We also give an estimate of the hitting probabilities for the support of super-Brownian motion at fixed time

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∞ ×R^d, 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 forA^c] 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 [8]. Thus we say that a pointr x ∈R×R^d isG-regular for a Borel setA⊂R×R^dif a.s. the graph of a super-Brownian motion started at timer with the Dirac mass atximmediately intersectsA^c, 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. [8]).

Let E = 0∞ ×R^d. We prove that 00 is G-regular for a Borel set A⊂R×R^d if and only if the capacity of A^c∩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 ×R^d

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.

731

[ · denotes the Euclidean norm onR^d.] 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 [6] 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 [5] 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 [10], [12] for detailed presentations. We will use time inhomoge- neous notations.

Letr x ∈R×R^dbe a fixed point. We denote by_{r x}the set of all stopped paths in R^d started at x at time r. An element w of _{r x} is a continuous mapping w r ζ → R^d such thatwr = x, and ζ =ζ_w ∈ r∞ is called its lifetime. We denote by wˆ the end point wζ. With the metricdw w = ζ_w−ζ_w+sup_s≥rws∧ζ_w−ws∧ζ_w, the space_{r x}is a Polish space.

The Brownian snake started at x at time r is a continuous strong Markov process W= W_s s≥ 0 with values in _{r x}, whose law is characterized by the following two properties.

1. The lifetime processζ= ζ_s=ζ_Ws s≥0is a reflecting Brownian motion inr∞.

2. Conditionally given ζ_s s ≥ 0, the process W_s s ≥ 0 is a time-inhomo- geneous continuous Markov process, such that fors≥s,

W_st =W_stforr≤t≤ms s =inf_{v∈s s}ζ_v.

W_sms s+t−W_sms s, 0≤t≤ζ_s−ms sis a Brownian motion inR^dindependent of W_s.

From now on we shall consider the canonical realization of the processW defined on the space=CR⁺_{r x}, and denote by_wthe law ofWstarted at w∈_{r x}. The trivial pathx_rsuch thatζ_xr=r,x_rr =xis clearly a regular point for the processW_w. We denote byN_{r x}the excursion measure outside x_r, normalized by the following: for everyε >0,

N_{r x}

sups≥0ζ_s> ε+r

= 1 2ε

Notice thatN_{r x}is an infinite measure. The distribution ofWunderN_{r x}can 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 >0ζ_s=rdenote the duration of the excursion of ζ

underN_{r x}. The graph^∗ ofWis defined underN_{r x}by ^∗ =

t W_str < t≤ζ_s0< s < σ =

ζ_sWˆ_s0< 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 byM_ff the space of all finite measures on R^d, 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∈M_f, andf∈_b+R^d, we shall writeν f =

fyνdy. We also denote by suppν the closed support of the measureν.

We consider underN_{r x}the continuous versionl^t_s t > r s≥0of the local time ofζat leveltand timesand define the measure-valued processYonR^d by setting for everyt > r, for everyϕ∈_b+R^d,

Y_t ϕ =σ

0 dl^t_sϕ ˆW_s Let _r =

x∈R^d_{r x}. Let µ be a finite measure on R^d and

i∈Iδ_Wⁱ be a Poisson measure onCR⁺_rwith intensity

µdxN_{r x}·. Then the process X defined by X_r = µ and X_t =

i∈IY_tWⁱ if t > r, is a super-Brownian motion started at timerat µ(see [10], [12]). 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 massδ_x). We deduce from the normalization of N_{r x}that, for every t > r,N_{r x}Y_t=0 =1/2t−r<∞. This implies that, fort > r, there is only a finite number of indicesi∈Isuch that^∗Wⁱ ∩ t∞ ×R^d is nonempty.

We consider the graph ofX, X =

ε>r t≥εt ×suppX_t

=

i∈I

^∗Wⁱ

where A¯ denotes the closure of A. A set A⊂R×R^dis called G-polar if Pr xX ∩A = ⵰ = 0 for every r x ∈ R×R^d. From Poisson measure theory, we have

Pr x

X ∩A=⵰

=1−exp

−N_{r x}^∗∩A=⵰

HenceAisG-polar if and only ifN_{r x}^∗∩A=⵰ =0 for allr x ∈R×R^d. We consider the capacity defined by the following: forA∈R×R^d,

capA =

inf R×R^d ds dy

s∞×R^d νdt dxpt−s x−yexp−t−s/2 ₂₋₁

where the infimum is taken over all probability measuresνonR×R^dsuch that νA =1. Dynkin proved (see Theorem 3.2 in [7]) that A∈R×R^dis G- polar if and only if capA =0. (We have capA =0⇔N_{r x}^∗∩A=⵰ =0 for allr x ∈R×R^d.) 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 [13]). 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×R^d, it is well known that the function r x → N_{r x}^∗∩Q^c=⵰ is the maximal nonnegative solutionu_Mof ∂u/∂t+¹₂u= 2u² inQ. Hence, we deduce from Theorem 1 that if00 ∈Q, thenu_M00 and capQ^c ∩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 [5].

Now, forA∈R×R^d, we consider underPr xthe random time τ_A=inf

t > rt ×suppX_t ∩A=⵰

Arguments similar to those of [5] yield that τ_A is a stopping time for the natural filtration ofXcompleted the usual way. Thus we havePr xτ_A=r = 1 or 0. Following Dynkin [8], Section II-6, we say a pointr x ∈R×R^disG- regular forA^cifPr x-a.s.τ_A=r. LetA^Gr denote the set of all points that are G-regular forA^c. From the known path properties of super-Brownian motion it is obvious that intA ⊂A^Gr ⊂ ¯A, where intA denotes the interior ofA.

We set T_A=infs >0ζ_sWˆ_s ∈A. Following [5], it is easy to deduce from Theorem 1 the next result.

Proposition 2. Let A∈R×R^d. The following properties are equiva- lent:

(i) r xisG-regular forA^c; (ii) N_{r x}^∗∩A=⵰ = ∞;

(iii) _xr-a.s. T_A=0;

(iv) capA_{r x}∩E = ∞, whereA_{r 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) capQ^c∩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 [8], 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 R^r, we denote by C^∞₀S the set of all functions of classC^∞ defined onSwith compact support. Iffis a measurable function defined onSthenf_∞=sup_x∈Sfs. We consider the Hilbert spaceL²p = f∈E f_p<∞, where f²_p=

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 [9], 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 [13], Theorem 1). Moreover, it coincides with the capacity cap on the analytic sets (see [13], Theorem 14). Now, we want to connect this capacity to an analytic capacity (see [3] for similar results but with different norms). Therefore we

consider the weighted Sobolev space W_D which is the completion ofC^∞₀E with respect to the norm · _D, defined by

ϕ²

D =∂_tϕ²

p+^d

i=1

∂_ilogp∂_iϕ²

p+^d

i=1

∂²_iiϕ²

p ϕ∈C^∞₀ E with the usual notations ∂_tgt x = ∂g/∂tt x, ∂_igt x = ∂g/∂x_it x for x= x₁ x_d ∈R^d and ∂²_ii =∂_i∂_i. Notice the nonzero constants do not belong to W_D. We can introduce the outer capacity cap_D associated to W_D defined as follows. For any compact set K⊂E, we set

cap_DK =infϕ²

Dϕ∈C^∞₀E ϕ≥0 ϕ≥1 onK

=infϕ²

Dϕ∈C^∞₀E ϕ≥0 ϕ≥1 on a neighborhood ofK Then we set for any open setG⊂E,

cap_DG =sup

cap_DKK⊂G Kcompact (1)

and, for any analytic setA⊂E, cap_DA =inf

cap_DGA⊂G G open Notice the definition is consistent (see [2], for example).

Proposition 4. There exists a constantCsuch that for any setA⊂E, CapA ≤cap_DA ≤C CapA

Proof. Since the two capacities are outer capacities, it is enough to con- sider open sets. Now, using (1) and [13], 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×R^d. 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

f

p ≤∂_tϕ−¹₂ϕ− ∇logp∇ϕ

p ≤ϕ

D

Hence we have CapK ≤ ϕ²_D. The first inequality follows by taking a se- quenceϕ_nsuch thatϕ_n²_D converges to cap_DK.

To prove the other inequality, let us consider a nonnegative functionf₁ ∈ L²p, such that 1f₁ ≥1 onK. Notice this impliesf_1p >0. Let δ >0.

It is easy to construct a functionε∈L²psuch thatε >0 onEandε_p≤ δf_1p. 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 x1_δ<t<δ−1_x<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 h_θt x =θ^−d−1ht/θ x/θ. Now using the uniform continuity ofp onδ/2∞×R^d, it is easy to see that iff=h_θ∗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 thatf_p≤2f_1p.

Let α ∈ C^∞₀ 0∞ such that 0 ≤ α ≤ 1, α =1 on 01/2 and α =0 on 1∞. Letξ∈C^∞₀ R^dsuch that 0≤ξ≤1 andξ=1 in a neighborhood of 0.

We define α_nt = αt/n and ξ_nx = ξx/n. The function ϕ_n = α_nξ_n1f belongs to C^∞₀ E, is nonnegative, andϕ_n ≥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 M_p=supMf_pf∈L²pf_p =1 its norm. We define the operator 1₀: for f ∈ E nonnegative, 1₀ft x = t⁻¹1ft x,t x ∈E. ForT >0, let us introduceE_T= 0 T ×R^d.

Lemma 5. The operators1_ET1and1₀are bounded operators from L²p into itself. Furthermore, we have1_ET1_p≤T/√

2and1_0p≤2.

Proof. Let f∈L²p. We have 1₀f²

p=

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

×

s^−1/21_s≤t1/2

s^1/2fs y²1_s≤t1/22

≤ Edt dx t⁻²pt x⁻¹

× Edsdypt−s x−yps ys^−1/21_s≤t

× Eds dy pt−s x−yps ys^1/2fs y²1_s≤t

≤ Edt dx t⁻²_t

0 dss^−1/2

× Eds dy pt−s x−yps ys^1/2fs y²1_s≤t

=2

Edt dx t⁻²t^1/2

Eds dy pt−s x−yps ys^1/2fs y²1_s≤t

≤2

Eds dy ps yfs y²s^1/2_∞

s t^−3/2dt≤4f²

p

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 1_{0 p}≤ 2. The operator 1_ET1 can be handled in a very similar way. ✷

Lemma 6. The operators defined onC^∞₀ Ebyg∈C^∞₀ E 1₁g =∂_t1g fori∈ 1 d 1_{2 i}g = ¹₂∂²_ii1g and fori∈ 1 d 1_{3 i}g =∂_ilogp∂_i1g

can be uniquely extended into bounded operators fromL²pinto itself. More- over we have

1_1p≤1+3d (2)

fori∈ 1 d 1_{2 ip}≤1 (3)

fori∈ 1 d 1_{3 ip}≤4 (4)

The proof of this lemma is given in the Appendix.

We now boundϕ_nD. Lemma 5 provides an upper bound for∂_tϕ_np ∂_tϕ_n

p≤∂_tα_n

∞1_En1f

p+1₁f

p

≤∂_tα

∞2^−1/2+ 1_1pf

p (5)

Using Lemma 5 we derive an upper bound for_d

i=1∂_ilogp∂_iϕ_np d

i=1

∂_ilogp∂_iϕ_n

p≤^d

i=1

1_{3 i}f

p+sup

x∈R^d

x_i∂_iξx1₀f

p

≤^d

i=1

1_{3 ip} +sup

x∈R^d

x_i∂_iξx1_0pf

p (6)

In order to give an upper bound for _d

i=1∂²_iiϕ_n

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,

1_En∂_iξ_n∂_i1g

p≤c₁n^−1/2g

p

Proof. Recall thatξ_nhas compact support. Then, an integration by parts, the Cauchy–Schwarz inequality and Lemma 5 give for 1≤i≤d,

1_En∂_iξ_n∂_i1g²

p = −

Ep1_En1g∂_iξ_n²∂²_ii1g

− Ep1_En1g∂_iξ_n²∂_i1g∂_ilogp

−2

Ep1_En1g∂_iξ_n∂_i1g∂²_iiξ_n

≤∂_iξ_n²

∞1_En1g

p∂²_ii1g

p

+∂_iξ_n²

∞1_En1g

p∂_i1g∂_ilogp

p

+2∂²_iiξ_n

∞1_En1g

p1_En∂_iξ_n∂_i1g

p

≤2^−1/2n⁻¹

21_{2 ip}+ 1_{3 ip}∂_iξ²

∞g²

p

+2^1/2n⁻¹∂²_iiξ

∞g

p1_En∂_iξ_n∂_i1g

p

Notice that ifa b care positive thena²≤c²+baimpliesa≤c+b. Thus we get 1_En∂_iξ_n∂_i1g

p

≤2^−1/4n^−1/2

21_{2 ip}+ 1_{3 ip1/2}∂_iξ

∞g

p

+2^1/2n⁻¹∂²_{i i}ξ

∞g

p which, thanks to Lemma 6, ends the proof. ✷

Using this lemma and Lemma 5, we get that d

i=1

∂²_iiϕ_n

p≤^d

i=1

21_{2 i}f

p+∂²_iiξ_n

∞1_En1f

p

+21_En∂_iξ_n∂_i1f

p

≤^d

i=1

21_{2 ip}+2^−1/2n⁻¹∂²_iiξ

∞+2c₁n^−1/2f

p (7)

Then we deduce from (5), (6), (7) and Lemma 6 that there exists a constantc₂ independent of fandn≥1 such that

ϕ_n

D≤c₂f

p

Thus we haveϕ_nD≤2c₂f_1p. 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 cap_DK > 0, there exists ϕ∈C^∞₀Esuch that:

(i) 0≤ϕ≤1;

(ii) ϕ=1on a neighborhood ofK;

(iii) ϕ²_D≤γcap_DK,

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 cap_DK > 0, there exists g ∈ C^∞₀E such that g ≥0, g≥ 1 in a neighborhood ofK, and 2 cap_DK ≥ g²_D. Letϕ=h◦g.

The function ϕ∈C^∞₀Esatisfies (i) and (ii). Let us check (iii). We have ∂_tϕ

p≤h

∞∂_tg

p ∂_ilogp∂_iϕ

p≤h

∞∂_ilogp∂_ig ∂²_iiϕ p

p≤h

∞∂²_iig

p+h◦g∂_ig²

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+ϕ_1p, whereϕ₁∈C^∞₀Eis a nonnegative function. An integration by parts and the Cauchy–Schwarz inequality give

Ep ∂_iϕ₁⁴ 1+ϕ₁² =3

Ep∂²_iiϕ₁∂_iϕ₁² 1+ϕ₁ +

Ep∂_ilogp∂_iϕ₁∂_iϕ₁² 1+ϕ₁

≤

3∂²_iiϕ₁

p+∂_ilogp∂_iϕ₁

p

∂_iϕ₁²/1+ϕ₁

p Thus we get

∂_iϕ₁²/1+ϕ₁

p≤3∂²_iiϕ₁

p+∂_ilogp∂_iϕ₁

p

(8)

Since we haveht≤21+t⁻¹h_∞, takingϕ₁=gin the above inequality we deduce that

h◦g∂_ig²

p≤2∂_ig²/1+g

ph

∞

≤6∂²_iig

p+∂_ilogp∂_ig

p h

∞

The previous inequalities imply there exists a constantcdepending only onh anddsuch thatϕ_D≤cg_D. 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⊂E_T, a compact set such that cap_DK> 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×R^d by ut x = N_{t x}^∗∩K =

⵰ (∈ 0∞). With the convention 0∞ = 0, the function uψ⁴ is bounded nonnegative and of class C^∞ on R×R^d. Let B_t 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 B_t =uψ⁴00 +_t

0 ∂_tuψ⁴s B_sds +_t

0

2uψ⁴s B_sds+_t

0 ∇uψ⁴s B_sdB_s

Consider the stopping timeT_a =T∧inft >0B_t≥a. We can then apply the optional stopping theorem at timeT_a and get

E0uψ⁴T_a B_Ta

=u00 +E0

_Ta

0 ∂_tuψ⁴s B_sds+E0

_Ta

0

2uψ⁴s B_sds

=u00 +E0

T_a 0

2u²ψ⁴+4uψ³∂_tψ+4∇u∇ψψ³ +6uψ²∇ψ∇ψ +2uψ³ψ

s B_sds 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,

u00 +2uψ²1_ET²

p

=E0uψ⁴T B_T

− E_Tp

4uψ³∂_tψ+4∇u∇ψψ³+6uψ²∇ψ∇ψ +2uψ³ψ

SinceK⊂E_T, we deduce thatut x =0 fort≥T. Thus we have u00 +2uψ²²

p

= − 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

− Epuψ³∂_tψ≤uψ²

p∂_tϕ

p

− Epuψ³∂²_iiψ≤uψ²

p∂²_iiϕ

p

and

− Epuψ²∇ψ∇ψ ≤uψ²

p

d i=1

∂_iϕ²

p

≤2uψ²

p

d i=1

∂_iϕ²/1+ϕ

p

≤6uψ²

p

d i=1

∂²_iiϕ

p+∂_ilogp∂_iϕ

p

where we have used (8) withϕ₁=ϕfor the last inequality. Now an integration by parts and the Cauchy–Schwarz inequality give

− Epψ³∇u∇ψ

= Epuψ²

ψ∇logp∇ψ +3∇ψ∇ψ +ψψ

≤uψ²

p

d i=1

∂_ilogp∂_iϕ

p+3∂_iϕ²

p+∂²_iiϕ

p

≤19uψ²

p

d i=1

∂_ilogp∂_iϕ

p+∂²_iiϕ

p

where we have used again (8) for the last inequality. Taking those results together, we deduce from (9) that

u00 +2uψ²²

p≤c₃uψ²

pϕ

D

where the constantc₃depends only ond. Sinceuψ²_pis finite (recalluψis bounded, and zero onT∞ ×R^d), this implies thatuψ²_p ≤c₃ϕ_D and henceu00 ≤c²₃ϕ²_D. This last inequality and the definition ofϕimply that

N₀₀

^∗∩K=⵰

=u00 ≤c²₃γcap_DK ≤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⊂E_T. We consider the probability measure µ defined on ₀₀by

µdw =

Eνdt dxP^{t x}₀ dw

where P^{t x}0 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 P^{t x}₀ can also be viewed as a probability measure on the canonical space CR⁺R^d endowed with the filtration_tgenerated by the coordinate mappings. Let P0 be the law on the canonical space of the standard Brownian motion. Fors∈ 0 t, we have

P^{t x}₀ dw_s= pt−s x−ws

pt x P₀dw_s

We consider the energy of µ with respect to the process W_s (see [11] for a precise description and definition). Thanks to [11], Proposition 1.1, we have

µ =2∞ 0 dsP0

Eνdt dxpt−s x−ws/pt x ₂

=2Iν Now, using [5], 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∞

0 FW_sdA_s

=

µdwFw 2. N₀₀A²_∞ =2µ.

We deduce from (1) that the additive functional increases only when Wˆ_s ∈ 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

N₀₀

^∗∩K=⵰

≥4⁻¹ CapK =4⁻¹ capK

ProofofTheorem 1. Notice the application defined onEbyTA = N₀₀^∗∩A =⵰ forA∈ E is a Choquet capacity (see [4], Th´eor`eme 1).

Since the capacity cap is an inner capacity (see [13], 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 cap_d−2 the usual Newtonian (logarithmic if d=2) capacity inR^d,

cap_d−2A =

inf R^d×R^dρdxρdyh_d−2x−y ₋₁

with h_γr =r^−γ if γ > 0 and h₀r =log₊1/r. The infimum is taken over all probability measuresρonR^d such thatρA =1. LetB0 hbe the open ball ofR^d 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

aµ1cap_d−2A ≤P0 µ

suppX₁∩A=⵰

≤bµ1cap_d−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 onR^dsuch thatρA =1. We get

Iν = 01×R^dds 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

a₁Iν ≤

A×Aρdxρdxh_d−2x−x ≤b₁Iν This implies that for any Borel setA⊂B02,

a₁cap_d−2A ≤cap1 ×A ≤b₁cap_d−2A

Since the capacity cap_d−2is invariant by translation, we get that for any Borel setA⊂B01, for anyx∈B01,

a₁cap_d−2A ≤cap1 ×A_x ≤b₁cap_d−2A whereA_x = yy−x∈A. We deduce from Theorem 1 that

4⁻¹a₁cap_d−2A ≤N_{0 x}

^∗∩ 1 ×A =⵰

≤C₀b₁cap_d−2A Since X₁=

i∈IY₁Wⁱ, where

i∈Iδ_Wi is a Poisson measure onCR⁺₀ with intensity

µdxN_{0 x}·, we have P0µ

suppX₁∩A=⵰

=1−exp

−

µdxN_{0 x}

suppY₁∩A=⵰ Notice thatN_{0 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 [15]. Two random Borel sets F₁ andF₂ inR^d 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 motionB_t t≥0started with the lawπ. Ford≥3 the range of Brownian motion is defined by_B= B_t t≥0inR^d. Ford=2, we also denote by_B the set_B= B_t t∈ 0 ξ, whereξis an exponential random variable of parameter 1 independent ofB_t 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

aµ1Pπ_B∩A=⵰ ≤P0µ

suppX₁∩A=⵰

≤bµ1Pπ_B∩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,

a₂cap_d−2A ≤Pπ

_B∩A=⵰

≤b₂cap_d−2A (see, e.g., [15], Proposition 3.2, ford≥3 and [14] ford=2). ✷

APPENDIX

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 n_d ∈ N^d, we set n = _d

i=1n_i,n!=_d

i=1n_i! and

n≥0 =_d

i=1_∞

ni=0. Forj∈ 1 d, letδjbe the element of N^d such that δj_i =δ_{i j}, the standard Kronecker symbol. If z= z₁ z_dis an element ofR^d, then we setzⁿ=_d

i=1zⁿ_iⁱ. Let··be the Euclidean product onR^d.

The function ϕz =exp−z²−2z x/2 is an entire function defined onR^d. We have

exp

−z²−2x z/2

=

n≥0

1

n!zⁿHe_nx (10)

where thenth termHe_nxis a polynomial ofx₁ x_dof degreencalled thenth Hermite polynomial. Those polynomials can easily be expressed with the usual one-dimensional Hermite polynomials He¹_k k ∈ N: He_nx = _d

i=1He¹nix_i, wherex= x₁ x_d.

Now let us recall some basic properties of the polynomialsHe_n. The follow- ing recurrence formula can be deduced from (10) by differentiating w.r.t. z_i: for alln∈N^d such thatn_i>0,

He_nx =x_iHe_n−δix − n_i−1He_n−2δix ∀x∈R^d (11)

where by conventionHe_n−kδi =0 ifn_i−k <0. The differential formula can be deduced from (10) by differentiating w.r.t.x_i: for alln∈N^d,

∂_iHe_n=n_iHe_n−δi (12)

We also recall the upper bound for He_n(see [1], 22.14.17); there exists a uni- versal constant 1< c₀<2 such that

He_nx≤c^d₀√

n! expx²/4

for allx∈R^d n∈N^d (13)

Using the definition of the Hermite polynomials, it is also easy to prove that dx pt xHe_nx/√

tHe_mx/√

t =n!^d

i=1

δ_{ni mi} (14)

It is also well known that the Hermite polynomials are a complete orthogonal system in L²R^dexp

−x²/2

dx. Finally, standard arguments on Hilbert spaces show that iff∈L²pthen

ft x =

n≥0

f_nt x =

n≥0

He_nx/√

tg_nt where g_nt = n!⁻¹

dx pt xHe_nx/√

tft xandg_n∈L²0∞. Fur- thermore, we have

f²

p=

n≥0

n!∞

0 dt g_nt² (15)

SinceC^∞₀ 0∞is dense inL²0∞, it is clear that the set of functions ft x =

n≥0He_nx/√

tg_nt whereg_n∈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, 1_{2 i} and 1_{3 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, 1_{2 i} and1_{3 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∈N^d, andt x ∈E, we set h_{n g}t x =He_n

x/√ t

t⁻_n/2 gt Let us prove that

1h_{n g} =h_{n G} (16)

Forz∈R, we introduce the function _{g z} defined onEby

g zt x =

n≥0

1

n!zⁿh_{n g}t x =gtexp

−z²−2x z/2t

Then we have

1 _{g z}t x = 1 pt x

_t

0 ds

dy pt−s x−yps y _{g z}s y

= 1 pt x

_t

0 ds dy p

st−s

t y−sx

t −t−sz t

×pt z−xgs

=exp

−z²−2x z/2t ^t

0 ds gs = _{G z}t x

Using (13), the Chapman–Kolmogorov equation, and that suppg ∈ α β, we get

1h_{n g}t x ≤√

n!c^d₀pt x⁻¹

E ds dy pt−s x−yps y

×expy²/4s

s⁻_n/2gs

≤√ n!√

2c₀^dpt x⁻¹_t

0 ds pt+s xs⁻_n/2gs

≤√ n!√

2c₀^dexpx²/4t ^t

0 ds s⁻_n/2gs

≤√ n!√

2c₀^dexpx²/4t g

∞β−αα⁻_n/2

The radius of the seriesa^kk!α^k−1/2 is infinite. Thus for anyt x ∈Ethe seriesn!⁻¹zⁿ1h_ngt xare convergent. Fubini’s theorem implies that

n≥0

1

n!zⁿ1h_{n g}t x =1

n≥0

1

n!zⁿh_{n g}

t x = _{G z}t x

Hence the two series n!⁻¹zⁿ1h_{n g}t x and n!⁻¹zⁿh_{n G}t x agree.

Since their radius of convergence is positive (in fact infinite), we get that (16) is true.

Let us prove that1_{2 i} has a bounded extension onL²p. We deduce from (12) that

1_{2 i}h_ngt x = 1

2∂²_{i i}1h_{n g}t x

= 1

2tn_in_i−1He_n−2δix/√

tt⁻_n/2t

0 ds gs (17)

Let us introducef∈, that is, fort x ∈E,ft x =

n≥0He_nx/√

tg_nt, where g_n ∈ C^∞₀ 0∞ and g_n =0 except for a finite number of terms. By linearity, we have

1_{2 i}ft x =

n≥0

2⁻¹n_in_i−1He_n−2δix/√

tt⁻¹⁻_n/2t

0 ds s_n/2g_ns Thus, using (14), we have

1_{2 i}f²

p=

n≥0

n−2δi!4⁻¹n²_in_i−1²

×∞

0 dt t⁻²⁻_n^t

0 ds s_n/2 g_ns

₂

≤

n≥0

n!n_in_i−1 4

4 n+1²

_∞

0 dt g_nt²

≤f²

p

where we used the Hardy inequality; fork >−1, _∞

0 dt t^−2−k_t

0 s^k/2hsds ₂

≤ 4 k+1²

_∞

0 dt ht²

for the first inequality and (15) for the second one. This means that 1_{2 i}, defined on , can be uniquely extended into a bounded operator 1˜_{2 i} from L²pinto itself. The above inequality implies ˜1_{2 ip}≤1.

Fori∈ 1 d, we set1_{4 i}=1_{3 i}+21_{2 i}. Using (12) and (11), we deduce from (16) that

1_{4 i}h_ngt x

=

−x_i/√

t∂_iHe_nx/√

t +n_in_i−1He_n−2δix/√ t

×t⁻¹⁻_n/2t

0 ds gs

= −n_iHe_nx/√

tt⁻¹⁻_n/2t

0 ds gs (18)

Arguing as above, we get, for f∈, 1_{4 i}f²

p=

n≥0

n!n²_i∞

0 dt t⁻²⁻_n^t

0 ds s_n/2 g_ns

₂

≤

n≥0

n!n²_i 4 n+1²

_∞

0 dt g_nt²

≤4f²

p

Thus the operators1_{4 i} and1_{3 i}, defined on, can be uniquely extended in bounded operators1˜_{4 i} and1˜_{3 i} fromL²pinto itself. Furthermore we have ˜1_{4 ip}≤2 and ˜1_{3 ip}≤ ˜1_{4 ip}+2 ˜1_{2 ip}≤4.

The proof concerning1₁easily follows from the previous results. From (16), we get

1₁h_{n g}t x

=h_{n g}t x −1 2

nHe_nx/√ t +^d

i=1

x_i

√t∂_iHe_nx/√ t

t⁻¹⁻_n/2t

0 ds gs Then using (17) and (18), we get

1₁h_{n g} =

I+¹₂^d

i=1

1_{4 i}+1_{3 i}

h_{n g} =

I+^d

i=1

1_{4 i}−1_{2 i} h_{n g} This means that1₁=I+_d

i=11_{4 i}−1_{2 i}on. Hence1₁can be uniquely ex- tended in a bounded operator1˜₁fromL²pinto itself and1˜₁=I+_d

i=1 ˜1_{4 i}− 1˜_{2 i}. We deduce that ˜1_1p≤1+3d.