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RANDOM WALKS IN RANDOM ENVIRONMENT: WHAT A SINGLE TRAJECTORY TELLS

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Niveau: Supérieur, Doctorat, Bac+8
RANDOM WALKS IN RANDOM ENVIRONMENT: WHAT A SINGLE TRAJECTORY TELLS OMER ADELMAN AND NATHANAEL ENRIQUEZ Abstract: We present a procedure that determines the law of a random walk in an iid random environment as a function of a single “typical” trajectory. We indicate when the trajectory characterizes the law of the environment, and we say how this law can be determined. We then show how independent trajectories having the distribution of the original walk can be generated as functions of the single observed trajectory. 1. Introduction Suppose you are given a “typical” trajectory of a random walk in an iid random environment. Can you say what the law of the environment is on the basis of the information supplied by this single trajectory? Can you determine the law of the walk? Such questions may arise if one intends to use the random environment model in applications. These questions are essentially pointless if the group is finite (in which case the environment at each of the finitely many sites that happen to be visited infinitely many times can of course be determined, but it is hard to say much more). So we assume that the group is infinite, and we go a little further: we assume that the (random) set of sites visited by the walk is almost surely infinite. (See remark 5.1.) Questions of this kind have been studied in the context of random walks in ran- dom scenery by Benjamini and Kesten [1], Lowe and Matzinger [3], and Matzinger [7].

  • random walk

  • oriented edges

  • random environment

  • walk can

  • pµ-almost surely

  • infinitely many

  • transition reinforced

  • times


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MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS COMPACT RIEMANNIAN MANIFOLDS ´ ˆ ´ JEROME VETOIS
Abstract.Let (M, g) be a smooth, compact Riemanniann-manifold, andhH¨olderbea continuous function onM prove the existence of multiple changing sign solutions for. We 22heΔg operator and theis ami equations likeΔgu+hu=|u|u re, w Laplace–Beltr the exponent 2= 2n/(n2) is critical from the Sobolev viewpoint.
1.1.Statement of the results
1.Introduction
ON
Let (M, g) be a smooth, compact Riemannian manifold of dimensionn3 andhbe a Ho¨ldercontinuousfunctiononM, namely a function which belongs toC0(M) for some real numberθin (0,1). We consider equations like
Δgu+hu=|u|22u ,(1.1) whereΔg=divgris the Laplace–Beltrami operator, and 2= 2n/(n2). IfH12(M) stands for the Sobolev space of all functions inL2(M) with one derivative inL2(M), then 2is the critical exponent for the embeddings ofH12(M) into Lebesgue spaces. We provide H12(M) with the scalar product hu, viH12(M)=ZMhru,rvigdvg+ΛZMuvdvg,(1.2) whereΛtyofniiuoctndlreHeo¨Thn.rotelagearnlesohcebottnatsnosipasotivicehprovides the regularity of weak solutions of equation (1.1). In case there holdsh4(nn21)Scalg, where Scalgis the scalar curvature of the manifold (M, g), equation (1.1) is the intensively studied Yamabe equation whose positive solutionsuare such that the scalar curvature of the conformal metricu22g this InSchoen [49], Trudinger [58], and Yamabe [59]).is constant (see Aubin [3], paper, we deal with multiplicity of solutions for equation (1.1) when the functionhis locally n2 less than4(n1)Scalgin Theorem 1.1, and globally less than4(nn2)1Scalgin Theorems 1.2 and 1.3. We define the energy of a solutionuof equation (1.1) to be the real numberE(u) given by E(u) =ZM|u|2dvg,(1.3) wheredvgis the volume element of the manifold (M, g). We say that an operator likeΔg+h is coercive onH12(M) if the energy associated to this operator controls theH12-norm. We Date: July 3, 2006. Published inInternational Journal of Mathematics18 1071–1111. 9,(2007), no. 1
MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS 2 letD1,2(Rn) be the homogeneous Sobolev space defined as the completion of the space of all smooth functions onRnwith compact support with respect to the scalar product hu, viD1,2(Rn)=ZRnhru,rvidx . We let alsoKnthe sharp constant for the embedding ofbe D1,2(Rn) intoL2(Rn), namely Kn=sn(n)42ω2n/n, whereωnis the volume of the unitn associate each solution of equation (1.1) with-sphere. We its opposite one, and call that a pair of solutions. We state our first result as follows.
Theorem 1.1.Let(M, g)be a smooth, compact Riemannian manifold of dimensionn4 andhcnitsuufniouoctnlderaH¨obeononMsuch that the operatorΔg+his coercive on H12(M). If there exists a pointx0inMsuch thath(x0)<4(nn21)Scalg(x0), then equation (1.1)admits at least(n+ 2)/2pairs of nontrivial solutions with energy less than2Knn. More precisely, we prove that either we do have infinitely many solutions of equation (1.1) or the (n+ 2)/we get in Theorem 1.1 have distinct energies.2 pairs of nontrivial solutions In the particular case where the manifold is locally conformally flat,n7, andhis aC1-function less than4(nn)12Scalg Inmanifold, the above result can be improved.on the whole such a setting, we establish two results. We first consider families of equations like Δgu+hu=|u|pα2u ,(1.4) where (pα)αis a sequence in [2,2] converging to 2. A sequence (uα)αis said to be a sequence of solutions for the family (1.4) if for anyα,uα weis a solution of equation (1.4). First, prove a compactness result for the family of equations (1.4) similar to the one proved by Devillanova–Solimini [17] in the case of smooth, bounded domains of the Euclidean space. Our compactness result is as follows.
Theorem 1.2.Let(M, g)be a smooth, compact, locally conformally flat Riemannian manifold of dimensionn7andhbe aC1-function onM. We let(pα)αbe a sequence in[2,2] converging to2, and we consider the family of equations(1.4) there holds. Ifh <4(nn2)1Scalg inM, then any bounded sequence inH12(M)of solutions for this family of equations remains bounded inC0(M).
An equivalent conclusion of Theorem 1.2 is that any bounded sequence inH12(M) of solu-tions for the family of equations (1.4) is compact inH12(M). In particular, such a sequence converges up to a subsequence inH12(M This) to a solution of the critical equation (1.1). easily follows from standard elliptic estimates (see, for instance, Gilbarg–Trudinger [28] Theo-rem 9.11) and the compactness of the embedding ofH2p(M) intoH12(M) for all real numbers p >2n/(n2). As a remark, note thatpα2is the only interesting difficult case for com-pactness since the embeddings ofH12(M) intoLp(M) are compact forp <2. Theorem 1.2 is the key argument in the proof of our last result which states as follows.
Theorem 1.3.Let(M, g)be a smooth, compact, locally conformally flat Riemannian manifold of dimensionn7andhbe aC1-function onM. If there holdsh <4(nn2)1ScalginM, then equation(1.1)infinitely many solutions with unbounded energies.admits
MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS
3
There are several situations where we do know that the solutions we get in Theorems 1.1 and 1.3 truly change sign. Such changing sign solutions are referred to as nodal solutions. Let us assume, for instance, that the Ricci curvature Ricgof the manifold (M, g) satisfies Ricg4n((nn)2)1λg(1.5) for some positive real numberλof bilinear forms, the inequality being strict, in the sense when the manifold is conformally diffeomorphic to the sphere. Then, as proved by Bidaut-Ve´ronV´eron[6],equation(1.1)withhλhas a unique positive solution, which turns out to beu=λ(n2)/4all but one pairs of solutions we get in particular, in such a situation, . In Theorem 1.1 are nodal. Concerning Theorem 1.3, it has been proved by Druet [21] that there is ana prioribound on the energy of positive solutions of equation (1.1) whenh <4(nn)12Scalg inM precisely, for any smooth, compact Riemannian manifold (. MoreM, g) of dimension n3, there exists a real numberE0such that ifuis a positive solution of equation (1.1), thenE(u)E0whereE(uas a direct consequence of the particular,  In) is as in (1.3). existence of thisa prioribound for positive solutions, Theorem 1.3 provides infinitely many nodal solutions for equation (1.1). Summarizing, the following corollary holds true.
Corollary 1.4.Let(M, g)be a smooth, compact Riemannian manifold of dimensionnandh be aC1-function onMsuch thath <4(nn2)1ScalginM. Ifn7and the manifold is locally conformally flat, then equation(1.1) Ifadmits infinitely many nodal solutions.n4, the manifold is arbitrary,hλfor someλ >0, and(1.5)holds true, the inequality being strict when the manifold is conformally diffeomorphic to the sphere, then equation(1.1)admits at leastn/2pairs of nodal solutions.
Compactness of positive solutions of equations like (1.1) have been intensively studied in recent years. Possible references on this topic, in the case of manifolds, are Druet [21, 22], Li– Zhang [40–42], Li–Zhu [43], Marques [45], and Schoen [50–52]. A survey reference on the sub-ject is Druet–Hebey [23]. We refer also to Hebey [31, 32] for compactness of positive solutions of critical elliptic systems in potential form and to Hebey–Robert–Wen [33] for compactness of positive solutions of critical fourth order equations. Compactness of changing sign solutions of equations like (1.1), in the case of smooth, bounded domains of the Euclidean space, have been studied in Devillanova–Solimini [17]. We follow this reference by Devillanova–Solimini [17] in several places in Section 3, as well as we follow the reference Clapp–Weth [15] in several places in Section 2. Possible other references on the existence of multiple nodal solutions for equations like (1.1) are Atkinson–Brezis–Peletier [2], Bahri–Lions [4], Capozzi–Fortunato–Palmieri [8], Castro–Cossio–Neuberger [9], Cerami–Fortunato–Struwe [10], Cerami–Solimini–Struwe [11], Devillanova–Solimini [18], Ding [19], Djadli–Jourdain [20], Fortunato–Jannelli, [26], Hebey– Vaugon [34], Holcman [35], Jourdain [36], Solimini [53], Tarantello [57], and Zhang [60]. Need-less to say, the above list does not pretend to exhaustivity. We refer also to the recent very nice paper by Ammann–Humbert [1] where the question of the existence of at least one changing sign solution to the Yamabe equation is addressed. A final remark in this introduction concerns the condition h <4(nn1)2Scalgin Theorem 1.2. Let (Sn,std) be the unitn holds Scal-sphere. Therestdn(n1), and the Yamabe equation on the unitn-sphere reads as
n n Δstdu+(42)u=u21.
(1.6)