国煤化工MHCNM HGNo.4Li & Liu: AN ESTIMATE ON THE DISTRIBUTION AND MOMENTS643Similarly, we have :C((B)= C(0)(B).It is well known that for the ball B(0,r)C(0)(B(0,r))=-2π告prd-2(3.7)(-1)(See, for example, [13]). Thus, by (3.4)- (3.6) and (3.7), we obtain (3.2).By (3.4)- (3.5) and (3.6), we can see that X has the same null capacity sets as Brownianmotion.Let a三3sup{p2(1,x): x∈R'} and β=六inf{p1(1,x): |x|≤2}. It is clear thata<∞andβ > 0. We have the following estimate for the distribution of Lp(0,r).Theorem 3.1 Let X be the lliptic difusion process in Rd (d≥3). Then for any x∈Rdwe havep (ao.n>)SaN1+n+r(告- 1)(tr-2)-世and when t(Ix| +r)-2≥2-2βM-1-dπfr(2-1) (1r-2)-*≤p° (La0.,)>).(3.9)Proof By (3.1)-(3.2) and the scaling property of Brownian motion, we haveP2 (Lg(0,r)>t)=P*(3s>t s.t. |Xs|≤r)= PPr(0,r>1(x)= lat,r,y)Pr(o.)(y)dy= lap(t,x,y) u(y),)p(0,r(dz)dy) B(0,r)=h(t,x,) / 。p(.y, z)dsu B0.n(d2)dy) B(0,r) Io= 1B(0,r)(dz) | p(s +t,x, z)ds .) B(0,r.≤2-2 (pn0.r)(d2)p2(u,x - z)duJ B(0,r)=γ2-[(HB(0,r)(dz)p2(1,u- t(x-z))u tdu≤aηz-'t- ++1C(B(0,))≤aMr1+insr(0-1) (r-)-号。Thus (3.8) is proved. .Similarly, we see thatif t(|x| +r)-2≥2-2, then .p" (Lg(o,)>t)= 1 HB(0.r(d2)/p(s + t,x, z)ds中国煤化工MHCNM HG644ACTA MATHEMATICA SCIENTIAVol.26 Ser.B≥η=1 1B(,r(dz) pr(u,x- z)du .=7-1_ μB(0,r)(dz) | pr(1,u +(x-z))u tdu≥βr-'t ++1C(B(0,r))2-n+r(-1)(tr-2)-4。This proves (3.9).As an application of Theorem 3.1, we can prove the following sufficient and necessarycondition for the existence of the k-moments of the last exit time for elliptic diffusion.Theorem 3.2 Let X be an elliptic diffusion process in Rd (d≥3), which is specified as .above. Then for any x∈Rd, k≥1, the following statements are equivalent:(1) 2(k+1)t)at=]* P" (re0.n>调)d+f p”(rn0on>调)d≤K1/ t(-出)dt +1<∞.2)= 3): Trivial.3)→1):∞>E° ([le0.r)= 。P" (IE0.r)>t)dt> x P* (LB0.,)> Vt)dt(x|+r)2k2-2k≥K2 |t(1-出)dtThis implies that卡(1-告)< -1, which is equivalent to 2(k:+ 1) < d.Remark 3.1 From Theorem 3.2, we can see that Er (L(0,r) <∞ifand only ifd> 4.So, when d = 3 or d = 4, elliptic diffusion X is transient but not strongly transient; whend≥5, X is strongly transient. This fact and Theorem 3.2 show that lliptic diffusion has someproperties that are similar to those of Brownian motion.References1 Aronson D G. Bounds for the fundamental solution of a parabolic equation. Bull Amer Soc, 1967, 73:2 Blumenthal R M, Getoor R K. Markov processes and potential theory. Academic Press, 1968中国煤化工MHCNM HGNo.4Li & Liu: AN ESTIMATE ON THE DISTRIBUTION AND MOMENTS6453 Chen Dianfa. The potential of Bessel processes and related problems. Acta Math Sinica, 1985, 28: 536- -5444 Chung K L. 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