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1、arXiv:0708.1671v1 math.PR 13 Aug 2007 The Annals of Probability 2007, Vol. 35, No. 4, 13331350 DOI: 10.1214/009117906000001204 c? Institute of Mathematical Statistics, 2007 HARNACK INEQUALITY AND APPLICATIONS FOR STOCHASTIC GENERALIZED POROUS MEDIA EQUATIONS1 By Feng-Yu Wang Beijing Normal Universit
2、y By using coupling and Girsanov transformations, the dimension- free Harnack inequality and the strong Feller property are proved for transition semigroups of solutions to a class of stochastic general- ized porous media equations. As applications, explicit upper bounds of the Lp-norm of the densit
3、y as well as hypercontractivity, ultra- contractivity and compactness of the corresponding semigroup are derived. 1. Introduction. The dimension-free Harnack inequality, fi rst introduced by the author in 19 for diff usions on Riemannian manifolds, has been ap- plied and extended intensively in the
4、study of fi nite- and infi nite-dimensional diff usion semigroups; see, for example, 16, 17, 20, 22 for applications to con- tractivity properties and functional inequalities, 1, 2, 11 for applications to short-time behaviors of infi nite-dimensional diff usions, and 7, 8 for ap- plications to the t
5、ransportation-cost inequality and heat kernel estimates. To establish the dimension-free Harnack inequality, the gradient estimate of the type |Ptf| eKtPt|f| has played a key role in the above men- tioned references, where the gradient is induced by the underlying diff usion coeffi cient. On the oth
6、er hand, however, in many cases the semigroup is not regular enough to satisfy this gradient estimate; indeed, this gradient esti- mate is equivalent to BakryEmerys curvature condition for a very general framework as in 5. To establish the dimension-free Harnack inequality on manifolds with unbounde
7、d below curvatures, a new approach is developed in the recent work 3 by using coupling and Girsanov transformations. Received October 2005; revised June 2006. 1Supported in part by National Natural Science Foundation of China Grant 10121101 and Rural Finance Development Program Grant 20040027009. AM
8、S 2000 subject classifi cations. Primary 60H15; secondary 76S05. Key words and phrases. Harnack inequality, stochastic generalized porous medium equation, ultracontractivity. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of
9、Probability, 2007, Vol. 35, No. 4, 13331350. This reprint diff ers from the original in pagination and typographic detail. 1 2F.-Y. WANG In this paper, we intend to study the transition semigroup for solutions to a class of stochastic generalized porous media equations, for which the semigroup is me
10、rely known to be Lipschitzian in the natural norm rather than in the intrinsic distance (cf. 6). So, we are not able to prove the Harnack inequality by using intrinsic gradient estimates. On the other hand, since the intrinsic distance is usually too big to be exponential integrable w.r.t. the under
11、lying reference measure, we prefer to establish a Harnack inequality depending only on the natural norm. Such a stronger inequality will provide more information including the strong Feller property and the ultracontractivity of the semigroup. To modify the argument in 3, we shall construct a new co
12、upling which only depends on the natural distance rather than the intrinsic one between the marginal processes (see Section 2 below). Strong solutions of the stochastic generalized porous medium equation have been studied intensively in recent years; see 6 for the existence, unique- ness and long-ti
13、me behavior of some stochastic generalized porous media equations with fi nite reference measures, see 12 for the stochastic porous media equation on Rd where the reference (Lebesgue) measure is infi nite; and see 18 for large deviation principles. Recently, a general result con- cerning existence a
14、nd uniqueness was presented in 15 for strong solutions of stochastic generalized porous media and fast diff usion equations. Let (E,M,m) be a separable probability space and (L,D(L) a negative defi nite self-adjoint linear operator on L2(m) having discrete spectrum. Let (0 )1 2 be all eigenvalues of
15、 L with unit eigenfunctions eii1. To state our equation, we fi rst introduce the state space of the solutions. Let H be the completion of L2(m) under the inner product hx,yiH:= X i=1 1 i hx,eiihy,eii, where h,i is the inner product in L2(m). It is well known that H is the dual space of the Sobolev s
16、pace H1:= D(L)1/2) and hence, is often denoted by H1in the literature. Let LHSdenote the space of all HilbertSchmidt operators from L2(m) to H. Let Wtbe the cylindrical Brownian motion on L2 (m) w.r.t. a complete fi ltered probability space (,Ft,P); that is, Wt:= Bi teii1for a sequence of independen
17、t one-dimensional Ft-Brownian motions Bi t. Let ,:0,) R R be progressively measurable and continuous in the second variable, and let Q:0,) LHS STOCHASTIC POROUS MEDIUM EQUATION3 be progressively measurable such that E Z T 0 kQtk2 LHSdt 0.(1.1) We consider the equation dXt= L(t,Xt)+(t,Xt)dt +QtdWt.(1
18、.2) In particular, if = 0,Q = 0 and (t,s):= |s|r1s for some r 1, then (1.2) reduces back to the classical porous medium equation (see, e.g., 4). In general, for a fi xed number r 1, we assume that there exist functions , C(0,) with 0 such that |(t,s)|+|(t,s)ts| t(1 +|s|r),s R, t 0, 2h(t,x) (t,y),y x
19、i2h(t,x) (t,y),L1(xy)i(1.3) 2 tkxyk r+1 r+1+ tkx yk 2 H, x,y Lr+1(m), t 0, where and in the sequel, kkpdenotes the norm in Lp(m) for p 1. A very simple example satisfying (1.3) is that (t,s) := |s|r1s and (t,s) := ts. By the fi rst inequality in (1.3), the fi rst term in the left-hand side of the se
20、cond inequality makes sense for any x,y Lr+1(m). Since L1is bounded in L2(m), if |(t,s)| t(1 + |s|(r+1)/2) for some positive C(0,), then the another term h(t,x) (t,y),L1(xy)i makes sense too. Oth- erwise, since the fi rst condition in (1.3) only implies |(t,s)| t(1 + |s|r), in general, to make the s
21、econd condition in (1.3) meaningful, we should and do assume that L1is bounded in Lr+1(m). In particular, this assumption holds automatically if L is a Dirichlet operator (cf., e.g., 14). Recall that an adapted continuous process Xtis called a solution to (1.2) if (cf. 6) E Z T 0 kXtkr+1 r+1dt 0, an
22、d for any f Lr+1(m), hXt,fiH= hX0,fiH Z t 0 m(f(s,Xs)+(s,Xs)L1f)ds + Z t 0 hQ(s,Xs)dWs,fiH,t 0. Due to (1.1), (1.3) and Theorems II.2.1 and II.2.2 in 13, for any X0 L2( H;F0,P) the equation (1.2) has a unique solution (cf. Theorem A.2 below). For any x H, let Xt(x) be the unique solution to (1.2) wi
23、th X0= x. Defi ne PtF(x) := EF(Xt(x),x H, 4F.-Y. WANG for any bounded measurable function F on H. We fi rst study Harnack inequalities for Pt. To this end, we assume that Qt() is nondegenerate for t 0 and ; that is, Qt()x = 0 implies x = 0. Let kxkQt:= ? kyk2,if y L2(m),Qty = x, ,otherwise. We call
24、k kQtthe intrinsic distance induced by Qt. Theorem 1.1.Assume (1.1) and (1.3). If there exists a nonnegative constant r 3 such that kxkr+1 r+1 2 tkxk 2+ Qt kxkr1 H ,x Lr+1(m), t 0,(1.4) holds on for some strictly positive function C(0,), then for any t 0, Ptis strong Feller and for any positive boun
25、ded measurable function F on H, any 1 and any x,y H, (PtF)(y) (PtF(x)exp ?c(,t)kx yk2(3r+)/(2+) H ( 1) ? ,(1.5) where c(,t) := ? 2(4+ )(6+2)/(2+) ?Z t 0 2 s 2 sexp ? 3r + 4 + Z s 0 udu ? ds ?/(2+)? ? (3r +)(6+2)/(2+) ?Z t 0 ssexp ? 3 r + 4+ Z s 0 udu ? ds ?2?1 . Unlike known Harnack inequalities est
26、ablished in 1, 2, 11 where the in- volved distance is almost surely infi nite, (1.5) only includes the usual norm on the state space H. This enables one to derive stronger regularity proper- ties of the semigroup, such as the strong Feller property of Ptand estimates of its transition density pt(x,y
27、). Moreover, as was done in 16, 19, 20, this inequality can also be applied to derive the hypercontractivity and ultra- contractivity of the semigroup (cf. Theorem 1.2 below). To apply Theorem 1.1 to contractivity properties of Pt, we consider the following time-homogenous case. Theorem 1.2.Assume (
28、1.1), (1.3) and (1.4) for some nonnegative con- stant r 3. Furthermore, let , and Q be deterministic and time-free such that , 0 and are constant with 1r=1 21. STOCHASTIC POROUS MEDIUM EQUATION5 (1) The Markov semigroup Pthas an invariant probability measure with full support on H and (e0kk r+1 H +k
29、kr+1 r+1) 0. If in addition 0, then the invariant probability measure is unique. (2) For any x H, any t 0 and any 1, the transition density pt(x,y) of Pt w.r.t. satisfi es kpt(x,)kLp() ?Z H exp ? (c()kx yk2(3r+)/(2+) H )(1.6) ? 1 ? 1 exp ? 3r + 4+ t ?(4+)/(2+)?1? (dy) ?(1)/ , where c() := 2(4+)/(3r
30、+)()4/(4+)and when = 0, the right-hand side means its limit as 0. (3) If r = 1, then Ptis hyperbounded (i.e., kPtkL2()L4() 0. If moreover 0, then Ptis hypercon- tractive, that is, kPtkL2()L4() 1 for large t 0. (4) If r 1, then Ptis ultracontractive and compact on L2() for any t 0. More precisely, th
31、ere exists c 0 such that kPtkL2()L() expc(1+t(1+r)/(r1),t 0.(1.7) To apply Theorems 1.1 and 1.2, one has to verify condition (1.4). To this end, we present below some simple suffi cient conditions for (1.4) to hold. Corollary 1.3.Let Qei:= qieifor i 1 with P i=1 q2 i i 0, then (1.4) holds for any no
32、nnegative constant (r 3,r 1 and a constant function 0. Consequently, if moreover and are deterministic and time-free such that (1.3) holds with 1r=1 12, then all assertions in Theorems 1.1 and 1.2 hold for (r 3,r 1 0,). Proof.Simply note that kk2 r+1 kk22 1 infiq2 i kk2 Q. ? Remark 1.1.In Corollary
33、1.3 there are two conditions on qi, where P i1 q2 i i means that q2 i should be small enough as i but the other says that the sequence should be at least uniformly positive. In partic- ular, such sequence exists if the spectrum of L is discrete enough such that P i1 1 i 0 the Hausdorff 6F.-Y. WANG d
34、imension of the fractal in the eff ective resistance metric (see 10). In the fi rst case it is well known that i ci2for some c 0 and all i 1, while according to Theorem 2.11, for the second case one has i ci(s+1)/sfor some c 0 and all i 1. See Section 3 below for more examples of L in an abstract fr
35、amework including high-order elliptic diff erential operators on Rd. Complete proofs of the above two theorems will be presented in Section 2. Assertions in Theorem 1.2 are direct consequences of Theorem 1.1 as soon as the desired concentration of is confi rmed. To prove the fi rst theorem, we adopt
36、 the coupling method and Girsanov transformations as in 3. Com- paring to the argument developed in 19, this method enables one to avoid verifying (intrinsic) gradient estimates of the semigroup. In Section 3, concrete suffi cient conditions for Corollary 1.3 to hold are provided for a large class o
37、f linear operators L in a rather abstract frame- work. Finally, in the Appendix we confi rm the existence and uniqueness of the solution to (1.2) as well as the existence and uniqueness of our coupling constructed below cf. (2.2). 2. Proofs of Theorems 1.1 and 1.2. 2.1. The main idea. To make the pr
38、oofs easy to follow, let us fi rst briefl y explain the main idea to obtain a Harnack inequality using coupling. Let x 6= y be two fi xed points in H, and let T 0 be a fi xed time. Let Xt(x) and Xt(y) be the solutions to (1.2) with initial data x and y, respectively. If (x,y) := inft 0:Xt(x) = Xt(y)
39、 Ta.s.,(2.1) then by the uniqueness of the solution, we have XT(x) = XT(y) a.s. Thus, for any nonnegative measurable function F on H, PTf(x) := EF(XT(x) = EF(XT(y) = PTF(y). This is much more than the Harnack inequality we wanted. Of course, in general (2.1) is wrong since it is so strong that PTmap
40、s any bounded func- tion to constant. What we can hope is that (x,y) T happens in a high probability (for x and y close enough). This is, however, not suffi cient to imply the Harnack inequality. To ensure that (x,y) T happens in probability 1, we shall add a strong enough drift term which forces Xt
41、(y) to move to Xt(x). To this end, let us take a constant (0,1) and a reference function C(0,);R+), and consider the modifi ed equation dYt= ? L(t,Yt)+(t,Yt)+ t(XtYt) kXtYtk H 1t 0 and nonnegative function tsuch that: (i) T a.s. (ii) EexpR T 0 2 t 2 kXtYtk2 H kXtYtk2 Qtdt . Let t:= tQ1 t (Xt Yt) kXt
42、 Ytk H 1t. Once (i) and (ii) are confi rmed, we may rewrite (2.2) as dYt= (L(t,Yt)+(t,Yt)dt + QtdWt,Y0= y, where Wt:= Wt+ Z t 0 sds,t 0,T. By (ii) and Girsanovs theorem, it is easy to see that Wtt0,Tis a cylin- drical Brownian motion on L2(m) under the weighted probability measure RP, where R := exp
43、 ?Z T 0 hdWt,ti 1 2 Z T 0 ktk2 2dt ? . Thus, by the uniqueness of the solution, the distribution of Ytt0,Tunder RP coincides with that of Xt(y)t0,Tunder P. Therefore, combining this with (i) we arrive at PTF(y) = ERF(YT) = ERF(XT) (ER/(1)(1)/(EF(XT)1/(2.3) = (ER/(1)(1)/(PTF(x)1/. Then the desired Ha
44、rnack inequality follows by estimating the moments of R. 2.2. Proofs. We fi rst study (i). By (1.3) and the It o formula due to 13, Theorem I.3.2, we have dkXt Ytk2 H (2 tkXtYtk r+1 r+1+tkXt Ytk 2 H tkXtYtk 2 H )dt,t T. 8F.-Y. WANG Then dkXtYtk2 He Rt 0 sds (2.4) (2 tkXtYtk r+1 r+1+tkXt Ytk 2 H )e R
45、t 0 sds dt,t T. Lemma 2.1. If satisfi es Z T 0 exp ? 2 Z t 0 sds ? tdt 2 kxyk H, (2.5) then XT= YT. Proof.By (2.4), 2 dkXt Ytk2 He Rt 0 sds/2 te/2 Rt 0 sds dt,t . If T , then it follows from this and (2.5) that kXTYTk2 He RT 0 sds/2 kx yk H 2 Z T 0 te/2 Rt 0 sds dt kx yk H. This implies XT= YTand he
46、nce, is contradictory to T r 3, we obtain dkXtYtk2 He Rt 0 sds 2 tkXtYtk 2(1) H e Rt 0 sdskXt Ytkr+1 r+1dt (2.6) 2 t 2 tkXt Ytk 2+ Qt e Rt 0 sdskXt Ytk2(1)+r1 H dt = 2 t 2 te Rt 0 sds kXtYtk2+ Qt kXtYtk(2+) H dt. Let 2 t := c22 t 2 te Rt 0 sds, c := 2kx yk H RT 0 ttexp Rt 0sds .(2.7) Then (2.5) hold
47、s so that XT= YTaccording to Lemma 2.1. So, (2.6) implies c2 Z T 0 2 tkXt Ytk 2+ Qt kXtYtk(2+) H dt kxyk2 H. STOCHASTIC POROUS MEDIUM EQUATION9 By this and the H older inequality, Z T 0 2 tkXtYtk2Qt kXtYtk2 H dt ?Z T 0 2 tkXt Ytk 2+ Qt kXtYtk(2+) H dt ?2/(2+) ?Z T 0 2 t dt ?/(2+) (2.8) (1c2kxyk2 H)
48、2/(2+) ?Z T 0 2 t dt ?/(2+) . This implies, for := /( 1), that ER = Eexp ? Z T 0 hdWt,ti 2 Z T 0 ktk2 2dt ? = Eexp ?( 1) 2 Z T 0 ktk2 2dt ? (2.9) exp ?( 1) 2 (1c2kxyk2 H) 2/(2+) ?Z T 0 2 t dt ?/(2+)? . Combining (2.9) with (2.3), we arrive at (PTF(y) (PTF)(x)exp ? 2( 1)( 1c2kx yk2 H) 2/(2+) ?Z T 0 2
49、 t dt ?/(2+)? . Taking (2.7) into account, we obtain (1.5). We now prove the strong Feller property. Since PTF(y) = ERF(YT) = ERF(XT), we have |PTF(y)PTF(x)| = |E(R 1)F(XT)| kFkE|R 1|.(2.10) From (2.9) we know that R is uniformly integrable for bounded kx ykH. Therefore, by (2.8) and the dominated convergence theorem we obtain lim yxE|R 1| = E limyx|R 1| = 0. Combining this with (2.10) we see that PTF Cb(H). Thus,