CHARACTERIZATION OF G-REGULARITY FOR SUPER-BROWNIAN

MOTION AND CONSEQUENCES FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

JEAN-FRANÇOIS DELMAS AND JEAN-STÉPHANE DHERSIN

Abstract. We give a characterization of G-regularity for super-Brownian motion and the Brownian snake. More precisely, we define a capacity on E = (0, ∞)×ℝ^{d}, which is not invariant by translation. We then prove that the hitting probability of a Borel set A ⊂ E for the graph of the Brownian snake starting at (0, 0) is comparable, up to multiplicative constants, to its capacity. This implies that super-Brownian motion started at time 0 at the Dirac mass δ_{0} hits immediately A (that is (0, 0) is G-regular for A^{c}) if and only if its capacity is infinite. As a direct consequence, if Q ⊂ E is a domain such that (0, 0) ∈ ∂Q, we give a necessary and sufficient condition for the existence on Q of a positive solution of ∂_{t}u + \frac{1}{2} Δu = 2u^{2} which blows up at (0, 0). We also give an estimation 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 characterization of the so called G-regularity for super-Brownian motion introduced by Dynkin [8]. Let us recall that a point (r, x) ∈ ℝ×ℝ^{d} is G-regular for an open set Q ⊂ ℝ × ℝ^{d} if a.s. the graph of a super-Brownian motion started at time r with the Dirac mass at x immediately intersects Q^{c}, the complementary of Q. (This definition can be extended to any Borel set.) We also recall that this is equivalent to the existence of nonnegative solutions of the equation \frac{∂u}{∂t} + \frac{1}{2}Δu = 2 u^{2} on the open set Q, which blow up at (r, x) ∈ ∂Q.

Let E = (0, ∞) × ℝ^{d}. We prove that (0, 0) is G-regular for a Borel set A ⊂ ℝ × ℝ^{d} if and only if the capacity of A^{c} ∩ E is infinite, for the following capacity: for any Borel set A′ ⊂ E,

cap(A′) = [inf I(ν)]^{−1} , where

I(ν) = ∫∫ _{E} dsdy p(s, y) (∫∫ _{E} ν(dt, dx)\frac{p(t − s, x − y)}{p(t, x)})^{2} ,

Date: May 20, 1998. 1991 Mathematics Subject Classification. 60G57, 35K60. Key words and phrases. Super-Brownian motion, Brownian snake, G-regularity, parabolic nonlinear

PDE, hitting probabilities, capacity, intersection-equivalence.

The research of the first author was done at the École Nationale des Ponts et Chaussées and at MSRI, supported by NSF grant DMS-9701755.

and p denotes the heat kernel:

p(t, x) = \{ \begin{matrix}(2πt)^{−d/2} e^{− |x|^{2} /2t} & if (t, x) ∈ E, \\ 0 & if (t, x) ∈ (−∞, 0] × ℝ^{d}. \end{matrix}

(|·| denotes the Euclidean norm on ℝ^{d}.) The infimum is taken over all probability measures ν on E such that ν(A′) = 1. Notice this capacity is not invariant by translation in time or space. This capacity arises naturally when one consider the Brownian snake, a useful tool to study super-Brownian motion. Indeed, using potential theory of symmetric Markov process, I(ν) 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 (0, 0) for sets Q = \{(s, y) ∈ E; |y| < \sqrt{s} h(s)\}, where h is a positive decreasing function defined on (0, ∞). Our result can also be viewed as a parabolic extension of the Wiener’s test proved by Dhersin and Le Gall [5] in an elliptic setting.

The proof of our results relies on the Brownian snake introduced by Le Gall. We only give definition and some properties for completeness in this paper, and refer to Le Gall [10, 12] for a detailed presentation. We will use time inhomogeneous notations.

Let (r, x) ∈ ℝ×ℝ^{d} be a fixed point. We denote by W_{r,x} the set of all stopped paths in ℝ^{d} started at x at time r. An element w of W_{r,x} is a continuous mapping w : [r, ζ] → ℝ^{d} such that w(r) = x, and ζ = ζ_{(w)} ∈ [r, ∞) is called its lifetime. We denote by ŵ the end point w(ζ). With the metric d(w, w′) = |ζ_{(w)} − ζ_{(w′)}| + sup_{s≥r} |w(s ∧ ζ_{(w)}) − w′(s ∧ ζ_{(w′)})|, the space W_{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 W_{r,x}, whose law is characterized by the following two properties.

(i) The lifetime process ζ = (ζ_{s} = ζ_{(W_{s})}, s ≥ 0) is a reflecting Brownian motion in [r, ∞). (ii) Conditionally given (ζ_{s}, s ≥ 0), the process (W_{s}, s ≥ 0) is a time-inhomogeneous con-

tinuous Markov process, such that for s′ ≥ s: · W_{s′}(t) = W_{s}(t) for r ≤ t ≤ m(s, s′) = inf_{v∈[s,s′]} ζ_{v}. · (W_{s′}(m(s, s′) + t) − W_{s′}(m(s, s′)), 0 ≤ t ≤ ζ_{s′} − m(s, s′)) is a Brownian motion in

ℝ^{d} independent of W_{s}.

From now on we shall consider the canonical realization of the process W defined on the space Ω = C(ℝ^{+}, W_{r,x}), and denote by E_{w} the law of W started at w ∈ W_{r,x}. The trivial path x_{r} such that ζ_{(x_{r})} = r, x_{r}(r) = x is clearly a regular point for the process (W, E_{w}). We denote by ℕ_{r,x} the excursion measure outside {x_{r}}. Notice that ℕ_{r,x} is an infinite measure. The distribution of W under ℕ_{r,x} can be characterized as above, except that now the lifetime process ζ is distributed according to the Itô measure of excursions of linear reflecting Brownian motion in [r, ∞). We normalize ℕ_{r,x} so that, for every ε > 0,

ℕ_{r,x} [sup_{s≥0} ζ_{s} > ε + r] = \frac{1}{2ε} .

Let σ = inf {s > 0; ζ_{s} = r} denote the duration of the excursion of ζ under ℕ_{r,x}. The graph G^{∗} of W is defined under ℕ_{r,x} by

G^{∗} = {(t, W_{s}(t)); r < t ≤ ζ_{s}, 0 < s < σ} = {(ζ_{s}, Ŵ_{s}); 0 < s < σ} .

We write G^{∗}(W) for G^{∗} when there is a risk of confusion.

Let us now explain the connection between the Brownian snake and super-Brownian motion. First of all, we introduce some notations. We denote by (M_{f}, M_{f}) the space of all finite measures on ℝ^{d}, endowed with the topology of weak convergence. We denote by B(S) (resp. B_{b+}(S)) the set of all real measurable (resp. bounded nonnegative measurable) functions defined on a polish space S. We also denote by B(S) the Borel σ-field on S. For every measure ν ∈ M_{f}, and f ∈ B_{b+}(ℝ^{d}), we shall write (ν, f) = ∫ f(y)ν(dy). We also denote by supp ν the closed support of the measure ν.

We consider under ℕ_{r,x} the continuous version (l_{s}^{t}, t > r, s ≥ 0) of the local time of ζ at level t and time s, and define the measure valued process Y on ℝ^{d} by setting for every t > r, for every φ ∈ B_{b+}(ℝ^{d}),

(Y_{t}, φ) = ∫_{0}^{σ} dl_{s}^{t} φ(Ŵ_{s}).

Let W_{r} = ⋃_{x∈ℝ^{d}} W_{r,x}. Let μ be a finite measure on ℝ^{d}, and ∑_{i∈I} δ_{W^{i}} be a Poisson measure on C(ℝ^{+}, W_{r}) with intensity ∫ μ(dx)ℕ_{r,x}[·]. Then the process X defined by X_{r} = μ and X_{t} = ∑_{i∈I} Y_{t}(W^{i}) if t > r, is a super-Brownian motion started at time r at μ (see [10, 12]). We shall denote by P_{r,μ} (resp. P_{r,x}) the law of the super-Brownian motion started at time r at μ (resp. at the Dirac mass δ_{x}). We deduce from the normalization of ℕ_{r,x} that, for every t > r, ℕ_{r,x} [Y_{t} ≠ 0] = 1/2(t − r) < ∞. This implies that there is only a finite number of indices i ∈ I such that G^{∗}(W^{i}) ∩ [t, ∞) × ℝ^{d} is non empty for t > r.

We consider the graph of X:

G(X) = ⋃_{ε>r} (\overline{⋃_{t≥ε}\{t\} × supp X_{t}}) = ⋃_{i∈I} G^{∗}(W^{i}),

where Ā denotes the closure of A. A set A ⊂ B(ℝ×ℝ^{d}) is called G-polar if P_{r,x}[G(X)∩A ≠ ∅] = 0 for every (r, x) ∈ ℝ × ℝ^{d}. From Poisson measure theory, we have

P_{r,x}[G(X) ∩ A ≠ ∅] = 1 − e^{−ℕ_{r,x}[G^{∗}∩A≠∅]} .

Hence A is G-polar if and only if ℕ_{r,x}[G^{∗} ∩ A ≠ ∅] = 0 for all (r, x) ∈ ℝ × ℝ^{d}. We consider the capacity defined by: for A ∈ B(ℝ × ℝ^{d}),

cap′(A) = [inf ∫∫ _{ℝ×ℝ^{d}} dsdy (∫∫ _{(s,∞)×ℝ^{d}} ν(dt, dx)p(t − s, x − y) e^{−(t−s)/2})^{2}]^{−1} ,

where the infimum is taken over all probability measures ν on ℝ × ℝ^{d} such that ν(A) = 1. Dynkin proved (see Theorem 3.2 in [7]) that A ∈ B(ℝ × ℝ^{d}) is G-polar if and only if cap′(A) = 0. (We have cap′(A) = 0 ⇔ ℕ_{r,x}[G^{∗} ∩ A ≠ ∅] = 0 for all (r, x) ∈ ℝ × ℝ^{d}.) It is easy to check that if A ⊂ E is a compact set then

cap′(A) = 0 ⇔ cap(A) = 0.

This can be extended to all Borel subsets of E since the two capacities are inner capacities (see Meyers [13]). In fact it seems more relevant to consider the capacity cap to characterize G-regularity, as we shall see. We have the following quantitative theorem.

Theorem 1. There exists a constant C_{0} such that for any A ∈ B(E),

4^{−1} cap(A) ≤ ℕ_{0,0}[G^{∗} ∩ A ≠ ∅] ≤ C_{0} cap(A).

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 obtain in section 4, by using additive functionals of the Brownian snake introduced in [5].

Now, for A ∈ B(ℝ × ℝ^{d}), we consider under P_{r,x} the random time

τ_{A} = inf{t > r, ({t} × supp X_{t}) ∩ A ≠ ∅}.

Arguments similar as those of [5] yield that τ_{A} is a stopping time for the natural filtration of X completed the usual way. Thus we have P_{r,x}(τ_{A} = r) = 1 or 0. Following Dynkin [8, section II-6], we say a point (r, x) ∈ ℝ×ℝ^{d} is G-regular for A^{c} if P_{r,x}-a.s. τ_{A} = r. Let A^{Gr} denote the set of all points that are G-regular for A^{c}. From the known path properties of super-Brownian motion it is obvious that int(A) ⊂ A^{Gr} ⊂ Ā, where int(A) denotes the interior of A. We set T_{A} = inf {s > 0, (ζ_{s}, Ŵ_{s}) ∈ A}. Following [5] it is easy to deduce from Theorem 1 the next result.

Proposition 2. Let A ∈ B(ℝ × ℝ^{d}). The following properties are equivalent: 1. (r, x) is G-regular for A^{c};

2. ℕ_{r,x} [G^{∗} ∩ A ≠ ∅] = ∞;

3. E_{x_{r}}-a.s. T_{A} = 0;

4. cap(A_{r,x} ∩ E) = ∞, where A_{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 equation.

Corollary 3. Let Q a domain in E such that (0, 0) ∈ ∂Q. The following three conditions are equivalent.

1. (0, 0) is G-regular for Q ;

2. cap(Q^{c} ∩ E) = ∞ ;

3. There exists a nonnegative solution of \frac{∂u}{∂t} + \frac{Δ}{2}u = 2u^{2} in Q such that

lim_{(s,y)→(0,0), (s,y)∈Q} u(s, y) = ∞.

The equivalence of assertions 1) and 3) is due to Dynkin [8, Theorem II.6.1]. The equivalence of 1) and 2) is given by Proposition 2.

Finally, using Theorem 1 we give in section 5 an estimation 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 of d-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^{p}, we denote by C_{0}^{∞}(S) the set of all functions of class C^{∞} defined on S with compact support. If f is a measurable function defined on S then ∥f ∥_{∞} = sup_{x∈S} |f(s)|. We consider the Hilbert space L^{2}(p) = {f ∈ B(E); ∥f ∥_{(p)} < ∞},

where ∥f ∥_{(p)}^{2} = ∫∫_{E} dtdx p(t, x)f(t, x)^{2}.

Notice the kernel defined on E×E by k(t, x; s, y) = p(t−s, x−y)p(t, x)^{−1} is nonnegative and lower semi-continuous. Thus we can introduce the operator Λ defined on the set of nonnegative functions f ∈ B(E) by:

Λ(f) = p^{−1}[p ∗ (pf)] = ∫∫ _{E} dsdy k(·, ·; s, y)p(s, y)f(s, y),

where ∗ denotes the usual convolution product on E. Furthermore, the function Λ(f) is even lower semi-continuous (see [9, Lemma 2.2.1]).

We define the capacity Cap on E in the following way: if A ⊂ E, then

Cap(A) = inf {∥f ∥_{(p)}^{2}; f ≥ 0, f ∈ L^{2}(p), Λ(f) ≥ 1 on A} ,

with the convention inf ∅ = ∞. Notice this capacity is not invariant by translation in time or space. This capacity is an outer capacity (see Meyers [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 Baras and Pierre [3] for similar results but with different norms). Therefore we consider the weighted Sobolev space W_{D} which is the completion of C_{0}^{∞}(E) with respect to the norm ∥·∥_{D}, defined by

∥φ∥_{D}^{2} = ∥∂_{t}φ∥_{(p)}^{2} + ∑_{i=1}^{d} ∥∂_{i}(log p) ∂_{i}φ∥_{(p)}^{2} + ∑_{i=1}^{d} ∥∂_{ii}^{2}φ∥_{(p)}^{2}, φ ∈ C_{0}^{∞}(E),

with the usual notations ∂_{t}g(t, x) = \frac{∂g}{∂t} (t, x), ∂_{i}g(t, x) = \frac{∂g}{∂x_{i}} (t, x) for x = (x_{1}, ⋯, x_{d}) ∈ ℝ^{d} and ∂_{ii}^{2} = ∂_{i}∂_{i}. Notice the non zero 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) = inf {∥φ∥_{D}^{2}; φ ∈ C_{0}^{∞}(E), φ ≥ 0, φ ≥ 1 on K}
= inf {∥φ∥_{D}^{2}; φ ∈ C_{0}^{∞}(E), φ ≥ 0, φ ≥ 1 on a neighborhood of K} .

Then we set for any open set G ⊂ E,

(1) cap_{D}(G) = sup {cap_{D}(K); K ⊂ G, K compact} ,

and, for any analytic set A ⊂ E,

cap_{D}(A) = inf {cap_{D}(G); A ⊂ 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,

Cap(A) ≤ cap_{D}(A) ≤ C Cap(A).

Proof. Since the two capacities are outer capacity, it is enough to consider open sets. Now, using (1) and [13, Theorem 8], we see it is enough to consider compact sets.

Let us introduce the operator H = ∂_{t} − \frac{1}{2} Δ. We consider a non empty compact set K ⊂ E. Let φ ∈ C_{0}^{∞}(E) be such that φ ≥ 0 and φ ≥ 1 on K. Notice that (in the distribution sense) Hp = δ_{(0,0)}, where δ_{(0,0)} is the Dirac mass at (0, 0) ∈ ℝ × ℝ^{d}. Then we have p ∗ [H(pφ)] = (Hp) ∗ (pφ) = pφ. The function f = p^{−1} |H(pφ)| = |H(φ) − (∇ log p, ∇φ)| is nonnegative and Λ(f) ≥ φ. Thus we have Λ(f) ≥ 1 on K. We also have ∥f ∥_{(p)} ≤ ∥φ∥_{D}. Hence we have Cap(K) ≤ ∥φ∥_{D}^{2}. The first inequality follows by taking a sequence (φ_{n}) such that ∥φ_{n} ∥_{D}^{2} converges to cap_{D}(K).

To prove the other inequality, let us consider a nonnegative function f_{1} ∈ L^{2}(p), such that Λ(f_{1}) ≥ 1 on K. Notice it implies ∥f_{1} ∥_{(p)} > 0. Let δ > 0. It is easy to construct a function ε ∈ L^{2}(p) such that ε > 0 on E and ∥ε∥_{(p)} ≤ δ ∥f_{1} ∥_{(p)}. We set f_{2} = f_{1} + ε. Since the function Λ(f_{2}) is lower semi-continuous, the set {(t, x) ∈ E; Λ(f_{2}) > 1} is open and it also contains K. It is then obvious that for δ′ > 0 small enough, if we set f_{3}(t, x) = f_{2}(t, x)1_{{δ′<t<δ′^{−1},|x|<δ′^{−1}}} for (t, x) ∈ E, we get Λ(f_{3}) > 1 on an open set containing K. Let us introduce a nonnegative function h ∈ C_{0}^{∞}(E) such that ∫∫_{E} h(t, x)dtdx = 1. For θ > 0, we write h_{θ}(t, x) = θ^{−d−1}h(t/θ, x/θ). Now using the uniform continuity of p on [δ′/2, ∞)×ℝ^{d}, it is easy to see that if f = h_{θ} ∗f_{3} , then Λ(f) > 1 on an open set containing K for θ small enough. The function f is nonnegative, belongs to C_{0}^{∞}(E) and the function Λ(f) is of class C^{∞}. We can choose δ and θ small enough so that ∥f ∥_{(p)} ≤ 2 ∥f_{1} ∥_{(p)}.

Let α ∈ C_{0}^{∞}([0, ∞)) such that 0 ≤ α ≤ 1, α = 1 on [0, 1/2] and α = 0 on [1, ∞). Let ξ ∈ C_{0}^{∞}(ℝ^{d}) such that 0 ≤ ξ ≤ 1 and ξ = 1 in a neighborhood of 0. We define α_{n}(t) = α(t/n) and ξ_{n}(x) = ξ(x/n). The function φ_{n} = α_{n}ξ_{n}Λ(f) belongs to C_{0}^{∞}(E), is nonnegative and φ_{n} ≥ 1 on a neighborhood of K for n great enough.

Let us now give two key lemmata. If M is a bounded operator from L^{2}(p) into itself, we denotes by ∥M ∥_{(p)} = sup{∥M(f)∥_{(p)}; f ∈ L^{2}(p), ∥f ∥_{(p)} = 1} its norm. We define the operator Λ_{0}: for f ∈ B(E) nonnegative, Λ_{0}(f)(t, x) = t^{−1}Λ(f)(t, x), (t, x) ∈ E. For T > 0, let us introduce E_{T} = (0, T) × ℝ^{d}.

Lemma 5. The operators 1_{E_{T}} Λ and Λ_{0} are bounded operators from L^{2}(p) into itself. Furthermore, we have ∥1_{E_{T}} Λ∥_{(p)} ≤ T/\sqrt{2} and ∥Λ_{0} ∥_{(p)} ≤ 2.

Proof of Lemma 5. Let f ∈ L^{2}(p). The Cauchy-Schwarz inequality implies:

∥Λ_{0}(f)∥_{(p)}^{2} = ∫∫ _{E} dtdx t^{−2}p(t, x)^{−1} [∫∫ _{E} dsdy p(t − s, x − y)p(s, y)f(s, y)]^{2}

≤ ∫∫ _{E} dtdx t^{−2}p(t, x)^{−1} ∫∫ _{E} ds′dy′ p(t − s′, x − y′)p(s′, y′)s′^{−1/2}1_{s′≤t}

∫∫ _{E} dsdy p(t − s, x − y)p(s, y)s^{1/2}f(s, y)^{2}
= 2 ∫∫ _{E} dtdx t^{−2}t^{1/2} ∫∫ _{E} dsdy p(t − s, x − y)p(s, y)s^{1/2}f(s, y)^{2}
≤ 2 ∫∫ _{E} dsdy p(s, y)f(s, y)^{2}s^{1/2} ∫_{s}^{∞} t^{−3/2}dt ≤ 4 ∥f ∥_{(p)}^{2} .

Hence the operator Λ_{0} is a bounded operator from L^{2}(p) into itself. And we have ∥Λ_{0} ∥_{(p)} ≤ 2. The operator 1_{E_{T}} Λ can handled in a very similar way. □

Lemma 6. The operators defined on C_{0}^{∞}(E) by: g ∈ C_{0}^{∞}(E)

Λ_{1}(g) = ∂_{t}Λ(g),
for i ∈ \{1, ⋯, d\}, Λ_{2,i}(g) = \frac{1}{2} ∂_{ii}^{2}Λ(g),
and for i ∈ {1, ⋯, d}, Λ_{3,i}(g) = ∂_{i}(log p) ∂_{i}Λ(g),

can be uniquely extended into bounded operators from L^{2}(p) into itself. And we have

(2)  ∥Λ_{1} ∥_{(p)} ≤ 1 + 3d,
(3)  for i ∈ {1, ⋯, d}, ∥Λ_{2,i} ∥_{(p)} ≤ 1,
(4)  for i ∈ {1, ⋯, d}, ∥Λ_{3,i} ∥_{(p)} ≤ 4.

The proof of this lemma is given in appendix.

We now bound ∥φ_{n} ∥_{D}. Lemma 5 provides an upper bound for ∥∂_{t}φ_{n} ∥_{(p)}:

∥∂_{t}φ_{n} ∥_{(p)} ≤ ∥∂_{t}α_{n} ∥_{∞} ∥1_{E_{n}}Λ(f)∥_{(p)} + ∥Λ_{1}f ∥_{(p)}

≤ (∥∂_{t}α∥_{∞} 2^{−1/2} + ∥Λ_{1} ∥_{(p)}) ∥f ∥_{(p)} . (5)

Using Lemma 5 we derive an upper bound for ∑_{i=1}^{d} ∥∂_{i} log p ∂_{i}φ_{n} ∥_{(p)}:

∑_{i=1}^{d} ∥∂_{i} log p ∂_{i}φ_{n} ∥_{(p)} ≤ ∑_{i=1}^{d} (∥Λ_{3,i}(f)∥_{(p)} + sup_{x∈ℝ^{d}} |x_{i}∂_{i}ξ(x)| ∥Λ_{0}(f)∥_{(p)})

≤ ∑_{i=1}^{d} (∥Λ_{3,i} ∥_{(p)} + sup_{x∈ℝ^{d}} |x_{i}∂_{i}ξ(x)| ∥Λ_{0} ∥_{(p)}) ∥f ∥_{(p)} . (6)

In order to give an upper bound for ∑_{i=1}^{d} ∥∂_{ii}^{2}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}Λ(g)∥_{(p)} ≤ c_{1}n^{−1/2} ∥g ∥_{(p)} .

Proof. Recall that ξ_{n} has compact support. Then, an integration by parts, Cauchy-Schwarz inequalities and Lemma 5 give for 1 ≤ i ≤ d,

∥1_{E_{n}}∂_{i}ξ_{n}∂_{i}Λ(g)∥_{(p)}^{2} = − ∫∫ _{E} p1_{E_{n}}Λ(g)(∂_{i}ξ_{n})^{2}∂_{ii}^{2}Λ(g)
− ∫∫ _{E} p1_{E_{n}}Λ(g)(∂_{i}ξ_{n})^{2}∂_{i}Λ(g)∂_{i} log p
− 2 ∫∫ _{E} p1_{E_{n}}Λ(g)∂_{i}ξ_{n}∂_{i}Λ(g)∂_{ii}^{2}ξ_{n}
≤ ∥∂_{i}ξ_{n} ∥_{∞}^{2} ∥1_{E_{n}}Λ(g)∥_{(p)} ∥∂_{ii}^{2}Λ(g)∥_{(p)}
+ ∥∂_{i}ξ_{n} ∥_{∞}^{2} ∥1_{E_{n}}Λ(g)∥_{(p)} ∥∂_{i}Λ(g) ∂_{i} log p∥_{(p)}
+ 2 ∥∂_{ii}^{2}ξ_{n} ∥_{∞} ∥1_{E_{n}}Λ(g)∥_{(p)} ∥1_{E_{n}}∂_{i}ξ_{n}∂_{i}Λ(g)∥_{(p)}
≤2^{−1/2}n^{−1} [2 ∥Λ_{2,i} ∥_{(p)} + ∥Λ_{3,i} ∥_{(p)}] ∥∂_{i}ξ ∥_{∞}^{2} ∥g ∥_{(p)}^{2}
+ 2^{1/2}n^{−1} ∥∂_{ii}^{2}ξ ∥_{∞} ∥g ∥_{(p)} ∥1_{E_{n}}∂_{i}ξ_{n}∂_{i}Λ(g)∥_{(p)} .

Notice that if a, b, c are positive then a^{2} ≤ c^{2} + ba implies a ≤ c + b. Thus we get

∥1_{E_{n}}∂_{i}ξ_{n}∂_{i}Λ(g)∥_{(p)}

≤ 2^{−1/4}n^{−1/2} [2 ∥Λ_{2,i} ∥_{(p)} + ∥Λ_{3,i} ∥_{(p)}]^{1/2} ∥∂_{i}ξ ∥_{∞} ∥g ∥_{(p)} +2^{1/2}n^{−1} ∥∂_{i,i}^{2}ξ ∥_{∞} ∥g ∥_{(p)},

which ends the proof. □

Using this lemma and Lemma 5, we get that

∑_{i=1}^{d} ∥∂_{ii}^{2}φ_{n} ∥_{p} ≤ ∑_{i=1}^{d} [2 ∥Λ_{2,i}(f)∥_{(p)} + ∥∂_{ii}^{2}ξ_{n} ∥_{∞} ∥1_{E_{n}}Λ(f)∥_{(p)} +2 ∥1_{E_{n}}∂_{i}ξ_{n} ∂_{i}Λ(f))∥_{(p)}]

≤ ∑_{i=1}^{d} [2 ∥Λ_{2,i} ∥_{(p)} +2^{−1/2}n^{−1} ∥∂_{ii}^{2}ξ ∥_{∞} +2c_{1}n^{−1/2}] ∥f ∥_{(p)} . (7)

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

∥φ_{n} ∥_{D} ≤ c_{2} ∥f ∥_{(p)} .

Thus we have ∥φ_{n} ∥_{D} ≤ 2c_{2} ∥f_{1} ∥_{(p)}. The second inequality of the proposition is then obvious with C = 4 c_{2}^{2}. □

We shall need the following lemma.

Lemma 8. There exists a constant γ such that if K ⊂ E is a compact set such that cap_{D}(K) > 0, there exists φ ∈ C_{0}^{∞}(E) which satisfies:

1) 0 ≤ φ ≤ 1,

2) φ = 1 on a neighborhood of K,

3) ∥φ∥_{D}^{2} ≤ γ cap_{D}(K).

The proof is classic, but we give it for completeness. Proof. Let h ∈ C^{∞}([0, ∞)) such that 0 ≤ h ≤ 1, h = 0 on [0, 1/4] 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 of K, and 2 cap_{D}(K) ≥ ∥g ∥_{D}^{2}. Let φ = h ∘ g. The function φ ∈ C_{0}^{∞}(E) satisfies 1) and 2). Let us check 3). We have

∥∂_{t}φ∥_{(p)} ≤ ∥h′ ∥_{∞} ∥∂_{t}g ∥_{(p)},
∥∂_{i} log p ∂_{i}φ∥_{(p)} ≤ ∥h′ ∥_{∞} ∥∂_{i} log p ∂_{i}g ∥_{(p)}
∥∂_{ii}^{2}φ∥_{(p)} ≤ ∥h′ ∥_{∞} ∥∂_{ii}^{2}g ∥_{(p)} + ∥(h′′ ∘ g) (∂_{i}g)^{2} ∥_{(p)} .

Only the upper bound for the second right hand-side term of the last inequality is not obvious. We first search an upper bound for ∥(∂_{i}φ_{1})^{2}/(1 + φ_{1})∥_{(p)}, where φ_{1} ∈ C_{0}^{∞}(E) is a nonnegative function. An integration by parts and Cauchy-Schwarz inequality give

∫∫ _{E} p\frac{(∂_{i}φ_{1})^{4}}{(1 + φ_{1})^{2}} = 3 ∫∫ _{E} p ∂_{ii}^{2}φ_{1} \frac{(∂_{i}φ_{1})^{2}}{1 + φ_{1}} + ∫∫ _{E} p ∂_{i} log p ∂_{i}φ_{1} \frac{(∂_{i}φ_{1})^{2}}{1 + φ_{1}}
≤ (3 ∥∂_{ii}^{2}φ_{1} ∥_{(p)} + ∥∂_{i} log p ∂_{i}φ_{1} ∥_{(p)}) ∥(∂_{i}φ_{1})^{2}/(1 + φ_{1})∥_{(p)} .

Thus we get

∥(∂_{i}φ_{1})^{2}/(1 + φ_{1})∥_{(p)} ≤ 3 (∥∂_{ii}^{2}φ_{1} ∥_{(p)} + ∥∂_{i} log p ∂_{i}φ_{1} ∥_{(p)}) . (8)

Since we have |h′′(t)| ≤ 2(1+t)^{−1} ∥h′′ ∥_{∞}, taking φ_{1} = g in the above inequality we deduce that

∥(h′′ ∘ g) (∂_{i}g)^{2} ∥_{(p)} ≤ 2 ∥(∂_{i}g)^{2}/(1 + g)∥_{(p)} ∥h′′ ∥_{∞}
≤ 6(∥∂_{ii}^{2}g ∥_{(p)} + ∥∂_{i} log p ∂_{i}g ∥_{(p)}) ∥h′′ ∥_{∞} .

The previous inequalities imply there exists a constant c depending only on h and d such that ∥φ∥_{D} ≤ c ∥g ∥_{D}. Thus 3) holds with γ = 2c^{2}. □

3. Upper bound for hitting probabilities

In this section we prove the second inequality of Theorem 1 for compact sets. Let us introduce K ⊂ E_{T} a compact set such that cap_{D}(K) > 0. Let φ be as in Lemma 8. We set φ = 0 outside E. We introduce the function ψ = 1 − φ, which takes values in [0, 1]. We consider the function u defined on ℝ × ℝ^{d} by u(t, x) = ℕ_{t,x}[G^{∗} ∩ 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 P_{0} a d-dimensional Brownian motion started from 0. Itô’s formula implies that for all t ≥ 0, P_{0}-a.s.,

uψ^{4}(t, B_{t}) = uψ^{4}(0, 0) + ∫_{0}^{t} ∂_{t}(uψ^{4})(s, B_{s})ds

+ ∫_{0}^{t} \frac{Δ}{2}(uψ^{4})(s, B_{s})ds + ∫_{0}^{t} ∇(uψ^{4})(s, B_{s})dB_{s}.

Consider the stopping time T_{a} = T ∧ inf{t > 0; |B_{t}| ≥ a}. We can then apply the stopping Theorem at time T_{a} and get

E_{0}uψ^{4}(T_{a}, B_{T_{a}}) = u(0, 0) + E_{0} ∫_{0}^{T_{a}} ∂_{t}(uψ^{4})(s, B_{s})ds + E_{0} ∫_{0}^{T_{a}} \frac{Δ}{2}(uψ^{4})(s, B_{s})ds
= u(0, 0) + E_{0} ∫_{0}^{T_{a}} [2u^{2}ψ^{4} + 4uψ^{3}∂_{t}ψ + 4(∇u, ∇ψ)ψ^{2}
+ 6uψ^{2}(∇ψ, ∇ψ) + 2uψ^{3}Δψ](s, B_{s})ds.

We have used that ∂_{t}u + \frac{1}{2} Δu = 2u^{2} to get the last equality. Notice that each integrand is either nonnegative or bounded. By dominated convergence and monotone convergence, we get as a goes to infinity

u(0, 0) + 2 ∥uψ^{2} 1_{E_{T}} ∥_{(p)}^{2} = E_{0}uψ^{4}(T, B_{T}) − ∫∫ _{E_{T}} p[4uψ^{3}∂_{t}ψ + 4(∇u, ∇ψ)ψ^{2}
+ 6uψ^{2}(∇ψ, ∇ψ) + 2uψ^{3}Δψ].

Since K ⊂ E_{T}, we deduce that u(t, x) = 0 for t ≥ T. Thus we have:

u(0, 0) + 2 ∥uψ^{2} ∥_{(p)}^{2} = − ∫∫ _{E} p[4uψ^{3}∂_{t}ψ + 4(∇u, ∇ψ)ψ^{2}

+ 6uψ^{2}(∇ψ, ∇ψ) + 2uψ^{3}Δψ]. (9)

We now bound the right hand side. Using Cauchy-Schwarz inequality, that 0 ≤ ψ ≤ 1, and that −φ and ψ have the same derivatives, we get

− ∫∫ _{E} puψ^{3}∂_{t}ψ ≤ ∥uψ^{2} ∥_{(p)} ∥∂_{t}φ∥_{(p)},
− ∫∫ _{E} puψ^{3}∂_{ii}^{2}ψ ≤ ∥uψ^{2} ∥_{(p)} ∥∂_{ii}^{2}φ∥_{(p)},

and − ∫∫ _{E} puψ^{2}(∇ψ, ∇ψ) ≤ ∥uψ^{2} ∥_{(p)} ∑_{i=1}^{d} ∥(∂_{i}φ)^{2} ∥_{(p)}

≤ 2 ∥uψ^{2} ∥_{(p)} ∑_{i=1}^{d} ∥(∂_{i}φ)^{2}/(1 + φ)∥_{(p)}

≤ 6 ∥uψ^{2} ∥_{(p)} ∑_{i=1}^{d}(∥∂_{ii}^{2}φ∥_{(p)} + ∥∂_{i} log p ∂_{i}φ∥_{(p)}),

where we have used (8) with φ_{1} = φ for the last inequality. Now an integration by parts and Cauchy-Schwarz inequality give

− ∫∫ _{E} pψ^{3}(∇u, ∇ψ) = ∫∫ _{E} puψ^{2} [ψ(∇ log p, ∇ψ) + 3(∇ψ, ∇ψ) + ψΔψ]

≤ ∥uψ^{2} ∥_{(p)} ∑_{i=1}^{d} [∥∂_{i} log p ∂_{i}φ∥_{(p)} +3 ∥(∂_{i}φ)^{2} ∥_{(p)} + ∥∂_{ii}^{2}φ∥_{(p)}]

≤ 19 ∥uψ^{2} ∥_{(p)} ∑_{i=1}^{d} [∥∂_{i} log p ∂_{i}φ∥_{(p)} + ∥∂_{ii}^{2}φ∥_{(p)}] ,

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

u(0, 0) + 2 ∥uψ^{2} ∥_{(p)}^{2} ≤ c_{3} ∥uψ^{2} ∥_{(p)} ∥φ∥_{D},

where the constant c_{3} depends only on d. Since ∥uψ^{2} ∥_{(p)} is finite (recall uψ is bounded, and zero on [T, ∞) × ℝ^{d}), this implies that ∥uψ^{2} ∥_{(p)} ≤ c_{3} ∥φ∥_{D} and hence u(0, 0) ≤ c_{3}^{2} ∥φ∥_{D}^{2}. This last inequality and the definition of φ imply that

ℕ_{0,0}[G^{∗} ∩ K ≠ ∅] = u(0, 0) ≤ c_{3}^{2}γ cap_{D}(K) ≤ c_{3}^{2}γC Cap(K) = c_{3}^{2}γC cap(K).

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 that K ⊂ E_{T}. We consider the probability measure μ defined on W_{0,0} by

μ(dw) = ∫∫ _{E} ν(dt, dx) P_{0}^{t,x}(dw),

where P_{0}^{t,x} is the law on W_{0,0} of the Brownian bridge starting at time 0 at point 0 and ending at time t at point x. Notice that the measure μ is in fact a measure on W_{0,0}^{∗}, the set of non trivial path in W_{0,0} (a trivial path is a path of lifetime zero). The measure P_{0}^{t,x} can also be viewed as a probability measure on the canonical space C(ℝ^{+}, ℝ^{d}) endowed with the filtration (C_{t}) generated by the coordinate mappings. Let P_{0} be the law on the canonical space of the standard Brownian motion. For s ∈ [0, t), we have

P_{0}^{t,x}(dw)_{|C_{s}} = \frac{p(t − s, x − w(s))}{p(t, x)}P_{0}(dw)_{|C_{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:

E(μ) = 2 ∫_{0}^{∞} ds P_{0} [(∫∫ _{E} ν(dt, dx)p(t − s, x − w(s))/p(t, x))^{2}] = 2I(ν).

Now, using [5, Proposition 5], we know there exists an additive functional A of the Brownian snake killed when its lifetime reaches 0 such that:

(i) For every Borel function F ≥ 0 on W_{0,0}^{∗}, ℕ_{0,0} [∫_{0}^{∞} F (W_{s})dA_{s}] = ∫ μ(dw)F (w).

(ii) ℕ_{0,0}[A_{∞}^{2}] = 2E(μ).

We deduce from (i) that the additive functional increases only when Ŵ_{s} ∈ supp ν ⊂ K. Therefore, using the Cauchy-Schwarz inequality, we get

ℕ_{0,0}[G^{∗} ∩ K ≠ ∅] ≥ ℕ_{0,0}[A_{∞} > 0] ≥ ℕ_{0}[A_{∞}]^{2}/ℕ_{0}[A_{∞}^{2}].

We get ℕ_{0,0}[G^{∗} ∩K ≠ ∅] ≥ [4I(ν)]^{−1}. Since the above inequality is true for any probability ν on K, we get that

ℕ_{0,0}[G^{∗} ∩ K ≠ ∅] ≥ 4^{−1} Cap(K) = 4^{−1} cap(K).

Proof of Theorem 1. Notice the application defined on B(E) by T (A) = ℕ_{0,0}[G^{∗} ∩ A ≠ ∅] for A ∈ B(E) is a Choquet capacity (see [4, théorème 1]). Since the capacity cap is an inner capacity (see [13, Theorem 12]), it is enough to prove the theorem for compact subsets of E. The result is then given by the previous section (with C_{0} = c_{3}^{2}γC) and the above result. □

5. Brownian range and support of X_{1}

In this section, we first give an estimation for the hitting probabilities of the support of X_{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 fix d ≥ 2. We denote by cap_{d−2} the usual Newtonian (logarithmic if d = 2) capacity in ℝ^{d}:

cap_{d−2}(A) = [inf ∫∫ _{ℝ^{d}×ℝ^{d}} ρ(dx)ρ(dy) h_{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 measures ρ on ℝ^{d} such that ρ(A) = 1. Let B(0, h) be the open ball of ℝ^{d} centered at 0 with radius h.

Proposition 9. Let M > 0. There exist two positive constants a and b such that for any Borel set A ⊂ B(0, 1), for any finite measure μ on B(0, 1), with (μ, 1) ≤ M, we have

a(μ, 1) cap_{d−2}(A) ≤ ℙ_{0,μ}[supp X_{1} ∩ A ≠ ∅] ≤ b(μ, 1) cap_{d−2}(A).

Proof. Let A ⊂ B(0, 2) be a Borel set. Let ν be a probability measure on E such that ν({1} × A) = 1. Then we have ν = δ_{{1}} × ρ, where ρ is a probability measure on ℝ^{d} such that ρ(A) = 1. We get

I(ν) = ∫∫ _{(0,1)×ℝ^{d}} dsdy p(s, y)

∫∫ _{A×A} ρ(dx)ρ(dx′)p(1 − s, x − y)p(1 − s, x′ − y)p(1, x)^{−1}p(1, x′)^{−1}.

Since x, x′ are in B(0, 2) and since s ∈ (0, 1) it is easy to see there exist two positive constants a_{1} and b_{1} (independent of A and ρ) such that

a_{1}I(ν) ≤ ∫∫ _{A×A} ρ(dx)ρ(dx′) h_{d−2}(|x − x′|) ≤ b_{1}I(ν).

This implies that for any Borel set A ⊂ B(0, 2),

a_{1} cap_{d−2}(A) ≤ cap({1} × A) ≤ b_{1} cap_{d−2}(A).

Since the capacity cap_{d−2} is invariant by translation, we get that for any Borel set A ⊂ B(0, 1), for any x ∈ B(0, 1),

a_{1} cap_{d−2}(A) ≤ cap({1} × A_{x}) ≤ b_{1} cap_{d−2}(A),

where A_{x} = {y; y − x ∈ A}. We deduce from Theorem 1 that

4^{−1}a_{1} cap_{d−2}(A) ≤ ℕ_{0,x}[G^{∗} ∩ ({1} × A) ≠ ∅] ≤ C_{0}b_{1} cap_{d−2}(A).

Since X_{1} = ∑_{i∈I} Y_{1}(W^{i}), where ∑_{i∈I} δ_{W^{i}} is a Poisson measure on C(ℝ^{+}, W_{0}) with intensity ∫ μ(dx)ℕ_{0,x}[·], we have

ℙ_{0,μ}[supp X_{1} ∩ A ≠ ∅] = 1 − e^{− ∫ μ(dx)ℕ_{0,x}[supp Y_{1}∩A≠∅]} .

Notice that ℕ_{0,x}-a.e., {1} × (supp Y_{1} ∩ A) = G^{∗} ∩ ({1} × A). Since (μ, 1) < M, we then easily get the result. □

Intersection-equivalence between random sets has been defined by Peres [15]. Two random Borel sets F_{1} and F_{2} in ℝ^{d} are intersection-equivalent in an open set U, if there exist positive constants a and b such that, for any Borel set A ⊂ U,

a P[A ∩ F_{1}] ≤ P[A ∩ F_{2}] ≤ b P[A ∩ F_{1}].

If π is a probability measure on B(0, 1), then we denote by P_{π} the law of a d-dimensional Brownian motion (B_{t}, t ≥ 0) started with the law π. For d ≥ 3 the range of Brownian motion is defined by R_{B} = {B_{t}, t ≥ 0} in ℝ^{d}. For d = 2, we also denote by R_{B} the set R_{B} = {B_{t}, t ∈ [0, ξ]}, where ξ is an exponential random variable of parameter 1 independent of (B_{t}, t ≥ 0).

Corollary 10. Let M > 0. There exist two positive constants a and b such that for any Borel set A ⊂ B(0, 1), for any absolutely continuous probability measure π on B(0, 1) with density bounded by M, for any finite measure μ on B(0, 1), with (μ, 1) ≤ M, we have

a(μ, 1)P_{π}[R_{B} ∩ A ≠ ∅] ≤ ℙ_{0,μ}[supp X_{1} ∩ A ≠ ∅] ≤ b(μ, 1)P_{π}[R_{B} ∩ A ≠ ∅].

Proof. This is a consequence of Proposition 9 and the fact that there exist two positive constants a_{2} and b_{2} such that for any Borel set A ⊂ B(0, 1), for any absolutely continuous probability measure π on B(0, 1) with density bounded by M,

a_{2} cap_{d−2}(A) ≤ P_{π}[R_{B} ∩ A ≠ ∅] ≤ b_{2} cap_{d−2}(A)

(see for example [15, Proposition 3.2] for d ≥ 3 and [14] for d = 2). □

6. 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.

6.1. Hermite polynomials. For n = (n_{1}, ⋯, n_{d}) ∈ ℕ^{d}, we set |n| = ∑_{i=1}^{d} n_{i}, n! = ∏_{i=1}^{d} n_{i}! and ∑_{n≥0} = ∑_{i=1}^{d} ∑_{n_{i}=0}^{∞}. For j ∈ {1, ⋯, d}, let δ(j) be the element of ℕ^{d} such that δ(j)_{i} = δ_{i,j}, the standard Kronecker symbol. If z = (z_{1}, ⋯, z_{d}) is an element of ℝ^{d}, then we set z^{n} = ∏_{i=1}^{d} z_{i}^{n_{i}}. Let (·, ·) be the Euclidean product on ℝ^{d}.

The function φ(z) = e^{−[|z|^{2} −2(z,x)]/2} is an entire function defined on ℝ^{d}. We have

e^{−[|z|^{2} −2(x,z)]/2} = ∑_{n≥0} \frac{1}{n!} z^{n}He_{n}(x), (10)

where the n-th term He_{n}(x) is a polynomial of (x_{1}, ⋯, x_{d}) of degree |n| called the nth Hermite polynomial. Those polynomials can easily be expressed with the usual one dimensional Hermite polynomials (He_{k}^{(1)}, k ∈ ℕ): He_{n}(x) = ∏_{i=1}^{d} He_{n_{i}}^{(1)}(x_{i}), where x = (x_{1}, ⋯, x_{d}).

Now, let us recall some basic properties of the polynomials He_{n}. The following recurrence formula can be deduced from (10) by derivating w.r.t. z_{i}: for all n ∈ ℕ^{d} such that n_{i} > 0,

He_{n}(x) = x_{i}He_{n−δ(i)}(x) − (n_{i} − 1)He_{n−2δ(i)}(x), ∀x ∈ ℝ^{d}, (11)

where by convention He_{n−kδ(i)} = 0 if n_{i} − k < 0. The derivative formula can be deduced from (10) by derivating w.r.t. x_{i}: for all n ∈ ℕ^{d},

∂_{i}He_{n} = n_{i}He_{n−δ(i)}. (12)

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

|He_{n}(x)| ≤ c_{0}^{d}\sqrt{n!} e^{|x|^{2} /4} for all x ∈ ℝ^{d}, n ∈ ℕ^{d}. (13)

Using the definition of the Hermite polynomials, it is also easy to prove that:

∫ dx p(t, x)He_{n}(x/\sqrt{t})He_{m}(x/\sqrt{t}) = n! ∏_{i=1}^{d} δ_{n_{i},m_{i}}. (14)

It is also well known that the Hermite polynomials is a complete orthogonal system in L^{2}(ℝ^{d}, e^{− |x|^{2} /2} dx). Finally, standard arguments on Hilbert spaces show that if f ∈ L^{2}(p) then

f(t, x) = ∑_{n≥0} f_{n}(t, x) = ∑_{n≥0} He_{n}(x/\sqrt{t})g_{n}(t),

where g_{n}(t) = (n!)^{−1} ∫ dx p(t, x)He_{n}(x/\sqrt{t})f(t, x) and g_{n} ∈ L^{2}((0, ∞)). Furthermore, we have

∥f ∥_{(p)}^{2} = ∑_{n≥0} n! ∫_{0}^{∞} dt g_{n}(t)^{2}. (15)

Since C_{0}^{∞}((0, ∞)) is dense in L^{2}((0, ∞)), it is clear that the set A of functions f(t, x) = ∑_{n≥0} He_{n}(x/\sqrt{t})g_{n}(t) where g_{n} ∈ C_{0}^{∞}((0, ∞)) is non zero for a finite number of indices n, is dense in L^{2}(p).

6.2. Proof of Lemma 6. In a first step we prove there exist unique bounded extensions Λ̃_{1}, Λ̃_{2,i} and Λ̃_{3,i} in L^{2}(p) of the operators Λ_{1}, Λ_{2,i} and Λ_{3,i} defined on A. Then in a second step we check that the extensions Λ̃_{1}, Λ̃_{2,i} and Λ̃_{3,i} and the operators Λ_{1}, Λ_{2,i} and Λ_{3,i}, which are also defined on C_{0}^{∞}(E), agree on C_{0}^{∞}(E).

First step. Let us compute Λ(f) for very particular functions f ∈ A. Let g ∈ C_{0}^{∞}((0, ∞)), α and β be two positive reals such that supp g ∈ [α, β], and G(t) = ∫_{0}^{t} ds g(s). For n ∈ ℕ^{d}, and (t, x) ∈ E, we set

h_{n,g}(t, x) = He_{n}(x/\sqrt{t})t^{− |n| /2}g(t).

Let us prove that

Λ(h_{n,g}) = h_{n,G}. (16)

For z ∈ ℝ, we introduce the function H_{g,z} defined on E by

H_{g,z}(t, x) = ∑_{n≥0} \frac{1}{n!} z^{n}h_{n,g}(t, x) = g(t) e^{−[|z|^{2} −2(x,z)]/2t} .

Then we have

Λ(H_{g,z})(t, x) = \frac{1}{p(t, x)} ∫_{0}^{t} ds ∫ dy p(t − s, x − y)p(s, y)H_{g,z}(s, y)

= \frac{1}{p(t, x)} ∫_{0}^{t} ds ∫ dy p (\frac{s(t − s)}{t}, y − \frac{sx}{t} − \frac{(t − s)z}{t} ) p(t, z − x)g(s)

= e^{−[|z|^{2} −2(x,z)]/2t} ∫_{0}^{t} ds g(s) = H_{G,z}(t, x).

Using (13), Chapman-Kolmogorov equation, and that supp g ∈ [α, β], we get

Λ(|h_{n,g}|)(t, x) ≤ \sqrt{n!} c_{0}^{d}p(t, x)^{−1} ∫∫ _{E} dsdy p(t − s, x − y)p(s, y) e^{|y|^{2} /4s} s^{− |n| /2} |g(s)|
≤ \sqrt{n!} (\sqrt{2}c_{0})^{d}p(t, x)^{−1} ∫_{0}^{t} ds p(t + s, x)s^{− |n| /2} |g(s)|
≤ \sqrt{n!} (\sqrt{2}c_{0})^{d} e^{|x|^{2} /4t} ∫_{0}^{t} ds s^{− |n| /2} |g(s)|
≤ \sqrt{n!} (\sqrt{2}c_{0})^{d} e^{|x|^{2} /4t} ∥g ∥_{∞}(β − α)α^{− |n| /2}.

The radius of the series ∑ a^{k}(k!α^{k})^{−1/2} is infinite. Thus for any (t, x) ∈ E the series ∑(n!)^{−1}z^{n}Λ(|h_{n,g}|)(t, x) are convergent. Fubini’s theorem implies that

∑_{n≥0} \frac{1}{n!} z^{n}Λ(h_{n,g})(t, x) = Λ (∑_{n≥0} \frac{1}{n!} z^{n}h_{n,g}) (t, x) = H_{G,z}(t, x).

Hence the two series ∑(n!)^{−1}z^{n}Λ(h_{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 that Λ_{2,i} has a bounded extension on L^{2}(p). We deduce from (12) that

Λ_{2,i}(h_{n,g})(t, x) = \frac{1}{2}∂_{i,i}^{2}Λ(h_{n,g})(t, x)

= \frac{1}{2t} n_{i}(n_{i} − 1)He_{n−2δ(i)}(x/\sqrt{t})t^{− |n| /2} ∫_{0}^{t} ds g(s). (17)

Let us introduce f ∈ A, i.e. for (t, x) ∈ E, f(t, x) = ∑_{n≥0} He_{n}(x/\sqrt{t})g_{n}(t), where g_{n} ∈ C_{0}^{∞}((0, ∞)) and g_{n} = 0 except for a finite number of terms. By linearity, we have

Λ_{2,i}(f)(t, x) = ∑_{n≥0} 2^{−1}n_{i}(n_{i} − 1)He_{n−2δ(i)}(x/\sqrt{t})t^{−1−|n| /2} ∫_{0}^{t} ds s^{|n| /2}g_{n}(s).

Thus, using (14), we have

∥Λ_{2,i}(f)∥_{(p)}^{2} = ∑_{n≥0}(n − 2δ(i))!4^{−1}n_{i}^{2}(n_{i} − 1)^{2} ∫_{0}^{∞} dt t^{−2−|n|} [∫_{0}^{t} ds s^{|n| /2}g_{n}(s)]^{2}

≤ ∑_{n≥0} n!\frac{n_{i}(n_{i} − 1)}{4} \frac{4}{(|n| +1)^{2}} ∫_{0}^{∞} dt g_{n}(t)^{2}

≤ ∥f ∥_{(p)}^{2},

where we used the Hardy inequality: for k > −1,

∫_{0}^{∞} dt t^{−2−k} [∫_{0}^{t} s^{k/2}h(s)ds]^{2} ≤ \frac{4}{(k + 1)^{2}} ∫_{0}^{∞} dt h(t)^{2}

for the first inequality and (15) for the second one. This means that Λ_{2,i}, defined on A, can be uniquely extended into a bounded operator Λ̃_{2,i} from L^{2}(p) into itself. The above inequality implies ∥Λ̃_{2,i} ∥_{(p)} ≤ 1.

For i ∈ {1, ⋯, d}, we set Λ_{4,i} = Λ_{3,i} + 2Λ_{2,i}. Using (12) and (11), we deduce from (16) that

(18) Λ_{4,i}(h_{n,g})(t, x)

\begin{matrix}= [−(x_{i}/\sqrt{t})∂_{i}He_{n}(x/\sqrt{t}) + n_{i}(n_{i} − 1)He_{n−2δ(i)}(x/\sqrt{t})] t^{−1−|n| /2} ∫_{0}^{t} ds g(s) \\ = −n_{i}He_{n}(x/\sqrt{t})t^{−1−|n| /2} ∫_{0}^{t} ds g(s). \end{matrix}

Arguing as above, we get for f ∈ A,

∥Λ_{4,i}(f)∥_{(p)}^{2} = ∑_{n≥0} n!n_{i}^{2} ∫_{0}^{∞} dt t^{−2−|n|} [∫_{0}^{t} ds s^{|n| /2}g_{n}(s)]^{2}

≤ ∑_{n≥0} n!n_{i}^{2} \frac{4}{(|n| +1)^{2}} ∫_{0}^{∞} dt g_{n}(t)^{2}

≤ 4 ∥f ∥_{(p)}^{2} .

Thus the operators Λ_{4,i} and Λ_{3,i}, defined on A, can be uniquely extended in bounded operators Λ̃_{4,i} and Λ̃_{3,i} from L^{2}(p) into itself. Furthermore we have ∥Λ̃_{4,i} ∥_{(p)} ≤ 2 and ∥Λ̃_{3,i} ∥_{(p)} ≤ ∥Λ̃_{4,i} ∥_{(p)} +2 ∥Λ̃_{2,i} ∥_{(p)} ≤ 4.

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

Λ_{1}(h_{n,g})(t, x)

= h_{n,g}(t, x) − \frac{1}{2} [|n| He_{n}(x/\sqrt{t}) + ∑_{i=1}^{d} \frac{x_{i}}{\sqrt{t}}∂_{i}He_{n}(x/\sqrt{t})] t^{−1−|n| /2} ∫_{0}^{t} ds g(s).

Then using (17) and (18), we get

Λ_{1}(h_{n,g}) = [I + \frac{1}{2} ∑_{i=1}^{d} [Λ_{4,i} + Λ_{3,i}]] (h_{n,g}) = [I + ∑_{i=1}^{d} [Λ_{4,i} − Λ_{2,i}]] (h_{n,g}).

This means that Λ_{1} = I + ∑_{i=1}^{d} [Λ_{4,i} − Λ_{2,i}] on A. Hence Λ_{1} can be uniquely extended in a bounded operator Λ̃_{1} from L^{2}(p) into itself and Λ̃_{1} = I + ∑_{i=1}^{d} [Λ̃_{4,i} − Λ̃_{2,i}]. We deduce that ∥Λ̃_{1} ∥_{(p)} ≤ 1 + 3d. □

Second step. We first consider the operators Λ_{3,i} for i ∈ {1, ⋯, d}. To check that Λ_{3,i} and Λ̃_{3,i} agree on C_{0}^{∞}(E), it is enough to check that for φ ∈ C_{0}^{∞}(E), Λ_{3,i}(φ)(t, x) = Λ̃_{3,i}(φ)(t, x) dtdx-a.e. Let φ ∈ C_{0}^{∞}(E). For k ∈ ℕ, we define,

φ_{k}(t, x) = ∑_{|n|≤k} He_{n}(x/\sqrt{t})(n!)^{−1} ∫_{ℝ^{d}} dy p(t, y)He_{n}(y/\sqrt{t})φ(t, y).

The sequence (φ_{k}, k ≥ 0) converges in L^{2}(p) to φ.

If x ∈ ℝ^{d}, y ∈ ℝ, i ∈ {1, ⋯, d}, we denote by z = x̂_{y}^{i} the element of ℝ^{d} such that z_{i} = y and z_{j} = x_{j} for j ≠ i. Since Λ_{3,i}(f)(t, x) = −t^{−1}x_{i}∂_{i}Λ(f)(t, x) for f ∈ A ∪ C_{0}^{∞}(E), we see that an integration by parts gives

∫_{0}^{x_{i}} dy Λ_{3,i}(f)(t, x̂_{y}^{i}) = −t^{−1}x_{i}Λ(f)(t, x) + ∫_{0}^{x_{i}} dy t^{−1}Λ(f)(t, x̂_{y}^{i}). (19)

For short we write P_{i}(f) for the operator P_{i}(f)(t, x) = ∫_{0}^{x_{i}} dy f(t, x̂_{y}^{i}). Let R > 0 and T > ε > 0 be fixed. Let Q = [ε, T] × [−R, R]^{d}. The heat kernel p is bounded below and above on Q by positive constant, say c_{Q} and C_{Q}. Using Cauchy-Schwarz inequality we have

∥1_{Q}P_{i}(f)∥_{(p)}^{2} ≤ C_{Q}R^{2} ∫∫ _{Q} dtdx f(t, x)^{2} ≤ C_{Q}c_{Q}^{−1}R^{2} ∥f ∥_{(p)}^{2} .

Thus the operator 1_{Q}P_{i} is continuous from L^{2}(p) to L^{2}(p). Thanks to Lemma 5 and the above first step, we get that the sequences (1_{Q}P_{i}(Λ_{0}(φ_{k})), k ≥ 0) and (1_{Q}P_{i}(Λ_{3,i}(φ_{k})), k ≥ 0) converge in L^{2}(p) respectively to 1_{Q}P_{i}(Λ_{0}(φ)) and 1_{Q}P_{i}(Λ̃_{3,i}(φ)). Notice also that (1_{Q}Λ(φ_{σ(k)}), k ≥ 0) converges in L^{2}(p) to 1_{Q}Λ(φ). Thus, there is a subsequence (σ(k), k ≥ 0) such that the sequences (1_{Q}P_{i}(Λ_{0}(φ_{σ(k)})), k ≥ 0), (1_{Q}P_{i}(Λ_{3,i}(φ_{σ(k)})), k ≥ 0) and (Λ(φ_{σ(k)}), k ≥ 0) converge dtdx-a.e. respectively to 1_{Q}P_{i}(Λ_{0}(φ)), 1_{Q}P_{i}(Λ̃_{3,i}(φ)) and Λ(φ). Now (19) holds for f = φ_{σ(k)}, this means that for (t, x) ∈ Q,

P_{i}(Λ_{3,i}(φ_{σ(k)}))(t, x) = −t^{−1}x_{i}Λ(φ_{σ(k)})(t, x) + P_{i}(Λ_{0}(φ_{σ(k)}))(t, x).

Taking the limit we get that dtdx-a.e. in Q,

P_{i}(Λ̃_{3,i}(φ))(t, x) = −t^{−1}x_{i}Λ(φ)(t, x) + P_{i}(Λ_{0}(φ))(t, x).

Since R, T, ε are arbitrary, the above equality holds dtdx-a.e. in E. Since (19) holds also for f = φ, we deduce that dtdx-a.e.,

∫_{0}^{x_{i}} dy Λ_{3,i}(φ)(t, x̂_{y}^{i}) = ∫_{0}^{x_{i}} dy Λ̃_{3,i}(φ)(t, x̂_{y}^{i}).

Hence we have dtdx-a.e., Λ_{3,i}(φ)(t, x) = Λ̃_{3,i}(φ)(t, x).

The proofs concerning the operators Λ_{1} and Λ_{2,i}, for i ∈ {1, ⋯, d}, and their extensions follow the same ideas. □

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