CHARACTERIZATION OF G-REGULARITY FOR SUPER-BROWNIAN MOTION AND CONSEQUENCES FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS
By Jean-Fran ¸cois Delmas1 and Jean-St´ephane Dhersin MSRI and Universit´e Ren´e Descartes
We give a characterization ofG-regularity for super-Brownian motion and the Brownian snake. More precisely, we define a capacity onE = 0∞ ×Rd, which is not invariant by translation. We then prove that the measure of hitting a Borel setA⊂Efor the graph of the Brownian snake excursion starting at00is comparable, up to multiplicative constants, to its capacity. This implies that super-Brownian motion started at time 0 at the Dirac massδ0hits immediatelyA[i.e.,00isG-regular forAc] if and only if its capacity is infinite. As a direct consequence, ifQ⊂Eis a domain such that00 ∈∂Q, we give a necessary and sufficient condition for the existence onQof a positive solution of∂tu+ 12u=2u2which blows up at 00. We also give an estimate of the hitting probabilities for the support of super-Brownian motion at fixed time. We prove that if d≥2, the support of super-Brownian motion is intersection-equivalent to the range of Brownian motion.
1. Introduction. The purpose of this paper is to give a characteriza- tion of the so-called G-regularity for super-Brownian motion introduced by Dynkin . Thus we say that a pointr x ∈R×Rd isG-regular for a Borel setA⊂R×Rdif a.s. the graph of a super-Brownian motion started at timer with the Dirac mass atximmediately intersectsAc, the complementary ofA.
In caseA=Qis an open set, this is equivalent to the existence of nonnegative solutions of the equation ∂u/∂t +12u=2u2 on the open setQ, which blow up atr x ∈∂Q(cf. ).
Let E = 0∞ ×Rd. We prove that 00 is G-regular for a Borel set A⊂R×Rd if and only if the capacity of Ac∩Eis infinite, for the following capacity: for any Borel setA⊂E,
capA = infIν−1 where
Iν = E ds dy ps y
Eνdt dxpt−s x−y pt x
2 andpdenotes the heat kernel,
pt x =
2πt−d/2exp−x2/2t ift x ∈E
0 ift x ∈ −∞0 ×Rd
Received May 1998; revised September 1998.
1Supported by NSF Grant DMS-97-01755.
AMS1991subject classifications. Primary 60G57, 35K60.
Key words and phrases. Super-Brownian motion, Brownian snake,G-regularity, parabolic non- linear PDE, hitting probabilities, capacity, intersection-equivalence.
[ · denotes the Euclidean norm onRd.] The infimum is taken over all prob- ability measuresν onEsuch thatνA =1. Notice that this capacity is not invariant by translation in time or space. This capacity arises naturally when one considers the Brownian snake, a useful tool to study super-Brownian mo- tion. Indeed, using potential theory of symmetric Markov process, Iν can be viewed as the energy, with respect to the Brownian snake, of a certain probability measure (see Section 4 for more details).
We extend a result due to Dhersin and Le Gall  where the authors study G-regularity of 00 for sets Q = s y ∈ E y < √shs, where h is a positive decreasing function defined on0∞. Our result can also be viewed as a parabolic extension of the Wiener’s test in  in an elliptic setting.
The proof of our results relies on the Brownian snake introduced by Le Gall.
We only give the definition and some properties for completeness in this paper and refer to ,  for detailed presentations. We will use time inhomoge- neous notations.
Letr x ∈R×Rdbe a fixed point. We denote byr xthe set of all stopped paths in Rd started at x at time r. An element w of r x is a continuous mapping w r ζ → Rd such thatwr = x, and ζ =ζw ∈ r∞ is called its lifetime. We denote by wˆ the end point wζ. With the metricdw w = ζw−ζw+sups≥rws∧ζw−ws∧ζw, the spacer xis a Polish space.
The Brownian snake started at x at time r is a continuous strong Markov process W= Ws s≥ 0 with values in r x, whose law is characterized by the following two properties.
1. The lifetime processζ= ζs=ζWs s≥0is a reflecting Brownian motion inr∞.
2. Conditionally given ζs s ≥ 0, the process Ws s ≥ 0 is a time-inhomo- geneous continuous Markov process, such that fors≥s,
Wst =Wstforr≤t≤ms s =infv∈s sζv.
Wsms s+t−Wsms s, 0≤t≤ζs−ms sis a Brownian motion inRdindependent of Ws.
From now on we shall consider the canonical realization of the processW defined on the space=CR+r x, and denote bywthe law ofWstarted at w∈r x. The trivial pathxrsuch thatζxr=r,xrr =xis clearly a regular point for the processWw. We denote byNr xthe excursion measure outside xr, normalized by the following: for everyε >0,
= 1 2ε
Notice thatNr xis an infinite measure. The distribution ofWunderNr xcan be characterized as above, except that now the lifetime processζis distributed according to the Itˆo measure of excursions of linear reflecting Brownian motion inr∞. Letσ =infs >0ζs=rdenote the duration of the excursion of ζ
underNr x. The graph∗ ofWis defined underNr xby ∗ =
t Wstr < t≤ζs0< s < σ =
ζsWˆs0< s < σ We write ∗Wfor∗ when there is a risk of confusion.
Let us now explain the connection between the Brownian snake and super- Brownian motion. First, we introduce some notations. We denote byMff the space of all finite measures on Rd, endowed with the topology of weak convergence. We denote by S (resp., b+S) the set of all real measur- able [resp., bounded nonnegative measurable] functions defined on a Polish space S. We also denote byS the Borel σ-field on S. For every measure ν∈Mf, andf∈b+Rd, we shall writeν f =
fyνdy. We also denote by suppν the closed support of the measureν.
We consider underNr xthe continuous versionlts t > r s≥0of the local time ofζat leveltand timesand define the measure-valued processYonRd by setting for everyt > r, for everyϕ∈b+Rd,
Yt ϕ =σ
0 dltsϕ ˆWs Let r =
x∈Rdr x. Let µ be a finite measure on Rd and
i∈IδWi be a Poisson measure onCR+rwith intensity
µdxNr x·. Then the process X defined by Xr = µ and Xt =
i∈IYtWi if t > r, is a super-Brownian motion started at timerat µ(see , ). We shall denote by Pr µ (resp., Pr x) the law of the super-Brownian motion started at time rat µ (resp., at the Dirac massδx). We deduce from the normalization of Nr xthat, for every t > r,Nr xYt=0 =1/2t−r<∞. This implies that, fort > r, there is only a finite number of indicesi∈Isuch that∗Wi ∩ t∞ ×Rd is nonempty.
We consider the graph ofX, X =
ε>r t≥εt ×suppXt
where A¯ denotes the closure of A. A set A⊂R×Rdis called G-polar if Pr xX ∩A = ⵰ = 0 for every r x ∈ R×Rd. From Poisson measure theory, we have
HenceAisG-polar if and only ifNr x∗∩A=⵰ =0 for allr x ∈R×Rd. We consider the capacity defined by the following: forA∈R×Rd,
inf R×Rd ds dy
s∞×Rd νdt dxpt−s x−yexp−t−s/2 2−1
where the infimum is taken over all probability measuresνonR×Rdsuch that νA =1. Dynkin proved (see Theorem 3.2 in ) that A∈R×Rdis G- polar if and only if capA =0. (We have capA =0⇔Nr x∗∩A=⵰ =0 for allr x ∈R×Rd.) It is easy to check that ifA⊂Eis a compact set, then
capA =0 ⇔ capA =0
This can be extended to all Borel subsets of Esince the two capacities are inner capacities (see ). In fact, it seems more relevant to consider the ca- pacity cap to characterizeG-regularity, as we shall see. We have the following quantitative theorem.
Theorem 1. There exists a constantC0such that for any A∈E, 4−1capA ≤N00∗∩A=⵰ ≤C0capA
If Q is a domain of R×Rd, it is well known that the function r x → Nr x∗∩Qc=⵰ is the maximal nonnegative solutionuMof ∂u/∂t+12u= 2u2 inQ. Hence, we deduce from Theorem 1 that if00 ∈Q, thenuM00 and capQc ∩E are comparable up to multiplicative constants which are independent of Q.
The proof of Theorem 1 is split in two parts. In Section 2, we introduce a capacity associated with a weighted Sobolev space, which is equivalent to the capacity cap. In Section 3, using the connections between super-Brownian motion and partial differential equations, we prove the upper bound with this new capacity and hence for the capacity cap. The lower bound is obtained in Section 4 by using additive functionals of the Brownian snake introduced in .
Now, forA∈R×Rd, we consider underPr xthe random time τA=inf
t > rt ×suppXt ∩A=⵰
Arguments similar to those of  yield that τA is a stopping time for the natural filtration ofXcompleted the usual way. Thus we havePr xτA=r = 1 or 0. Following Dynkin , Section II-6, we say a pointr x ∈R×RdisG- regular forAcifPr x-a.s.τA=r. LetAGr denote the set of all points that are G-regular forAc. From the known path properties of super-Brownian motion it is obvious that intA ⊂AGr ⊂ ¯A, where intA denotes the interior ofA.
We set TA=infs >0ζsWˆs ∈A. Following , it is easy to deduce from Theorem 1 the next result.
Proposition 2. Let A∈R×Rd. The following properties are equiva- lent:
(i) r xisG-regular forAc; (ii) Nr x∗∩A=⵰ = ∞;
(iii) xr-a.s. TA=0;
(iv) capAr x∩E = ∞, whereAr x= s y s+r y+x ∈A.
We can give a straightforward analytic consequence of Proposition 2 and the link between super-Brownian motion and nonlinear differential equations.
Corollary 3. LetQbe a domain inEsuch that00 ∈∂Q. The following three conditions are equivalent:
(i) 00isG-regular for Q;
(ii) capQc∩E = ∞;
(iii) there exists a nonnegative solution of∂u/∂t+12u=2u2inQsuch that
s y→00lims y∈Q us y = ∞
The equivalence of assertions (i) and (iii) is due to Dynkin , Theorem II.6.1. The equivalence of (i) and (ii) is given by Proposition 2.
Finally, using Theorem 1, we give in Section 5 an estimate of the hitting probability of the support of X1, and we prove that in dimension d≥ 2, the support of super-Brownian motion and the range ofd-dimensional Brownian motion are intersection-equivalent.
2. Equivalence of capacities for a weighted Sobolev space. In this section, we introduce a new capacity, associated with a weighted Sobolev space, which is equivalent to the capacity cap. This capacity will be very useful in the next section to prove the upper bound for Theorem 1.
If S is an open subset of Rr, we denote by C∞0S the set of all functions of classC∞ defined onSwith compact support. Iffis a measurable function defined onSthenf∞=supx∈Sfs. We consider the Hilbert spaceL2p = f∈E fp<∞, where f2p=
Edt dx pt xft x2.
Notice the kernel defined onE×Ebykt xs y =pt−s x−ypt x−1 is nonnegative and lower semicontinuous. Thus we can introduce the operator 1defined on the set of nonnegative functionsf∈Eby
1f =p−1p∗ pf =
Eds dy k··s yps yfs y
where ∗denotes the usual convolution product onE. Furthermore, the func- tion1fis even lower semicontinuous (see , Lemma 2.2.1).
We define the capacity Cap onEin the following way: ifA⊂E, then CapA =inff2
pf≥0 f∈L2p 1f ≥1 onA
with the convention inf⵰ = ∞. Notice this capacity is not invariant by translation in time or space. This capacity is an outer capacity (see , Theorem 1). Moreover, it coincides with the capacity cap on the analytic sets (see , Theorem 14). Now, we want to connect this capacity to an analytic capacity (see  for similar results but with different norms). Therefore we
consider the weighted Sobolev space WD which is the completion ofC∞0E with respect to the norm · D, defined by
p ϕ∈C∞0 E with the usual notations ∂tgt x = ∂g/∂tt x, ∂igt x = ∂g/∂xit x for x= x1 xd ∈Rd and ∂2ii =∂i∂i. Notice the nonzero constants do not belong to WD. We can introduce the outer capacity capD associated to WD defined as follows. For any compact set K⊂E, we set
Dϕ∈C∞0E ϕ≥0 ϕ≥1 onK
Dϕ∈C∞0E ϕ≥0 ϕ≥1 on a neighborhood ofK Then we set for any open setG⊂E,
capDKK⊂G Kcompact (1)
and, for any analytic setA⊂E, capDA =inf
capDGA⊂G G open Notice the definition is consistent (see , for example).
Proposition 4. There exists a constantCsuch that for any setA⊂E, CapA ≤capDA ≤C CapA
Proof. Since the two capacities are outer capacities, it is enough to con- sider open sets. Now, using (1) and , Theorem 8, we see it is enough to consider compact sets.
Let us introduce the operatorH=∂t−12. We consider a nonempty compact set K⊂E. Let ϕ∈ C∞0 E be such thatϕ≥ 0 andϕ≥ 1 on K. Notice that (in the distribution sense) Hp = δ00, where δ00 is the Dirac mass at 00 ∈R×Rd. Then we havep∗ Hpϕ = Hp ∗ pϕ =pϕ. The function f=p−1Hpϕ=Hϕ − ∇logp∇ϕis nonnegative and
1f =p−1p∗Hpϕ ≥p−1p∗Hpϕ =ϕ Thus we have1f ≥1 onK. We also have
p ≤∂tϕ−12ϕ− ∇logp∇ϕ
Hence we have CapK ≤ ϕ2D. The first inequality follows by taking a se- quenceϕnsuch thatϕn2D converges to capDK.
To prove the other inequality, let us consider a nonnegative functionf1 ∈ L2p, such that 1f1 ≥1 onK. Notice this impliesf1p >0. Let δ >0.
It is easy to construct a functionε∈L2psuch thatε >0 onEandεp≤ δf1p. We setf2=f1+ε. Since the function1f2is lower semicontinuous, the sett x ∈E1f2>1is open and it also containsK. It is then obvious
that for δ>0 small enough, if we setf3t x =f2t x1δ<t<δ−1x<δ−1 for t x ∈E, we get1f3>1 on an open set containing K. Let us introduce a nonnegative function h∈ C∞0E such that Eht xdt dx =1. For θ > 0, we write hθt x =θ−d−1ht/θ x/θ. Now using the uniform continuity ofp onδ/2∞×Rd, it is easy to see that iff=hθ∗f3, then1f>1 on an open set containingKfor θsmall enough. The function fis nonnegative, belongs toC∞0E and the function1fis of classC∞. We can chooseδandθsmall enough so thatfp≤2f1p.
Let α ∈ C∞0 0∞ such that 0 ≤ α ≤ 1, α =1 on 01/2 and α =0 on 1∞. Letξ∈C∞0 Rdsuch that 0≤ξ≤1 andξ=1 in a neighborhood of 0.
We define αnt = αt/n and ξnx = ξx/n. The function ϕn = αnξn1f belongs to C∞0 E, is nonnegative, andϕn ≥1 on a neighborhood of K forn large enough.
Let us now give two key lemmas. If Mis a bounded operator from L2p into itself, we denote by Mp=supMfpf∈L2pfp =1 its norm. We define the operator 10: for f ∈ E nonnegative, 10ft x = t−11ft x,t x ∈E. ForT >0, let us introduceET= 0 T ×Rd.
Lemma 5. The operators1ET1and10are bounded operators from L2p into itself. Furthermore, we have1ET1p≤T/√
Proof. Let f∈L2p. We have 10f2
Edt dx t−2pt x−1
Eds dy pt−s x−yps yfs y 2
≤ Edt dx t−2pt x−1
Eds dy pt−s x−yps y
≤ Edt dx t−2pt x−1
× Edsdypt−s x−yps ys−1/21s≤t
× Eds dy pt−s x−yps ys1/2fs y21s≤t
≤ Edt dx t−2t
× Eds dy pt−s x−yps ys1/2fs y21s≤t
Edt dx t−2t1/2
Eds dy pt−s x−yps ys1/2fs y21s≤t
Eds dy ps yfs y2s1/2∞
where we used the Cauchy–Schwarz inequality for the second inequality.
Hence the operator 10 is a bounded operator from L2p into itself. And
we have 10 p≤ 2. The operator 1ET1 can be handled in a very similar way. ✷
Lemma 6. The operators defined onC∞0 Ebyg∈C∞0 E 11g =∂t1g fori∈ 1 d 12 ig = 12∂2ii1g and fori∈ 1 d 13 ig =∂ilogp∂i1g
can be uniquely extended into bounded operators fromL2pinto itself. More- over we have
fori∈ 1 d 12 ip≤1 (3)
fori∈ 1 d 13 ip≤4 (4)
The proof of this lemma is given in the Appendix.
We now boundϕnD. Lemma 5 provides an upper bound for∂tϕnp ∂tϕn
Using Lemma 5 we derive an upper bound ford
13 ip +sup
In order to give an upper bound for d
p, we need an intermediary lemma.
Lemma 7. There exists a constant c1 (depending on ξ) such that for all n≥1,g∈C∞0E,i∈ 1 d,
Proof. Recall thatξnhas compact support. Then, an integration by parts, the Cauchy–Schwarz inequality and Lemma 5 give for 1≤i≤d,
p = −
212 ip+ 13 ip∂iξ2
Notice that ifa b care positive thena2≤c2+baimpliesa≤c+b. Thus we get 1En∂iξn∂i1g
212 ip+ 13 ip1/2∂iξ
p which, thanks to Lemma 6, ends the proof. ✷
Using this lemma and Lemma 5, we get that d
Then we deduce from (5), (6), (7) and Lemma 6 that there exists a constantc2 independent of fandn≥1 such that
Thus we haveϕnD≤2c2f1p. The second inequality of the proposition is then obvious withC=4c22. ✷
We shall need the following lemma.
Lemma 8. For any compact set K ⊂ E with capDK > 0, there exists ϕ∈C∞0Esuch that:
(ii) ϕ=1on a neighborhood ofK;
whereγis a constant independent ofKandϕ.
The proof is classic, but we give it for completeness.
Proof. Let h ∈ C∞0∞ such that 0 ≤ h ≤ 1, h = 0 on 01/4 and h = 1 on 3/4∞. Since capDK > 0, there exists g ∈ C∞0E such that g ≥0, g≥ 1 in a neighborhood ofK, and 2 capDK ≥ g2D. Letϕ=h◦g.
The function ϕ∈C∞0Esatisfies (i) and (ii). Let us check (iii). We have ∂tϕ
∞∂ilogp∂ig ∂2iiϕ p
Only the upper bound for the second right-hand side term of the last inequal- ity is not obvious. We first search an upper bound for ∂iϕ12/1+ϕ1p, whereϕ1∈C∞0Eis a nonnegative function. An integration by parts and the Cauchy–Schwarz inequality give
Ep ∂iϕ14 1+ϕ12 =3
Ep∂2iiϕ1∂iϕ12 1+ϕ1 +
p Thus we get
Since we haveht≤21+t−1h∞, takingϕ1=gin the above inequality we deduce that
The previous inequalities imply there exists a constantcdepending only onh anddsuch thatϕD≤cgD. Thus (iii) holds withγ=2c2. ✷
3. Upper bound for hitting probabilities. In this section we prove the second inequality of Theorem 1 for compact sets. Let us introduceK⊂ET, a compact set such that capDK> 0. Let ϕbe as in Lemma 8. We set ϕ =0 outsideE. We introduce the functionψ=1−ϕ, which takes values in 01.
We consider the function u defined on R×Rd by ut x = Nt x∗∩K =
⵰ (∈ 0∞). With the convention 0∞ = 0, the function uψ4 is bounded nonnegative and of class C∞ on R×Rd. Let Bt t ≥ 0 denote under P0 a d-dimensional Brownian motion started from 0. Itˆo’s formula implies that for allt≥0, P0-a.s.,
uψ4t Bt =uψ400 +t
0 ∂tuψ4s Bsds +t
0 ∇uψ4s BsdBs
Consider the stopping timeTa =T∧inft >0Bt≥a. We can then apply the optional stopping theorem at timeTa and get
0 ∂tuψ4s Bsds+E0
2u2ψ4+4uψ3∂tψ+4∇u∇ψψ3 +6uψ2∇ψ∇ψ +2uψ3ψ
s Bsds We have used that ∂tu+12u=2u2to get the last equality. Notice that each integrand is either nonnegative or bounded. By dominated convergence and monotone convergence, we get, asagoes to infinity,
SinceK⊂ET, we deduce thatut x =0 fort≥T. Thus we have u00 +2uψ22
= − Ep
4uψ3∂tψ+4∇u∇ψψ3+6uψ2∇ψ∇ψ +2uψ3ψ (9)
We now bound the right-hand side. Using the Cauchy–Schwarz inequality, that 0≤ψ≤1 and that−ϕandψhave the same derivatives, we get
− Epuψ2∇ψ∇ψ ≤uψ2
where we have used (8) withϕ1=ϕfor the last inequality. Now an integration by parts and the Cauchy–Schwarz inequality give
ψ∇logp∇ψ +3∇ψ∇ψ +ψψ
where we have used again (8) for the last inequality. Taking those results together, we deduce from (9) that
where the constantc3depends only ond. Sinceuψ2pis finite (recalluψis bounded, and zero onT∞ ×Rd), this implies thatuψ2p ≤c3ϕD and henceu00 ≤c23ϕ2D. This last inequality and the definition ofϕimply that
=u00 ≤c23γcapDK ≤c23γCCapK =c23γCcapK 4. Lower bound for hitting probabilities and proof of Theorem 1.
In this section, we prove the first inequality of Theorem 1 for compact sets.
Let us introduce a compact set K ⊂ E, ν a probability measure on K and T >0 such thatK⊂ET. We consider the probability measure µ defined on 00by
Eνdt dxPt x0 dw
where Pt x0 is the law on 00 of the Brownian bridge starting at time 0 at point 0 and ending at timetat pointx. Notice that the measureµis in fact a measure on00∗ , the set of nontrivial path in00 (a trivial path is a path of lifetime zero). The measure Pt x0 can also be viewed as a probability measure on the canonical space CR+Rd endowed with the filtrationtgenerated by the coordinate mappings. Let P0 be the law on the canonical space of the standard Brownian motion. Fors∈ 0 t, we have
Pt x0 dws= pt−s x−ws
pt x P0dws
We consider the energy of µ with respect to the process Ws (see  for a precise description and definition). Thanks to , Proposition 1.1, we have
µ =2∞ 0 dsP0
Eνdt dxpt−s x−ws/pt x 2
=2Iν Now, using , Proposition 5, we know there exists an additive functionalA of the Brownian snake killed when its lifetime reaches 0 such that:
1. For every Borel functionF≥0 on00∗ , N00∞
µdwFw 2. N00A2∞ =2µ.
We deduce from (1) that the additive functional increases only when Wˆs ∈ suppν⊂K. Therefore, using the Cauchy–Schwarz inequality, we get
N00∗∩K=⵰ ≥N00A∞>0 ≥N0A∞2/N0A2∞
We get N00∗∩K=⵰ ≥ 4Iν−1. Since the above inequality is true for any probabilityν onK, we get that
≥4−1 CapK =4−1 capK
ProofofTheorem 1. Notice the application defined onEbyTA = N00∗∩A =⵰ forA∈ E is a Choquet capacity (see , Th´eor`eme 1).
Since the capacity cap is an inner capacity (see , Theorem 12), it is enough to prove the theorem for compact subsets ofE. The result is then given by the previous section (withC0=c23γC) and the above result. ✷
5. Brownian range and support of X1. In this section, we first give an estimate for the hitting probabilities of the support ofX1. Then we prove that the range of Brownian motion and the support of super-Brownian motion at fixed time are intersection-equivalent.
Let us fixd≥2. We denote by capd−2 the usual Newtonian (logarithmic if d=2) capacity inRd,
inf Rd×Rdρdxρdyhd−2x−y −1
with hγr =r−γ if γ > 0 and h0r =log+1/r. The infimum is taken over all probability measuresρonRd such thatρA =1. LetB0 hbe the open ball ofRd centered at 0 with radiush.
Proposition 9. LetM >0. There exist two positive constantsaandbsuch that for any Borel set A ⊂B01, for any finite measureµ onB01, with µ1 ≤M, we have
aµ1capd−2A ≤P0 µ
≤bµ1capd−2A Proof. Let A⊂B02be a Borel set. Letν be a probability measure on Esuch thatν1×A =1. Then we haveν=δ1×ρ, whereρis a probability measure onRdsuch thatρA =1. We get
Iν = 01×Rdds dy ps y
× A×Aρdxρdxp1−s x−yp1−s x−yp1 x−1p1 x−1
Since x x are inB02and sinces∈ 01, it is easy to see there exist two positive constantsa1andb1 (independent ofAandρ) such that
A×Aρdxρdxhd−2x−x ≤b1Iν This implies that for any Borel setA⊂B02,
a1capd−2A ≤cap1 ×A ≤b1capd−2A
Since the capacity capd−2is invariant by translation, we get that for any Borel setA⊂B01, for anyx∈B01,
a1capd−2A ≤cap1 ×Ax ≤b1capd−2A whereAx = yy−x∈A. We deduce from Theorem 1 that
4−1a1capd−2A ≤N0 x
∗∩ 1 ×A =⵰
≤C0b1capd−2A Since X1=
i∈IδWi is a Poisson measure onCR+0 with intensity
µdxN0 x·, we have P0µ
suppY1∩A=⵰ Notice thatN0 x-a.e.,1 × suppY1∩A =∗∩ 1 ×A. Sinceµ1< M, we then easily get the result. ✷
Intersection-equivalence between random sets has been defined by Peres . Two random Borel sets F1 andF2 inRd are intersection-equivalent in an open setU, if there exist positive constantsaandbsuch that, for any Borel setA⊂U,
aPA∩F1 ≤PA∩F2 ≤bPA∩F1
If π is a probability measure on B01, then we denote by Pπ the law of a d-dimensional Brownian motionBt t≥0started with the lawπ. Ford≥3 the range of Brownian motion is defined byB= Bt t≥0inRd. Ford=2, we also denote byB the setB= Bt t∈ 0 ξ, whereξis an exponential random variable of parameter 1 independent ofBt t≥0.
Corollary 10. LetM >0. There exist two positive constantsaandbsuch that for any Borel setA ⊂B01, for any absolutely continuous probability measureπ onB01with density bounded byM, for any finite measureµon B01, withµ1 ≤M, we have
≤bµ1PπB∩A=⵰ Proof. This is a consequence of Proposition 9 and the fact that there exist two positive constantsa2andb2such that for any Borel setA⊂B01,
for any absolutely continuous probability measureπ onB01 with density bounded byM,
≤b2capd−2A (see, e.g., , Proposition 3.2, ford≥3 and  ford=2). ✷
In this section, we give the proof of Lemma 6, which relies on the properties of the Hermite polynomials. We first recall the definition and some properties of those polynomials.
A.1. Hermite polynomials. For n = n1 nd ∈ Nd, we set n = d
ni=0. Forj∈ 1 d, letδjbe the element of Nd such that δji =δi j, the standard Kronecker symbol. If z= z1 zdis an element ofRd, then we setzn=d
i=1znii. Let··be the Euclidean product onRd.
The function ϕz =exp−z2−2z x/2 is an entire function defined onRd. We have
where thenth termHenxis a polynomial ofx1 xdof degreencalled thenth Hermite polynomial. Those polynomials can easily be expressed with the usual one-dimensional Hermite polynomials He1k k ∈ N: Henx = d
i=1He1nixi, wherex= x1 xd.
Now let us recall some basic properties of the polynomialsHen. The follow- ing recurrence formula can be deduced from (10) by differentiating w.r.t. zi: for alln∈Nd such thatni>0,
Henx =xiHen−δix − ni−1Hen−2δix ∀x∈Rd (11)
where by conventionHen−kδi =0 ifni−k <0. The differential formula can be deduced from (10) by differentiating w.r.t.xi: for alln∈Nd,
We also recall the upper bound for Hen(see , 22.14.17); there exists a uni- versal constant 1< c0<2 such that
for allx∈Rd n∈Nd (13)
Using the definition of the Hermite polynomials, it is also easy to prove that dx pt xHenx/√
δni mi (14)
It is also well known that the Hermite polynomials are a complete orthogonal system in L2Rdexp
dx. Finally, standard arguments on Hilbert spaces show that iff∈L2pthen
ft x =
fnt x =
tgnt where gnt = n!−1
dx pt xHenx/√
tft xandgn∈L20∞. Fur- thermore, we have
0 dt gnt2 (15)
SinceC∞0 0∞is dense inL20∞, it is clear that the set of functions ft x =
tgnt wheregn∈C∞0 0∞ is nonzero for a finite number of indices n, is dense inL2p.
A.2. Proof of Lemma 6. First step. We prove there exist unique bounded extensions 1˜1,1˜2 i and 1˜3 i in L2p of the operators11, 12 i and 13 i defined on . Then in a second step we check that the extensions 1˜1, 1˜2 i and 1˜3 i and the operators 11, 12 i and13 i, which are also defined on C∞0E, agree onC∞0 E.
First step. Let us compute 1f for very particular functions f ∈ . Let g∈C∞0 0∞, αandβ be two positive reals such that suppg ∈ α β, and Gt =t
0 ds gs. Forn∈Nd, andt x ∈E, we set hn gt x =Hen
t−n/2 gt Let us prove that
1hn g =hn G (16)
Forz∈R, we introduce the function g z defined onEby
g zt x =
n!znhn gt x =gtexp
Then we have
1 g zt x = 1 pt x
dy pt−s x−yps y g zs y
= 1 pt x
0 ds dy p
t −t−sz t
−z2−2x z/2t t
0 ds gs = G zt x
Using (13), the Chapman–Kolmogorov equation, and that suppg ∈ α β, we get
1hn gt x ≤√
E ds dy pt−s x−yps y
0 ds pt+s xs−n/2gs
0 ds s−n/2gs
The radius of the seriesakk!αk−1/2 is infinite. Thus for anyt x ∈Ethe seriesn!−1zn1hngt xare convergent. Fubini’s theorem implies that
n!zn1hn gt x =1
t x = G zt x
Hence the two series n!−1zn1hn gt x and n!−1znhn Gt x agree.
Since their radius of convergence is positive (in fact infinite), we get that (16) is true.
Let us prove that12 i has a bounded extension onL2p. We deduce from (12) that
12 ihngt x = 1
2∂2i i1hn gt x
0 ds gs (17)
Let us introducef∈, that is, fort x ∈E,ft x =
tgnt, where gn ∈ C∞0 0∞ and gn =0 except for a finite number of terms. By linearity, we have
12 ift x =
0 ds sn/2gns Thus, using (14), we have
0 dt t−2−nt
0 ds sn/2 gns
0 dt gnt2
where we used the Hardy inequality; fork >−1, ∞
0 dt t−2−kt
0 sk/2hsds 2
≤ 4 k+12
0 dt ht2
for the first inequality and (15) for the second one. This means that 12 i, defined on , can be uniquely extended into a bounded operator 1˜2 i from L2pinto itself. The above inequality implies ˜12 ip≤1.
Fori∈ 1 d, we set14 i=13 i+212 i. Using (12) and (11), we deduce from (16) that
14 ihngt x
t +nini−1Hen−2δix/√ t
0 ds gs
0 ds gs (18)
Arguing as above, we get, for f∈, 14 if2
0 dt t−2−nt
0 ds sn/2 gns
n!n2i 4 n+12
0 dt gnt2
Thus the operators14 i and13 i, defined on, can be uniquely extended in bounded operators1˜4 i and1˜3 i fromL2pinto itself. Furthermore we have ˜14 ip≤2 and ˜13 ip≤ ˜14 ip+2 ˜12 ip≤4.
The proof concerning11easily follows from the previous results. From (16), we get
11hn gt x
=hn gt x −1 2
nHenx/√ t +d
0 ds gs Then using (17) and (18), we get
11hn g =
14 i+13 i
hn g =
14 i−12 i hn g This means that11=I+d
i=114 i−12 ion. Hence11can be uniquely ex- tended in a bounded operator1˜1fromL2pinto itself and1˜1=I+d
i=1 ˜14 i− 1˜2 i. We deduce that ˜11p≤1+3d.