CHARACTERIZATION OF *G-REGULARITY FOR*
SUPER-BROWNIAN MOTION AND CONSEQUENCES
FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

By Jean-Fran ¸cois Delmas^{1} 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 massδ_{0}hits 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∂_{t}u+ ^{1}_{2}u=2u^{2}which
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^{d}if 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 +^{1}_{2}u=2u^{2} 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/2}exp−x^{2}/2t ift x ∈E

0 ift x ∈ −∞0 ×R^{d}

Received May 1998; revised September 1998.

1Supported by NSF Grant DMS-97-01755.

*AMS*1991*subject 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^{d}be 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≥r}ws∧ζ_{w}−w^{}s∧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_{W}_{s}_{} 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_{s}^{}t =W_{s}tforr≤t≤ms s^{} =inf_{v∈s s}^{}_{}ζ_{v}.

W_{s}^{}ms s^{}+t−W_{s}^{}ms s^{}, 0≤t≤ζ_{s}^{}−ms s^{}is a Brownian motion
inR^{d}independent 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_{w}the law ofWstarted at
w∈_{r x}. The trivial pathx_{r}such thatζ_{x}_{r}_{}=r,x_{r}r =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_{s}tr < t≤ζ_{s}0< s < σ =

ζ_{s}Wˆ_{s}0< 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_{f}_{f}
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}_{s}ϕ ˆW_{s}
Let _{r} =

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

i∈Iδ_{W}^{i} be a
Poisson measure onCR^{+}_{r}with intensity

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

i∈IY_{t}W^{i} 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^{∗}W^{i} ∩ t∞ ×R^{d} is nonempty.

We consider the graph ofX, X =

ε>r t≥εt ×suppX_{t}

=

i∈I

^{∗}W^{i}

where A¯ denotes the closure of A. A set A⊂R×R^{d}is 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},

cap^{}A =

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^{d}such that
νA =1. Dynkin proved (see Theorem 3.2 in [7]) that A∈R×R^{d}is G-
polar if and only if cap^{}A =0. (We have cap^{}A =0⇔N_{r x}^{∗}∩A=⵰ =0
for allr x ∈R×R^{d}.) It is easy to check that ifA⊂Eis a compact set, then

cap^{}A =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 constant*C_{0}*such that for any* A∈E,
4^{−1}capA ≤N_{0}_{0}^{∗}∩A=⵰ ≤C_{0}capA

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_{M}of ∂u/∂t+^{1}_{2}u=
2u^{2} inQ. Hence, we deduce from Theorem 1 that if00 ∈Q, thenu_{M}00
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^{d}isG-
regular forA^{c}ifPr 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_{s}Wˆ_{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 x*is*G-regular forA^{c}*;*
(ii) N_{r x}^{∗}∩A=⵰ = ∞;

(iii) _{x}_{r}*-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. *Let*Q*be a domain in*E*such that*00 ∈∂Q. The following
*three conditions are equivalent:*

(i) 00*is*G-regular for Q;

(ii) capQ^{c}∩E = ∞;

(iii) *there exists a nonnegative solution of*∂u/∂t+^{1}_{2}u=2u^{2}*in*Q*such 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 X_{1}, 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^{∞}_{0}S the set of all functions
of classC^{∞} defined onSwith compact support. Iffis a measurable function
defined onSthenf_{∞}=sup_{x∈S}fs. We consider the Hilbert spaceL^{2}p =
f∈E f_{p}<∞, where f^{2}_{p}=

Edt dx pt xft x^{2}.

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^{−1}p∗ 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 =inff^{2}

pf≥0 f∈L^{2}p 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^{∞}_{0}E
with respect to the norm · _{D}, defined by

ϕ^{2}

D =∂_{t}ϕ^{2}

p+^{d}

i=1

∂_{i}logp∂_{i}ϕ^{2}

p+^{d}

i=1

∂^{2}_{ii}ϕ^{2}

p ϕ∈C^{∞}_{0} E
with the usual notations ∂_{t}gt x = ∂g/∂tt x, ∂_{i}gt x = ∂g/∂x_{i}t x
for x= x_{1} 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_{D}K =infϕ^{2}

Dϕ∈C^{∞}_{0}E ϕ≥0 ϕ≥1 onK

=infϕ^{2}

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

cap_{D}G =sup

cap_{D}KK⊂G Kcompact
(1)

and, for any analytic setA⊂E,
cap_{D}A =inf

cap_{D}GA⊂G G open
Notice the definition is consistent (see [2], for example).

Proposition 4. *There exists a constant*C*such that for any set*A⊂E,
CapA ≤cap_{D}A ≤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}−^{1}_{2}. 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 = δ_{00}, where δ_{00} is the Dirac mass at
00 ∈R×R^{d}. Then we havep∗ Hpϕ = Hp ∗ pϕ =pϕ. The function
f=p^{−1}Hpϕ=Hϕ − ∇logp∇ϕis nonnegative and

1f =p^{−1}p∗Hpϕ ≥p^{−1}p∗Hpϕ =ϕ
Thus we have1f ≥1 onK. We also have

f

p ≤∂_{t}ϕ−^{1}_{2}ϕ− ∇logp∇ϕ

p ≤ϕ

D

Hence we have CapK ≤ ϕ^{2}_{D}. The first inequality follows by taking a se-
quenceϕ_{n}such thatϕ_{n}^{2}_{D} converges to cap_{D}K.

To prove the other inequality, let us consider a nonnegative functionf_{1} ∈
L^{2}p, such that 1f_{1} ≥1 onK. Notice this impliesf_{1}_{p} >0. Let δ >0.

It is easy to construct a functionε∈L^{2}psuch thatε >0 onEandε_{p}≤
δf_{1}_{p}. We setf_{2}=f_{1}+ε. Since the function1f_{2}is lower semicontinuous,
the sett x ∈E1f_{2}>1is open and it also containsK. It is then obvious

that for δ^{}>0 small enough, if we setf_{3}t x =f_{2}t x1_{δ}_{}_{<t<δ}−1_{x}_{<δ}^{−1}_{} for
t x ∈E, we get1f_{3}>1 on an open set containing K. Let us introduce a
nonnegative function h∈ C^{∞}_{0}E such that _{E}ht xdt dx =1. For θ > 0,
we write h_{θ}t x =θ^{−d−1}ht/θ x/θ. Now using the uniform continuity ofp
onδ^{}/2∞×R^{d}, it is easy to see that iff=h_{θ}∗f_{3}, then1f>1 on an open
set containingKfor θsmall enough. The function fis nonnegative, belongs
toC^{∞}_{0}E and the function1fis of classC^{∞}. We can chooseδandθsmall
enough so thatf_{p}≤2f_{1}_{p}.

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

We define α_{n}t = αt/n and ξ_{n}x = ξx/n. The function ϕ_{n} = α_{n}ξ_{n}1f
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 L^{2}p
into itself, we denote by M_{p}=supMf_{p}f∈L^{2}pf_{p} =1 its
norm. We define the operator 1_{0}: for f ∈ E nonnegative, 1_{0}ft x =
t^{−1}1ft x,t x ∈E. ForT >0, let us introduceE_{T}= 0 T ×R^{d}.

Lemma 5. *The operators*1_{E}_{T}1*and*1_{0}*are bounded operators from* L^{2}p
*into itself. Furthermore, we have*1_{E}_{T}1_{p}≤T/√

2*and*1_{0}_{p}≤2.

Proof. Let f∈L^{2}p. We have
1_{0}f^{2}

p=

Edt dx t^{−2}pt x^{−1}

Eds dy pt−s x−yps yfs y
_{2}

≤ Edt dx t^{−2}pt x^{−1}

Eds dy pt−s x−yps y

×

s^{−1/2}1_{s≤t}_{1/2}

s^{1/2}fs y^{2}1_{s≤t}_{1/2}_{2}

≤ Edt dx t^{−2}pt x^{−1}

× Eds^{}dy^{}pt−s^{} x−y^{}ps^{} y^{}s^{−1/2}1_{s}≤t

× Eds dy pt−s x−yps ys^{1/2}fs y^{2}1_{s≤t}

≤ Edt dx t^{−2}_{t}

0 ds^{}s^{−1/2}

× Eds dy pt−s x−yps ys^{1/2}fs y^{2}1_{s≤t}

=2

Edt dx t^{−2}t^{1/2}

Eds dy pt−s x−yps ys^{1/2}fs y^{2}1_{s≤t}

≤2

Eds dy ps yfs y^{2}s^{1/2}_{∞}

s t^{−3/2}dt≤4f^{2}

p

where we used the Cauchy–Schwarz inequality for the second inequality.

Hence the operator 1_{0} is a bounded operator from L^{2}p into itself. And

we have 1_{0} _{p}≤ 2. The operator 1_{E}_{T}1 can be handled in a very similar
way. *✷*

Lemma 6. *The operators defined on*C^{∞}_{0} E*by*g∈C^{∞}_{0} E
1_{1}g =∂_{t}1g
*for*i∈ 1 d 1_{2 i}g = ^{1}_{2}∂^{2}_{ii}1g
*and for*i∈ 1 d 1_{3 i}g =∂_{i}logp∂_{i}1g

*can be uniquely extended into bounded operators from*L^{2}p*into itself. More-*
*over we have*

1_{1}_{p}≤1+3d
(2)

*for*i∈ 1 d 1_{2 i}_{p}≤1
(3)

*for*i∈ 1 d 1_{3 i}_{p}≤4
(4)

The proof of this lemma is given in the Appendix.

We now boundϕ_{n}_{D}. Lemma 5 provides an upper bound for∂_{t}ϕ_{n}_{p}
∂_{t}ϕ_{n}

p≤∂_{t}α_{n}

∞1_{E}_{n}1f

p+1_{1}f

p

≤∂_{t}α

∞2^{−1/2}+ 1_{1}_{p}f

p (5)

Using Lemma 5 we derive an upper bound for_{d}

i=1∂_{i}logp∂_{i}ϕ_{n}_{p}
d

i=1

∂_{i}logp∂_{i}ϕ_{n}

p≤^{d}

i=1

1_{3 i}f

p+sup

x∈R^{d}

x_{i}∂_{i}ξx1_{0}f

p

≤^{d}

i=1

1_{3 i}_{p} +sup

x∈R^{d}

x_{i}∂_{i}ξx1_{0}_{p}f

p (6)

In order to give an upper bound for _{d}

i=1∂^{2}_{ii}ϕ_{n}

p, we need an intermediary lemma.

Lemma 7. *There exists a constant* c_{1} *(depending on* ξ) such that for all
n≥1,g∈C^{∞}_{0}E,i∈ 1 d,

1_{E}_{n}∂_{i}ξ_{n}∂_{i}1g

p≤c_{1}n^{−1/2}g

p

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

1_{E}_{n}∂_{i}ξ_{n}∂_{i}1g^{2}

p = −

Ep1_{E}_{n}1g∂_{i}ξ_{n}^{2}∂^{2}_{ii}1g

− Ep1_{E}_{n}1g∂_{i}ξ_{n}^{2}∂_{i}1g∂_{i}logp

−2

Ep1_{E}_{n}1g∂_{i}ξ_{n}∂_{i}1g∂^{2}_{ii}ξ_{n}

≤∂_{i}ξ_{n}^{2}

∞1_{E}_{n}1g

p∂^{2}_{ii}1g

p

+∂_{i}ξ_{n}^{2}

∞1_{E}_{n}1g

p∂_{i}1g∂_{i}logp

p

+2∂^{2}_{ii}ξ_{n}

∞1_{E}_{n}1g

p1_{E}_{n}∂_{i}ξ_{n}∂_{i}1g

p

≤2^{−1/2}n^{−1}

21_{2 i}_{p}+ 1_{3 i}_{p}∂_{i}ξ^{2}

∞g^{2}

p

+2^{1/2}n^{−1}∂^{2}_{ii}ξ

∞g

p1_{E}_{n}∂_{i}ξ_{n}∂_{i}1g

p

Notice that ifa b care positive thena^{2}≤c^{2}+baimpliesa≤c+b. Thus we
get 1_{E}_{n}∂_{i}ξ_{n}∂_{i}1g

p

≤2^{−1/4}n^{−1/2}

21_{2 i}_{p}+ 1_{3 i}_{p}_{1/2}∂_{i}ξ

∞g

p

+2^{1/2}n^{−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}_{ii}ϕ_{n}

p≤^{d}

i=1

21_{2 i}f

p+∂^{2}_{ii}ξ_{n}

∞1_{E}_{n}1f

p

+21_{E}_{n}∂_{i}ξ_{n}∂_{i}1f

p

≤^{d}

i=1

21_{2 i}_{p}+2^{−1/2}n^{−1}∂^{2}_{ii}ξ

∞+2c_{1}n^{−1/2}f

p (7)

Then we deduce from (5), (6), (7) and Lemma 6 that there exists a constantc_{2}
independent of fandn≥1 such that

ϕ_{n}

D≤c_{2}f

p

Thus we haveϕ_{n}_{D}≤2c_{2}f_{1}_{p}. The second inequality of the proposition is
then obvious withC=4c^{2}_{2}. *✷*

We shall need the following lemma.

Lemma 8. *For any compact set* K ⊂ E *with* cap_{D}K > 0, there exists
ϕ∈C^{∞}_{0}E*such that:*

(i) 0≤ϕ≤1;

(ii) ϕ=1*on a neighborhood of*K;

(iii) ϕ^{2}_{D}≤γcap_{D}K,

*where*γ*is a constant independent of*K*and*ϕ.

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_{D}K > 0, there exists g ∈ C^{∞}_{0}E such that
g ≥0, g≥ 1 in a neighborhood ofK, and 2 cap_{D}K ≥ g^{2}_{D}. Letϕ=h◦g.

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

p≤h^{}

∞∂_{t}g

p
∂_{i}logp∂_{i}ϕ

p≤h^{}

∞∂_{i}logp∂_{i}g
∂^{2}_{ii}ϕ p

p≤h^{}

∞∂^{2}_{ii}g

p+h^{}◦g∂_{i}g^{2}

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 ∂_{i}ϕ_{1}^{2}/1+ϕ_{1}_{p},
whereϕ_{1}∈C^{∞}_{0}Eis a nonnegative function. An integration by parts and the
Cauchy–Schwarz inequality give

Ep ∂_{i}ϕ_{1}^{4}
1+ϕ_{1}^{2} =3

Ep∂^{2}_{ii}ϕ_{1}∂_{i}ϕ_{1}^{2}
1+ϕ_{1} +

Ep∂_{i}logp∂_{i}ϕ_{1}∂_{i}ϕ_{1}^{2}
1+ϕ_{1}

≤

3∂^{2}_{ii}ϕ_{1}

p+∂_{i}logp∂_{i}ϕ_{1}

p

∂_{i}ϕ_{1}^{2}/1+ϕ_{1}

p Thus we get

∂_{i}ϕ_{1}^{2}/1+ϕ_{1}

p≤3∂^{2}_{ii}ϕ_{1}

p+∂_{i}logp∂_{i}ϕ_{1}

p

(8)

Since we haveh^{}t≤21+t^{−1}h^{}_{∞}, takingϕ_{1}=gin the above inequality
we deduce that

h^{}◦g∂_{i}g^{2}

p≤2∂_{i}g^{2}/1+g

ph^{}

∞

≤6∂^{2}_{ii}g

p+∂_{i}logp∂_{i}g

p h^{}

∞

The previous inequalities imply there exists a constantcdepending only onh
anddsuch thatϕ_{D}≤cg_{D}. Thus (iii) holds withγ=2c^{2}. *✷*

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_{D}K> 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 uψ^{4} 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.,

uψ^{4}t B_{t} =uψ^{4}00 +_{t}

0 ∂_{t}uψ^{4}s B_{s}ds
+_{t}

0

2uψ^{4}s B_{s}ds+_{t}

0 ∇uψ^{4}s B_{s}dB_{s}

Consider the stopping timeT_{a} =T∧inft >0B_{t}≥a. We can then apply
the optional stopping theorem at timeT_{a} and get

E0uψ^{4}T_{a} B_{T}_{a}

=u00 +E0

_{T}_{a}

0 ∂_{t}uψ^{4}s B_{s}ds+E0

_{T}_{a}

0

2uψ^{4}s B_{s}ds

=u00 +E0

T_{a}
0

2u^{2}ψ^{4}+4uψ^{3}∂_{t}ψ+4∇u∇ψψ^{3}
+6uψ^{2}∇ψ∇ψ +2uψ^{3}ψ

s B_{s}ds
We have used that ∂_{t}u+^{1}_{2}u=2u^{2}to 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 +2uψ^{2}1_{E}_{T}^{2}

p

=E0uψ^{4}T B_{T}

− E_{T}p

4uψ^{3}∂_{t}ψ+4∇u∇ψψ^{3}+6uψ^{2}∇ψ∇ψ +2uψ^{3}ψ

SinceK⊂E_{T}, we deduce thatut x =0 fort≥T. Thus we have
u00 +2uψ^{2}^{2}

p

= − Ep

4uψ^{3}∂_{t}ψ+4∇u∇ψψ^{3}+6uψ^{2}∇ψ∇ψ +2uψ^{3}ψ
(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

− Epuψ^{3}∂_{t}ψ≤uψ^{2}

p∂_{t}ϕ

p

− Epuψ^{3}∂^{2}_{ii}ψ≤uψ^{2}

p∂^{2}_{ii}ϕ

p

and

− Epuψ^{2}∇ψ∇ψ ≤uψ^{2}

p

d i=1

∂_{i}ϕ^{2}

p

≤2uψ^{2}

p

d i=1

∂_{i}ϕ^{2}/1+ϕ

p

≤6uψ^{2}

p

d i=1

∂^{2}_{ii}ϕ

p+∂_{i}logp∂_{i}ϕ

p

where we have used (8) withϕ_{1}=ϕfor the last inequality. Now an integration
by parts and the Cauchy–Schwarz inequality give

− Epψ^{3}∇u∇ψ

= Epuψ^{2}

ψ∇logp∇ψ +3∇ψ∇ψ +ψψ

≤uψ^{2}

p

d i=1

∂_{i}logp∂_{i}ϕ

p+3∂_{i}ϕ^{2}

p+∂^{2}_{ii}ϕ

p

≤19uψ^{2}

p

d i=1

∂_{i}logp∂_{i}ϕ

p+∂^{2}_{ii}ϕ

p

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

u00 +2uψ^{2}^{2}

p≤c_{3}uψ^{2}

pϕ

D

where the constantc_{3}depends only ond. Sinceuψ^{2}_{p}is finite (recalluψis
bounded, and zero onT∞ ×R^{d}), this implies thatuψ^{2}_{p} ≤c_{3}ϕ_{D} and
henceu00 ≤c^{2}_{3}ϕ^{2}_{D}. This last inequality and the definition ofϕimply that

N_{00}

^{∗}∩K=⵰

=u00 ≤c^{2}_{3}γcap_{D}K ≤c^{2}_{3}γCCapK =c^{2}_{3}γCcapK
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
_{0}_{0}by

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 on_{00}^{∗} , the set of nontrivial path in_{00} (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_{t}generated
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 P_{0}dw_{}_{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 on_{00}^{∗} ,
N_{0}_{0}∞

0 FW_{s}dA_{s}

=

µdwFw
2. N_{00}A^{2}_{∞} =2µ.

We deduce from (1) that the additive functional increases only when Wˆ_{s} ∈
suppν⊂K. Therefore, using the Cauchy–Schwarz inequality, we get

N_{0}_{0}^{∗}∩K=⵰ ≥N_{00}A_{∞}>0 ≥N_{0}A_{∞}^{2}/N_{0}A^{2}_{∞}

We get N_{00}^{∗}∩K=⵰ ≥ 4Iν^{−1}. Since the above inequality is true for
any probabilityν onK, we get that

N_{0}_{0}

^{∗}∩K=⵰

≥4^{−1} CapK =4^{−1} capK

ProofofTheorem 1. Notice the application defined onEbyTA =
N_{00}^{∗}∩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 (withC_{0}=c^{2}_{3}γC) and the above result. *✷*

5. Brownian range and support of *X*_{1}. In this section, we first give an
estimate for the hitting probabilities of the support ofX_{1}. 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−2}A =

inf R^{d}×R^{d}ρdxρdyh_{d−2}x−y
_{−1}

with h_{γ}r =r^{−γ} if γ > 0 and h_{0}r =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. *Let*M >0. There exist two positive constantsa*and*b*such*
*that for any Borel set* A ⊂B01, for any finite measureµ *on*B01, with
µ1 ≤M, we have

aµ1cap_{d−2}A ≤P0 µ

suppX_{1}∩A=⵰

≤bµ1cap_{d−2}A
Proof. Let A⊂B02be a Borel set. Letν be a probability measure on
Esuch thatν1×A =1. Then we haveν=δ_{1}×ρ, whereρis a probability
measure onR^{d}such thatρA =1. We get

Iν = 01×R^{d}ds dy ps y

× A×Aρdxρdx^{}p1−s x−yp1−s x^{}−yp1 x^{−1}p1 x^{}^{−1}

Since x x^{} are inB02and sinces∈ 01, it is easy to see there exist two
positive constantsa_{1}andb_{1} (independent ofAandρ) such that

a_{1}Iν ≤

A×Aρdxρdx^{}h_{d−2}x−x^{} ≤b_{1}Iν
This implies that for any Borel setA⊂B02,

a_{1}cap_{d−2}A ≤cap1 ×A ≤b_{1}cap_{d−2}A

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

a_{1}cap_{d−2}A ≤cap1 ×A_{x} ≤b_{1}cap_{d−2}A
whereA_{x} = yy−x∈A. We deduce from Theorem 1 that

4^{−1}a_{1}cap_{d−2}A ≤N_{0 x}

^{∗}∩ 1 ×A =⵰

≤C_{0}b_{1}cap_{d−2}A
Since X_{1}=

i∈IY_{1}W^{i}, where

i∈Iδ_{W}i is a Poisson measure onCR^{+}_{0}
with intensity

µdxN_{0 x}·, we have
P0µ

suppX_{1}∩A=⵰

=1−exp

−

µdxN_{0 x}

suppY_{1}∩A=⵰
Notice thatN_{0 x}-a.e.,1 × suppY_{1}∩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 F_{1} andF_{2} inR^{d} are intersection-equivalent in
an open setU, if there exist positive constantsaandbsuch that, for any Borel
setA⊂U,

aPA∩F_{1} ≤PA∩F_{2} ≤bPA∩F_{1}

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. *Let*M >0. There exist two positive constantsa*and*b*such*
*that for any Borel set*A ⊂B01, for any absolutely continuous probability
*measure*π *on*B01*with density bounded by*M, for any finite measureµ*on*
B01, withµ1 ≤M, we have

aµ1Pπ_{B}∩A=⵰ ≤P0µ

suppX_{1}∩A=⵰

≤bµ1Pπ_{B}∩A=⵰
Proof. This is a consequence of Proposition 9 and the fact that there
exist two positive constantsa_{2}andb_{2}such that for any Borel setA⊂B01,

for any absolutely continuous probability measureπ onB01 with density bounded byM,

a_{2}cap_{d−2}A ≤Pπ

_{B}∩A=⵰

≤b_{2}cap_{d−2}A
(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 = n_{1} 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= z_{1} z_{d}is an element ofR^{d}, then we setz^{n}=_{d}

i=1z^{n}_{i}^{i}. Let··be the
Euclidean product onR^{d}.

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

exp

−z^{2}−2x z/2

=

n≥0

1

n!z^{n}He_{n}x
(10)

where thenth termHe_{n}xis a polynomial ofx_{1} x_{d}of degreencalled
thenth Hermite polynomial. Those polynomials can easily be expressed with
the usual one-dimensional Hermite polynomials He^{1}_{k} k ∈ N: He_{n}x =
_{d}

i=1He^{1}nix_{i}, wherex= x_{1} 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_{n}x =x_{i}He_{n−δi}x − n_{i}−1He_{n−2δi}x ∀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},

∂_{i}He_{n}=n_{i}He_{n−δi}
(12)

We also recall the upper bound for He_{n}(see [1], 22.14.17); there exists a uni-
versal constant 1< c_{0}<2 such that

He_{n}x≤c^{d}_{0}√

n! expx^{2}/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_{n}x/√

tHe_{m}x/√

t =n!^{d}

i=1

δ_{n}_{i}_{ m}_{i}
(14)

It is also well known that the Hermite polynomials are a complete orthogonal
system in L^{2}R^{d}exp

−x^{2}/2

dx. Finally, standard arguments on Hilbert
spaces show that iff∈L^{2}pthen

ft x =

n≥0

f_{n}t x =

n≥0

He_{n}x/√

tg_{n}t
where g_{n}t = n!^{−1}

dx pt xHe_{n}x/√

tft xandg_{n}∈L^{2}0∞. Fur-
thermore, we have

f^{2}

p=

n≥0

n!∞

0 dt g_{n}t^{2}
(15)

SinceC^{∞}_{0} 0∞is dense inL^{2}0∞, it is clear that the set of functions
ft x =

n≥0He_{n}x/√

tg_{n}t whereg_{n}∈C^{∞}_{0} 0∞ is nonzero for a finite
number of indices n, is dense inL^{2}p.

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 L^{2}p of the operators1_{1}, 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 1_{1}, 1_{2 i} and1_{3 i}, which are also defined on
C^{∞}_{0}E, 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!z^{n}h_{n g}t x =gtexp

−z^{2}−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

−z^{2}−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^{d}_{0}pt x^{−1}

E ds dy pt−s x−yps y

×expy^{2}/4s

s^{−}_{n}_{/2}gs

≤√ n!√

2c_{0}^{d}pt x^{−1}_{t}

0 ds pt+s xs^{−}_{n}_{/2}gs

≤√ n!√

2c_{0}^{d}expx^{2}/4t ^{t}

0 ds s^{−}_{n}_{/2}gs

≤√ n!√

2c_{0}^{d}expx^{2}/4t g

∞β−αα^{−}_{n}_{/2}

The radius of the seriesa^{k}k!α^{k}^{−1/2} is infinite. Thus for anyt x ∈Ethe
seriesn!^{−1}z^{n}1h_{ng}t xare convergent. Fubini’s theorem implies that

n≥0

1

n!z^{n}1h_{n g}t x =1

n≥0

1

n!z^{n}h_{n g}

t x = _{G z}t x

Hence the two series n!^{−1}z^{n}1h_{n g}t x and n!^{−1}z^{n}h_{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 onL^{2}p. We deduce from
(12) that

1_{2 i}h_{ng}t x = 1

2∂^{2}_{i i}1h_{n g}t x

= 1

2tn_{i}n_{i}−1He_{n−2δi}x/√

tt^{−}_{n}_{/2}t

0 ds gs (17)

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

n≥0He_{n}x/√

tg_{n}t,
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^{−1}n_{i}n_{i}−1He_{n−2δi}x/√

tt^{−1−}_{n}_{/2}_{t}

0 ds s_{n}_{/2}g_{n}s
Thus, using (14), we have

1_{2 i}f^{2}

p=

n≥0

n−2δi!4^{−1}n^{2}_{i}n_{i}−1^{2}

×∞

0 dt t^{−2−}_{n}^{t}

0 ds s_{n}_{/2}
g_{n}s

_{2}

≤

n≥0

n!n_{i}n_{i}−1
4

4
n+1^{2}

_{∞}

0 dt g_{n}t^{2}

≤f^{2}

p

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

0 dt t^{−2−k}_{t}

0 s^{k/2}hsds
_{2}

≤ 4
k+1^{2}

_{∞}

0 dt ht^{2}

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
L^{2}pinto itself. The above inequality implies ˜1_{2 i}_{p}≤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_{ng}t x

=

−x_{i}/√

t∂_{i}He_{n}x/√

t +n_{i}n_{i}−1He_{n−2δi}x/√
t

×t^{−1−}_{n}_{/2}t

0 ds gs

= −n_{i}He_{n}x/√

tt^{−1−}_{n}_{/2}_{t}

0 ds gs (18)

Arguing as above, we get, for f∈,
1_{4 i}f^{2}

p=

n≥0

n!n^{2}_{i}∞

0 dt t^{−2−}_{n}^{t}

0 ds s_{n}_{/2}
g_{n}s

_{2}

≤

n≥0

n!n^{2}_{i} 4
n+1^{2}

_{∞}

0 dt g_{n}t^{2}

≤4f^{2}

p

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

The proof concerning1_{1}easily follows from the previous results. From (16),
we get

1_{1}h_{n g}t x

=h_{n g}t x −1
2

nHe_{n}x/√
t +^{d}

i=1

x_{i}

√t∂_{i}He_{n}x/√
t

t^{−1−}_{n}_{/2}_{t}

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

1_{1}h_{n g} =

I+^{1}_{2}^{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 that1_{1}=I+_{d}

i=11_{4 i}−1_{2 i}on. Hence1_{1}can be uniquely ex-
tended in a bounded operator1˜_{1}fromL^{2}pinto itself and1˜_{1}=I+_{d}

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