Wiener s test for superbrownian motion and the Brownian snake


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1 Probab. Theory Relat. Fields 18, (1997 Wiener s test for superbrownian motion and the Brownian snake JeanStephane Dhersin and JeanFrancois Le Gall Laboratoire de Probabilites, Universite Paris 6, 4 Place Jussieu, F Paris Cedex 5, France ( Received: 5 January 1996 / In revised form: 3 October 1996 Summary. We prove a Wienertype criterion for superbrownian motion and the Brownian snake. If F is a Borel subset of R d and x R d, we provide a necessary and sucient condition for superbrownian motion started at x to immediately hit the set F. Equivalently, this condition is necessary and sucient for the hitting time of F by the Brownian snake with initial point x to be. A key ingredient of the proof is an estimate showing that the hitting probability of F is comparable, up to multiplicative constants, to the relevant capacity of F. This estimate, which is of independent interest, renes previous results due to Perkins and Dynkin. An important role is played by additive functionals of the Brownian snake, which are investigated here via the potential theory of symmetric Markov processes. As a direct application of our probabilistic results, we obtain a necessary and sucient condition for the existence in a domain D of a positive solution of the equation u = u 2 which explodes at a given point Mathematics Subject Classication (1991: Primary, 6J45, 6G57; Secondary, 6J55, 6J5, 35J65 1 Introduction This work is concerned with regularity problems for superbrownian motion and the Brownian snake. These problems are closely related to the characterization of polar sets for superbrownian motion. A sucient condition for nonpolarity, in terms of a suitable capacity, was rst given by Perkins [23], and later Dynkin [8] proved the necessity of this condition, by using analytic results of Baras and Pierre [2] concerning removable singularities of semilinear partial dierential equations. In the present work, we rene the Dynkin Perkins result by showing that the hitting probability of a set is comparable, up to
2 14 J.S. Dhersin, J.F. Le Gall multiplicative constants, to its capacity. We then apply these estimates to an analogue of the classical Wiener criterion for superbrownian motion. In the same way as for usual Brownian motion, this Wiener criterion has an interesting analytic counterpart. Namely, if D is a domain in R d and we obtain a necessary and sucient condition for the existence of a positive solution of the partial dierential equation u = u 2 in D that tends to at x. The existence of such solutions has been investigated in a more general setting by Marcus and Veron [18, 19]. However, in the special case of equation u = u 2, our results give much more precise information than the analytic methods. We use the pathvalued process called the Brownian snake as a basic tool. Let us briey describe this process, referring to [14, 15] for more detailed information. The Brownian snake is a Markov process taking values in the set of all stopped paths in R d. By denition, a stopped path in R d is a continuous mapping w :[;] R d. The number = w = is called the lifetime of the path. We sometimes write ŵ = w( w for the tip of the path w. Wedenote by W the set of all stopped paths in R d. This set is equipped with the distance d(w; w = + sup t = w(t w (t. Let us x x R d and denote by W x the set of all stopped paths with initial point w( = x. The Brownian snake with initial point x is the continuous strong Markov process W =(W s ;s= in W x whose law is characterized as follows. 1. If s denotes the lifetime of W s, the process ( s ;s= is a reecting Brownian motion in R Conditionally on ( s ;s=, the process W remains a (timeinhomogeneous Markov process. Its conditional transition kernels are described by the following properties: For s s, W s (t=w s (t for every t 5 m(s; s := inf [s; s ] r ; (W s (m(s; s +t W s (m(s; s ; 5t5 s m(s; s is a standard Brownian motion in R d independent of W s. Informally, one should think of W s as a Brownian path in R d with a random lifetime s evolving like (reecting linear Brownian motion. When s decreases, the path W s is erased from its tip. When s increases, the path W s is extended (independently of the past by adding small pieces of Brownian motion at its tip. We may and will assume that the process (W s ;s= is the canonical process on the space := C(R + ; W of all continuous functions from R + into W: The canonical ltration on is denoted by (Fs s =. For w W, we denote by P w the law of the Brownian snake started at w. We denote by x the trivial path in W x with lifetime and for simplicity we write P x instead of P x. It is immediate that x is a regular recurrent point for the Markov process W. We denote by N x the associated excursion measure. Then N x is a nite measure on the set of all! for which there exists a number = (! = such that s (! if and only if s. The law of W under N x is described by properties analogous to 1 and 2, with the only dierence that the law of reecting Brownian motion in 1 is replaced by the Itô measure
3 Wiener s test for superbrownian motion 15 of positive excursions of linear Brownian motion (see [15]. We can normalize N x so that, for every, N x (sup s = s =(2 1. The following scaling property of N x is often useful. For, dene W ( =(W s ( ;s= by W s ( (t=x+(w 4 s( 2 t x; 5t5 W ( = 2 s 4 s: Then the law of W ( under N x is 2 N x. As a consequence, one can check that, for every, ( N x sup Wˆ s x = = c(d 2 ; s= with a constant c(d depending only on d. We also introduce the random sets R = { ˆ W s ;s=}; R ={ˆ W s ;s=; s }; that will be of interest under the excursion measure N x. Notice that R = R {x}, N x a.e. Let us now explain the connection between the Brownian snake and super Brownian motion. Denote by (L t s(;s=;t= the jointly continuous collection of local times of the process ( s ;s=. Notice that (L s (;s= is also the local time of W at x. Weset 1 := inf {s = ;L s( 1} and, for every t =, we let X t be the random measure on R d dened by X t ; = 1 d s L t s( ( ˆ W s : Then (X t ;t= is under P x a superbrownian motion in R d started at x (see [14]. The associated historical process is the process (X t ;t= dened by 1 X t ; = d s L t s((w s : Let R X denote the range of X, dened as in [4]: R X = ( supp X t ; t= where supp X t is the topological support of X t, and A denotes the closure of a subset A of R d. Then, it is easily checked (cf. [14], proof of Proposition 2.2 that R X = { Wˆ s ; 5 s 5 1 ; s }: (1 A Borel subset F of R d is called Rpolar if P x (R X F = for some (or equivalently for every x R d. From (1 and excursion theory, we have P x (R X F =1 exp N x (R F ; (2 and therefore F is Rpolar if and only if N x (R F = for x R d. For every =, let cap (F be the capacity dened by cap (F = (inf ( (dy(dy (y; y 1 ;
4 16 J.S. Dhersin, J.F. Le Gall where the inmum is taken over all probability measures supported on F, and log + (y; y := y y if =; y y if : Perkins and Dynkin proved that there are no nonempty Rpolar sets when d 5 3 and, when d = 4, a set F is Rpolar if and only if cap d 4 (F=. We denote by B(y; r the open ball of radius r centered at y, and by B(y; r the closed ball. Theorem 1. Suppose that d = 4. There exist two positive constants ; such that; if F is a Borel set contained in the ball B(; 1=2; and if x = 1; then x 2 cap (F 1 + (log x cap (F 5 N x(r F 5 x 2 cap (F 1 + (log x cap (F if d =4; x 2 d cap d 4 (F 5 N x (R F 5 x 2 d cap d 4 (F; if d = 5: The same bounds hold with P x (R X F instead of N x (R F. Remarks. (i The last assertion is an immediate consequence of the rst part of the theorem and (2. (ii When d 5 3, since points are not Rpolar, a simple scaling argument shows that, under the assumptions of Theorem 1, if F, x 2 5 N x (R F 5 x 2 : (iii Bounds analogous to Theorem 1 can also be given for the probability that the ranges of p independent superbrownian motions have a common point in the set F. For simplicity, consider the case d = 5. Let k = 1 and let Y 1 ;:::;Y k be k independent superbrownian motions started respectively at x1 ;:::; xk, where 1 5 x j 5 2. There exist two positive constants k and k such that, if F is a Borel subset of B(; 1=2, then k cap k(d 4 (F 5 P(R Y 1 R Yk F 5 k cap k(d 4 (F; (3 where R Y j denotes the range of Y j. These bounds follow from Theorem 1 by using a technique of Peres [22]: By Theorem 1 and Corollary 4.3(i of [22], each set R Y j is intersectionequivalent in B(; 1=2 (in the sense of [22, Denition 1] to a percolation set Q d (2 (d 4 constructed as in [22], with the minor modication that one starts from the cube [ 1=2; 1=2] d rather than from [; 1] d. From Lemmas 2.2 and 2.3 of [22], one gets that R Y 1 R Yk is intersection equivalent in B(; 1=2 to Q d (2 k(d 4, and the bounds (3 then follow from Corollary 4.3(i of [22]. As was observed by Yuval Peres (personal communication, a qualitative version of the previous argument, using only the Dynkin Perkins characterization of polar sets, already gives the weaker fact that the probability considered in (3 is strictly positive if and only if cap k(d 4 (F.
5 Wiener s test for superbrownian motion 17 From (3, or from its weak version, one can derive the converse of a theorem of Perkins ([23, Theorem 5.9]. According to this theorem, the condition cap k(d 4 (F is sucient for the set F to contain kmultiple points of superbrownian motion with positive probability. The necessity of the condition follows from (3, using also Lemma 16 of Serlet [25] which allows us to derive information on kmultiple points of a single superbrownian motion from intersections of k independent superprocesses. As a special case, we recover the known fact that there are no kmultiple points when k(d 4 = d. The lower and upper bounds of Theorem 1 are proved in a very dierent way. The proof of the lower bound uses the familiar idea of constructing an additive functional of the Brownian snake that will be nonzero only if the process hits F. In Sect. 2, we develop a general theory of additive functionals of the Brownian snake. In view of forthcoming applications, we have presented this theory in more generality than was really needed here. Additive functionals of the Brownian snake play an important role in the probabilistic representation of solutions of u = u 2 in a domain (see [17]. Our construction of additive functionals should also be compared with the results of Dynkin and Kuznetsov [1], who introduce more general functionals of Markov snakes. When the relevant properties of additive functionals have been established, the proof of the lower bound of Theorem 1 becomes easy, and is presented in Sect. 3. The proof of the upper bound of Theorem 1 is given in Sect. 4. It relies on analytic estimates inspired from Baras and Pierre [2], along the lines of [16], Sect. 3. These analytic ingredients are here supplemented by a suitable application of Itô s formula. Let us now turn to the Wiener criterion. We set F := inf {t ; supp X t F }: Let (Gt t = denote the natural ltration of X, and let (G t t = be its augmentation with the class of all P x negligible sets of (; F. As X is Markov with respect to (Gt+, standard arguments show that the ltration (G t t = is rightcontinuous. In particular G + = G is P x trivial. We show in Sect. 5 that F is a stopping time of the ltration (G t. Thus, P x ( F == or 1. We say that x is superregular for F if P x ( F ==1. Let F sr denote the set of all points that are superregular for F. From the known path properties of superbrownian motion (see [4, 23] it is obvious that int(f F sr F, where int(f denotes the interior of F. We also set T F = inf {s ; Wˆ s F; s }. The following proposition shows that superregularity can be characterized from the behavior of the Brownian snake in dierent ways. Proposition 2. Let F be a Borel subset of R d. The following properties are equivalent: (i x is superregular for F; (ii P x (T F ==1; (iii N x (R F =.
6 18 J.S. Dhersin, J.F. Le Gall The next theorem, which is analogous to the classical Wiener criterion (see e.g. [24], provides a characterization of the set of superregular points. Theorem 3. Let F be a Borel subset of R d and; for every x R d ;n=1;let F n (x={y F; 2 n 5 y x 2 n+1 }: If d 5 3; F sr = F. Ifd=4;x F sr if and only if 2 n(d 2 cap d 4 (F n (x = : (4 n=1 The case d 5 3 of Theorem 3 is very easy: If x F, then we may nd a sequence (x n in F converging to x. By the scaling properties of N x, we have N x (R F = N x (x n R =c x x n 2, where c because points are not Rpolar in dimension d 5 3. The desired result then follows from Proposition 2. The proofs of Proposition 2 and Theorem 3 (when d = 4 are given in Sect. 5. The necessity of the condition (4 is easy from the upper bound in Theorem 1 and the characterization of superregularity given by Proposition 2(iii. The key idea of the suciency part is to introduce M = sup{ s ;s=} and to observe that N x (R F =N x (T F = N x (T F ; 2 2n 5 M 2 2n+2 : n=1 Using a renement of the lower bound of Theorem 1, one can check that each term N x (T F ; 2 2n 5 M 2 2n+2 is bounded below by a constant times the corresponding term in the series (4. The classical Wiener criterion was originally formulated in analytic terms [27]. Similarly, one can give an analytic formulation of Theorem 3. The proof of the next theorem, which is also given in Sect. 5, is easy from the connections between superbrownian motion or the Brownian snake and the partial dierential equation u = u 2 (see [8] or [15]. Theorem 4. Let D be a domain in R d such that Set F = R d \D; and let F n (x be as in the previous theorem. The following two conditions are equivalent. (i There exists a nonnegative solution of u = u 2 in D such that lim u(y= : D y x (ii Either d 5 3; or d = 4 and (4 holds. Notice that every nonnegative solution of u = u 2 is bounded above by the maximal solution, which has a simple probabilistic expression (see e.g. [15, Sect. 5]. Therefore, if d 5 3, or if d = 4 and (4 holds at every the maximal solution tends to at every The existence of positive solutions to the more general equation u = u p with innite boundary conditions has been recently investigated by Marcus and Veron ([18, Theorem 2.3] see also [26, Sect. 4.2]. Marcus and Veron show that such solutions exist under an exterior cone condition on D, which can
7 Wiener s test for superbrownian motion 19 be weakened to an exterior segment condition when 1 p (d 1=(d 3. Clearly, Theorem 4 gives a much more precise result in the special case p =2. In Sect. 6, as an application of Theorem 3, we treat the special case when D is the complement of a thorn with vertex at x. The superregularity of x is then characterized by an integral test analogous to the classical result for usual Brownian motion. 2 Additive functionals of the Brownian snake Our goal in this section is to construct a general class of additive functionals of the Brownian snake and to investigate their properties. Clearly our results are related to the theory of additive functionals for general Markov processes that has been developed by dierent authors under various assumptions (see, in particular, [3, 7, 12]. However, as we were unable to derive our results from a known theory, and also because we will need several specic facts, we will present a short selfcontained construction in the special case of the Brownian snake. We follow ideas from Dynkin [7] and rely on some results of Fitzsimmons and Getoor [11]. Throughout this section, x is a xed point of R d. For w W x, the law of the Brownian snake started at w and killed when it hits the trivial path x is denoted by Pw. We may consider (W s ; Pw as a Markov process in W x, with transition kernels Qs, and the trivial path x plays the role of a cemetery point for this Markov process. The set of nontrivial paths in W x is denoted as W Let U(w; dw denote the potential kernel of the killed process. According to [16], we have for w Wx U(w; dw w =2 da db R a; b (w; dw ; a where the kernel R a; b (w; dw can be dened as follows. Under R a; b (w; dw, the path w coincides with w on [;a], and is then distributed as a Brownian path started from w(a at time a, and stopped at time b. Let Px b (dw= R ;b (x;dw be the law of a Brownian path started at x and stopped at time b. Then, the process (W s ; Pw is symmetric with respect to the nite measure M x (dw= db Px b (dw: The measure M x and the excursion measure N x are linked by the formula ( N x (W s ds = M x ( ; for any nonnegative measurable function on W x. It is easy to verify that the process (W s ; Pw satises all assumptions of Fitzsimmons and Getoor [11]. Let F be a Borel subset of Wx and let H F = inf {s = ;W s F}. Then F is called M x polar if M x (dw a:e: Pw(H F =: x.
8 11 J.S. Dhersin, J.F. Le Gall A property is said to hold quasieverywhere (q.e. if it holds outside an M x  polar set. Let be a nite measure on Wx. It is straightforward to check that U is then nite (in fact U({w; w 5 b} for every b. When U is absolutely continuous with respect to M x, we may choose (see [11] and the references therein a Borel measurable excessive version U( ofthe Radon Nikodym derivative of U with respect to M x. We say that has nite energy if in addition E( := ; U( : This condition and the quantity E( do not depend on the choice of U( (see [11]. Recall that (Fs ;s= is the canonical ltration on. For every s =, the shift operator s on is dened by W r ( s!=w s+r (!. Finally, we extend the denition given in the introduction by setting (! = inf {s ; s (!=}, for every!. Denition. An (Ft adapted increasing process A on ; such that A =; is called an additive functional of the killed Brownian snake if the following properties hold. There exist a Borel subset N of Wx and a Borel subset of such that: (a N is M x polar, and for every w Wx \N; Pw(H N =; (b for every s = ; s ( ; \N; Pw(=1; (d for every! ; the function s A s (! is continuous, constant over [; and such that, for every r; s = ; A r+s (!=A r (!+A s ( r!. (c N x (\= and, for every w W x Theorem 5. Let be a nite measure on Wx with nite energy. There exists an additive functional A of the killed Brownian snake such that the following properties hold: (i E w(a =U((w; q.e.; (ii for every nonnegative Borel function f on W x ; N x ( f(w s da s = ; f ; (iii N x (A 2 =2E(. Moreover; if denotes the timereversal operator on (i.e. W s (!=W ( s+ (!; then A s = A A ( s+ for every s = ; N x a:e: Proof. Denote by B(E the Borel eld on a metric space E. Because E(, we may apply Proposition 3.7 of [11] to construct an M x exit law associated with U(. Precisely, there exists a family =( s s of nonnegative functions on Wx such that the mapping (s; x s (x is
9 Wiener s test for superbrownian motion 111 B(]; [ B(Wx measurable, for every s; s, Qs s = s+s, M x a.e., and U(= s ds; M x a:e: (5 Furthermore, s L 2 (M x for every s and ( M x 2 s ds = 1 E( : (6 2 Finally, we have also Qs = s M x for every s ([11, Theorem (3.16]. Notice that, in [11], the mapping (s; x s (x isb(]; [ E e measurable, where E e denotes the eld generated by excessive functions. Here however, as the excessive function U( is Borel measurable, the same argument as in [11] shows that this mapping can be taken to be B(]; [ B(Wx  measurable. For every, we set A s s = (W r dr: We will construct the additive functional A as a suitable limit of A when goes to. It will be convenient to use the Kuznetsov measure K x associated with M x. Let us briey describe the construction of K x (this construction is a trivial extension of the wellknown case of linear Brownian motion killed at. Let := C(R;W be the canonical space of all continuous functions from R into W, and denote again by W the canonical process on. For! and s R, let s! be dened by W r ( s!=w s+r (!, for every r =. The measure K x is characterized by the following two properties: (a K x (d! a.e., there exist (! (! such that W s (! x if and only if (! s (!; (b the law under K x (d! of the pair ((!; (!!is N x, where denotes Lebesgue measure on R. For every xed s R, the law of W s under K x ( {W s x}ism x. Furthermore, on the set {W s x}, the conditional distribution of (W s+r ;r= knowing (W u ;u5sispw s. For! and r s, weset A s [r; s](!= (W u (! du: r Lemma 6. For r s ; the limit A [r; s] = lim A [r; s] exists in L 2 (K x ; and ( s r K x ((A [r; s] 2 =2M x du (s r u 2 u=2 : (7
10 112 J.S. Dhersin, J.F. Le Gall Proof. We evaluate ( s K x (A [r; s]a [r; s]=k x r s = r s =2 r =2 s du dv (W u (W v r s du dv M x ( Q v u + s u r s r s du dv M x ( 2 (v u++ =2 u du (s r u M x ( 2 (u++ =2: s dv du M x ( Qu v v We used the symmetry of the operators Qs on L 2 (M x together with the equality Qs r = r+s. From (6, it follows that ( s r lim K x(a ; [r; s]a [r; s]=2m x du (s r u 2 u=2 ; which completes the proof. Lemma 7. There exists K x a.e. a nite measure A(ds on R; supported on ](!;(![ and without atoms, such that; for every r s; A [r; s] = s A(du; r K x a.e. Proof. Let m, n be two positive integers. By (7, we have ( 2 n m K x (A [(i 12 n ;i2 n ] =2m 2 2 n dv M x ( 2 v=22 n (2 n v; i= 2 n m+1 and by (6, this quantity converges to as n tends to, for any xed m. We conclude that, for every m = 1, sup A [(i 12 n ;i2 n ] 2 n m i 5 2 n m decreases to as n,k x a.e. For any real number s of the type s = i2 n, set A s = A [;s] if s =, A s = A [s; ] if s. The previous observation implies that K x a.e. the function s A s can be extended to a continuous nondecreasing function from R into R. By construction, the function A is constant over ] ;] and over [; [. It is then easy to check that the measure A(ds has the desired properties. We now choose a sequence ( p decreasing to in such a way that the convergence of Lemma 6 holds K x a.e. for all rational r s along the subsequence ( p. We set for every! and every s =, A s (! = lim inf p Ap s (!;
11 Wiener s test for superbrownian motion 113 so that the process (A s ;s= is clearly adapted with respect to the ltration (Fs. We let be the Borel subset of all! such that the functions A p s (! are nite and converge to A s (! uniformly on R +. For!, the function s A s (! is continuous, nondecreasing and constant over [; [. It is also clear that s (, and the additivity property A r+s (!=A r (!+A s ( r! holds for every!. From Lemma 7 and the choice of the sequence ( p, we see that K x a.e. A p [r; s] converges to A [r; s] uniformly on the set {(r; s R 2 ;r5s}. It follows that N x ( c = and PM x ( c =. We set N := {w Wx ; Pw( 1} Then M x (N =. Let w W x \N, and assume that Pw(H N. By the section theorem (see [5, p. 219], we may nd a stopping time T of the ltration (Fs such that Pw(T and W T N on the set {T }. By the strong Markov property, the conditional distribution of T! knowing FT is PW T. By the denition of N, we get that Pw(T ; T!. On the other hand, since w N, we know that, on {T }, A p s ( T!=A p T+s (! Ap T (! converges Pw a.s. uniformly on R +. We thus get a contradiction, which proves that Pw(H N = when w N. In particular, N is M x polar. At this point, we have checked that A satises all properties of the denition of an additive functional. Let us turn to the proof of (i. As a consequence of Lemma 6, we have A s = lim p Ap s ; in L 2 (PM x. Hence, M x (dw a.e., Ew(A s = lim p E w(a p s = lim p p+s = lim u (w du = p p s s Q r p (w dr u (w du: It follows that, M x a.e., Ew(A = u (wdu = U((w; using (5. We can then replace M x a.e. by q.e. To this end, note that by the property (a of the denition of an additive functional, we may restrict the state space of the killed Brownian snake to W x \N. The additivity property (d then shows that the function w E w(a is excessive. We have just checked that this function coincides with the excessive function U(, M x a.e. However two excessive functions that are equal M x a.e. must coincide q.e.
12 114 J.S. Dhersin, J.F. Le Gall Let us now check (iii. We observe that the convergence of A p towards A holds in L 2 (N x. Indeed ( N x (A A =2N x dr (W r ds (W s r ( =2N x dr (W r U (W r =2M x ( U =2 ds M x (dw (w Qs (w =2 ds M x ( 2 (s++ =2; which increases as ; to 2 ds M x ( 2 s=2 =2E(. Formula (iii follows. It remains to prove (ii. We may assume that f is bounded and continuous and that f(w= if w 5, for some xed. Clearly, the event { f(w s da s } has nite N x measure. Since A is bounded in L 2 (N x, we have N x ( ( f(w s da s = lim N x f(w s da p s p ( = lim N x f(w s p (W s ds p = lim Mx (dw f(w p (w p = lim p Q p ;f = lim p ; Q p f = ; f : In the fourth equality, we used the identity Qs = s M x. The last assertion in Theorem 5 is immediate from our construction since it is obvious that A s = A A ( s +, for every s =, N x a.e. For! and s =, we dene ˆ s! byw r (ˆ s!=w (s r+ (!. The following technical lemma will be useful in Sect. 3. Lemma 8. Let and A be as in Theorem 5, and let ; be two nonnegative Borel functions on. Then; ( N x s ˆ s da s = (dw Ew( Ew( :
13 Wiener s test for superbrownian motion 115 Proof. As a consequence of the strong Markov property under N x (see [15], we have for any progressively measurable nonnegative function g s (! on R +, ( ( N x g s s da s = N x g s EW s ( da s : On the other hand, the timereversal invariance of N x, and the timereversal property stated in Theorem 5 give for any Borel function f on W x ( ( N x f(w s ˆ s da s = N x f(w s s da s : We use both facts in the following calculation ( ( N x s ˆ s da s = N x EW s ( ˆ s da s = N x ( = N x ( E W s ( s da s E W s ( E W s ( da s = (dw E w( E w( ; by Theorem 5(ii. 3 Lower bounds for hitting probabilities Our main goal in this section is to prove the lower bound of Theorem 1. In this section and the next one, we assume that d=4. We denote by M the maximum of the lifetime process: M := sup{ s ; 5s5} : Lemma 9. There exists a positive constant = (d such that; if F is a Borel subset of B(; 1\B(; 1=3 and x B(; 1=4; then 1 N x (R F ; 4 M51= cap d 4 (F : Proof. We may suppose that cap d 4 (F, and then nd a probability measure on F such that (dy(dy d 4 (y y52 (cap d 4 (F 1 : (8 We denote by G the Green function of Brownian motion in R d, G(x; y = c d y x 2 d for a positive constant c d. For y x, let P xy denote the law of Brownian motion started at x and conditioned to die at y. In other words, P xy is the law of the htransform of Brownian motion started at x, with h = G( ;y
14 116 J.S. Dhersin, J.F. Le Gall (see [6]. We may and will consider P xy as a probability measure on the set {w W x ; ŵ=y}. Fix x B(; 1=4 and let be the nite measure on W x dened by (dw = (dy G(x; y P xy (dw : Notice that is supported on {w W x ; ŵ F}. According to [16], Sect. 3, the energy of is E(=2 dz G(x; z( (dy G(z; y 2 =2 (dy(dy ( dz G(x; zg(z; yg(z; y : On the other hand, an elementary calculation shows that there exists a constant C d (independent of the choice of x B(; 1=4 such that for y; y B(; 1\B(; 1 3, dz G(x; zg(z; yg(z; y 5C d d 4 (y y : Using (8, we now obtain E(54 C d (cap d 4 (F 1 : (9 Therefore we can introduce the additive functional A associated with by Theorem 5. From Theorem 5 (ii applied with f(w =1 F c(ŵ, it is obvious that the event {A } is N x a.e. contained in {R F }. Therefore, N x (R F ; 1 4 M51 = N x(a ; 1 4 M51 = N x(a 1 ; 4 M512 N x ((A ; (1 2 by the Cauchy Schwarz inequality. From Theorem 5(iii and (9 we have N x ((A 2 58 C d (cap d 4 (F 1 : (11 It remains to get a lower bound on N x (A ; 1=4 M51. To this end, we apply Lemma 8 with = =1 {1=4 M 5 1}, observing that = s ˆ s, for every s (;, N x a.e. We obtain N x (A ; 1 4 M51 = (dw(p w(1=4 M51 2 = (dw1 {1=4 w 5 1=2} (P w(m 51 2 = (dw1 {1=4 w 5 1=2}(1 w 2 = 1 4 ( 1 4 w5 1 2 : We have used the fact that ( s ;s= is distributed under Pw Brownian motion started at w and stopped when it hits. as a linear
15 Wiener s test for superbrownian motion 117 From the denition of, we have ( 1 4 w5 1 2 = (dy G(x; y P xy ( ; and, if p t stands for the Brownian transition kernel, P xy ( = 1=2 1=4 dt p t (x; y dt p t (x; y =C d ; for some positive constant C d independent of x B(; 1=4, y B(; 1\B(; 1 3. We thus obtain N x (A 1 ; 4 M51= 1 4 (c d2 2 d C d : Lemma 9 follows by combining this bound with (1 and (11. Proof of the lower bound of Theorem 1(case d=4. We use a simple scaling lemma whose proof follows immediately from the denition of cap d 4 (F. Lemma 1. If F is a Borel subset of B(; 1=2 and ]; 1]; (i if d =4;(cap (F 1 = (cap (F 1 + log 1 ; (ii if d=5; cap d 4 (F = 4 d cap d 4 (F. Then, let F be a Borel subset of B(; 1=2 and x =1. We set { } 1 F = z = x + (y x;y F ; x +1=2 in such a way that F B(x; 1\B(x; 1=3. By Lemma 9 and a trivial translation argument, we have N x (R F = cap d 4 ( F : (12 On one hand, Lemma 1 gives cap (F 1 + log( x +1=2cap cap d 4 ( F = (F ( x +1=2 4 d cap d 4 (F On the other hand, the scaling properties of N x give if d =4; if d=5: (13 N x (R F =( x N x (R F : (14 The lower bound of Theorem 1 follows immediately from (12 (14. Remark. In the previous proof, we did not use the exact statement of Lemma 9 but only the lower bound N x (R F = cap d 4 (F, which can be obtained more easily. The precise form of Lemma 9 will be needed in Sect. 5.
16 118 J.S. Dhersin, J.F. Le Gall 4 Upper bounds for hitting probabilities In this section, we prove the upper bound of Theorem 1. We use analytic techniques inspired from Baras and Pierre [2]. First recall that if K is a compact subset of R d, the capacity c 2; 2 (K is dened by c 2; 2 (K = inf { 2 2; 2; C (R d ; = 1 on a neighborhood of K} ; where C (Rd denotes the set of all C functions on R d with compact support, and 2;2 stands for the Sobolev norm: 2; 2 = : For the sake of completeness, we recall the relations between cap d 4 (K and c 2; 2 (K. Lemma 11. There exist two positive constants c 1 ;c 2 compact subset K of B(; 1; such that; for every c 1 cap d 4 (K5c 2; 2 (K5c 2 cap d 4 (K : Proof. We use some results about equivalence of capacities due to Adams and Polking [1] and Meyers [21]. Denote by g 2 (x; y the usual Bessel kernel g 2 (x; y = dt e t (4t d=2 y x 2 exp : 4t The Bessel capacity B 2 (K of a compact subset K of R d is then dened by B 2 (K = inf { f 2 2 ;f C (R d ; dy f(y g 2 (x; y=1; x K} : According to Theorem A of [1], the capacities c 2; 2 (K and B 2 (K are equivalent, in the sense that c 1 B 2 (K5c 2; 2 (K5c 2 B 2 (K for some positive constants c 1 ; c 2 independent of K. On the other hand, Theorem 14 of [21] shows that the Bessel capacity B 2 (K is equivalent to C 2 (K 2, where C 2 (K = sup{ ; 1 ; is a nite measure supported on K and g } : Clearly, we have also C 2 (K 2 = (inf { g ;is a probability measure supported on K} 1 : However, a few lines of calculations show that, if is supported on B(; 1, 1 (dy(dy 4 (y y 5 g = dz( (dy g 2 (y; z (dy(dy 4 (y y ; where 1, 2 are positive constants depending only on d. Thus, cap d 4 (K and C 2 (K 2 are equivalent in the previous sense, provided we consider only compact subsets of B(; 1. The desired result follows.
17 Wiener s test for superbrownian motion 119 Lemma 12. There exists a positive constant = (d such that; for every compact subset K of B(; 1=2; if c 2; 2 (K ; one can nd a function C (Rd such that (i 5 51; (ii =1on a neighborhood of K; (iii =on B(; 3=4 c ; (iv 2 2; 2 5 c 2; 2 (K. Proof. According to [1, p. 53], there exist two constants Q; 1 depending only on d, such that, for every compact subset K of B(; 1 2 with c 2; 2(K, we can nd a function C (R d which satises (a 55Q, (b =1 on a neighborhood of K, (c 2 2; c 2; 2 (K. We let H be a C function from R + into [; 1] such that H(t =t for 5t and H(t =1 for t=1. We also choose C (Rd such that 5(x51, (x =1 if x 5 2 3, (x= if x =3 4. We claim that the function (x =(xh((x satises the desired properties. Properties (i (iii are immediate. We then observe that 2 5C 2 ; 2 5C ( ; where the constant C only depends on the choice of and H. Furthermore, 2 2 5C ( : To handle the term 2 2 we use the easy inequality ( (y 4 dy 1=2 5C 2 2 5C Q 2 2 ; (15 by (a. Property (iv then follows from (c and the previous observations. Proof of the upper bound of Theorem 1. It is enough to consider the case when cap d 4 (F. We rst assume that F = K is a compact subset of B(; 1=2 such that c 2; 2 (K. We know from [15], Proposition 5.3 (see also [8] that u(x =N x (R K solves the equation u =4u 2 in K c. Then, let be as in Lemma 12 and set =1, so that is a C function which vanishes on a neighborhood of K and is equal to 1 on B(; 3=4 c. First step. We prove that there exists a constant A = A(d such that R d (y 4 u(y 2 dy5a 2 2; 2 : (16
18 12 J.S. Dhersin, J.F. Le Gall It will be convenient to replace u by a periodic function. To this end, for every R 2, we set and K R = K + RZ d = {z = y +(k 1 R;:::;k d R;y K; k 1 ;:::;k d Z}; u R (x =N x (R K R : Then u R solves u R =4u 2 R on K c R, u5u R and u R has period R in every direction. If U R =[ R=2;R=2] d, we have thus 4 U R (y 4 u R (y 2 dy = U R (y 4 u R (y dy = U R ( 4 (y u R (y dy ; where the last equality follows from integration by parts using the periodicity of u R and the fact that = 1 on a neighborhood R. Then, 1 ( 4 u R dy u R dy + 3 u R dy 4 U R U R U R 5 3( U R 4 u 2 R dy 1=2 ( 4 dy 1=2 +( U R 6 u 2 R dy 1=2 ( 2 dy 1=2 5 ( U R 4 u 2 R dy 1=2 (3( 4 dy 1=2 +( 2 dy 1=2 : (17 Since 5 51, the same argument as in (15 gives 4 dy5a A 2 2; 2, where the constant A only depends on d. The previous bound then gives U R 4 u 2 R dy5a 2 2; 2 again with a constant A that depends only on d. Since u5u R, the same bound holds with u R replaced by u, and (16 follows by letting R tend to innity. From (17, we have also, with a constant A = A (d, ( 4 u R dy5a 2 2; 2 ; U R and the same argument gives ( 4 udy5a 2 2; 2 : (18 R d Second step. Let x B(; 1 c and let (B t ;t= denote a ddimensional Brownian motion that starts at x under the probability P x. For a x, we set T a = inf {t =; B t =a}. Then from Itô s formula, P x a.s., ( 4 t u(b t =u(x+ ( 4 u(b s db s + 1 t ( 4 u(b s ds : 2
19 Wiener s test for superbrownian motion 121 By applying the optional stopping theorem at T a t, weget ( E x (( 4 u(b Ta t = u(x+ 1 Ta t 2 E x ( 4 u(b s ds = u(x+ 1 2 E x ( Ta t ( 4 u+2 ( 4 u+u( 4 (B s ds : Since u =4u 2 =onk c, we thus get ( u(x5e x (( 4 u(b Ta t 1 Ta t 2 E x (2 ( 4 u+u( 4 (B s ds : Note that both functions ( 4 and ( 4 vanish outside the ball B(; 3=4, and that u(y tends to as y. By letting t, and then a tend to innity, we easily obtain u(x 5 1 ( 2 E x (2 ( 4 u+u( 4 (B s ds = c d ( ( 4 u+ 1 R 2 u( 4 (y y x 2 d dy : (19 d We will now bound the righthand side of (19. Since x =1 and =1 outside B(; 3 4, we can nd a constant C depending only on d such that (u( 4 (y y x 2 d dy 5 C x 2 d u( 4 (y dy R d R d 5 CA x 2 d 2 2; 2 ; (2 by (18. We then consider the other term in the righthand side of (19. We observe that if h d (y = y x 2 d, we can nd a constant C (independent of x B(; 1 c such that h d (y 5C x 1 d, for every y B(; 3 4. Thus an integration by parts gives ( ( 4 u(y y x 2 d dy R d = (u( 4 (y y x 2 d dy + (u 4 h d (y dy R d R d 5CA x 2 d 2 2; 2 +4C x 1 d R d u 3 dy 5CA x 2 d 2 2; 2 +4C x 1 d ( R d u 2 4 dy 1=2 ( R d 2 dy 1=2 5(CA +4C A 1=2 x 2 d 2 2;2; (21
20 122 J.S. Dhersin, J.F. Le Gall using (16 once again. Then, by combining (19 (21, Lemma 12(iv and Lemma 11, we get N x (R K =u(x5 x 2 d cap d 4 (K ; with a constant depending only on d. This is the desired upper bound when d=5. When d = 4, we use a scaling argument and apply the previous bound to x = x= x, K = x 1 K. It follows that N x (R K = x 2 N x (R K 5 x 2 cap (K x 2 cap = (K 1 + (log x cap (K ; by Lemma 1. This gives the upper bound of Theorem 1, when F is compact. When F is not compact, we observe that the mapping F N x (R F is the restriction to Borel sets of a Choquet capacity (see e.g. [2, Ch. 2]. Therefore, N x (R F = sup{n x (R K ;K F; K compact} : This completes the proof of Theorem 1. 5 Regularity and Wiener s test 5.1 Regularity for superbrownian motion and for the Brownian snake In this subsection, we prove Proposition 2. We rst state a technical lemma. Lemma 13. For every t ; set Rt X = ( supp X t ; 5r5t and; for every Borel subset F of R d ; set F = inf {t ;R X t F } : Then F = F, P x a.s.; and F is a stopping time of the ltration (G t. Proof. From the denition of the process X in terms of W; it is easy to verify that = { Wˆ s ;5s5 1 ; s 5t}; P x a:s: (22 R X t We then check that, if K is compact, then {R X t K } = { K 5 t} P x a:s: (23 The inclusion { K 5 t} {Rt X K } a.s. is easy, as we know from Perkins [23] that the mapping t supp X t is a.s. rightcontinuous with respect to the
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