ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 10, Pages 68–77 (August 2, 2004) S 1079-6762(04)00131-3

A NEW INEQUALITY FOR SUPERDIFFUSIONS AND ITS APPLICATIONS TO NONLINEAR DIFFERENTIAL EQUATIONS

E. B. DYNKIN

(Communicated by Mark Freidlin)

Abstract. Our motivation is the following problem: to describe all positive solutions of a semilinear elliptic equation Lu = u^{α} with α > 1 in a bounded smooth domain E ⊂ ℝ^{d}. In 1998 Dynkin and Kuznetsov solved this problem for a class of solutions which they called σ-moderate. The question if all solutions belong to this class remained open. In 2002 Mselati proved that this is true for the equation Δu = u^{2} in a domain of class C^{4}. His principal tool— the Brownian snake—is not applicable to the case α ≠ 2. In 2003 Dynkin and Kuznetsov modified most of Mselati’s arguments by using superdiffusions instead of the snake. However a critical gap remained. A new inequality established in the present paper allows us to close this gap.

1. Introduction

1.1. Diffusions and superdiffusions. We denote by M(S) the set of all finite measures, and by P(S) the set of all probability measures on a measurable space S. B(E) stands for the set of all positive Borel functions on E. We use notation ⟨u, μ⟩ for the integral of u with respect to a measure μ, and notation P {A, Y } for the integral ∫_{A} Y dP.

Let L be an elliptic differential operator of the second order in ℝ^{d}. Under mild assumptions on the coefficients of L, there exists a continuous Markov process ξ = (ξ_{t}, Π_{x}) in ℝ^{d} whose transition density is a fundamental solution of the parabolic equation ∂u/∂t = Lu. We call this process a diffusion. For every open set D we denote by τ_{D} the first exit time of ξ from D.

Let ψ be a positive Borel function on ℝ_{+} = [0, ∞). Suppose that to every open set D and every μ ∈ M(ℝ^{d}) there corresponds a random measure (X_{D}, P_{μ}) on ℝ^{d} such that, for every f ∈ B(ℝ^{d}),

(1.1)  P_{μ}e^{−⟨f,X_{D}⟩} = e^{−⟨u,μ⟩}

where u satisfies the equation

(1.2)  u(x) + Π_{x} ∫_{0}^{τ_{D}} ψ[u(ξ_{t})]dt = Π_{x}f(ξ_{τ_{D}}).

Received by the editors April 23, 2004. 2000 Mathematics Subject Classification. Primary 60H30; Secondary 35J60, 60J60. Key words and phrases. Positive solutions of semilinear elliptic PDEs, superdiffusions, condi-

tional diffusions, ℕ-measures.

Partially supported by the National Science Foundation Grant DMS-0204237.

\begin{matrix}c \\ ◯ \end{matrix}2004 American Mathematical Society

Reverts to public domain 28 years from publication

We call the family X = (X_{D}, P_{μ}) a superdiffusion. [Heuristically, we have here a model of a random evolution of the cloud of particles, and X_{D} is a mass distribution on ∂D if each particle is frozen at the first exit from D.]

The existence of a superdiffusion is proved for a convex class of positive convex functions which contains the functions

(1.3)  ψ(u) = u^{α}, 0 < α ≤ 2.

[See, e.g., Chapter 4 in [Dy02].]

By restricting the family (X_{D}, P_{μ}) to D ⊂ E and μ ∈ M(E) we define a superdiffusion in an open set E.

A new tool—a family of measures ℕ_{x}, x ∈ E (defined on the same space OO as measures P_{μ})—was introduced in [DK04]. Our inspiration was the role played by an analog of these measures in Le Gall’s theory of the Brownian snake.^{1}

The range R_{E} of a superdiffusion in a domain E is a minimal closed set which supports, P_{x}-a.s. and ℕ_{x}-a.s., an exit measure X_{O} for an arbitrary open set O ⊂ E and for every x ∈ E.

1.2. Stochastic boundary values of harmonic functions. We say that a function h in E is harmonic in E if Lh = 0 in E and we use the notation H(E) for the set of all positive harmonic functions. If E is smooth,^{2} then there exists a 1-1 correspondence between H(E) and M(∂E). The harmonic function h_{ν} corresponding to ν ∈ M(∂E) is given by the formula

(1.4)  h_{ν}(x) = ∫_{∂E} k_{E}(x, y)ν(dy)

where k_{E}(x, y) is the Poisson kernel for L in E. For every ν ∈ M(∂E), there exists a random variable Z_{ν} such that

(1.5)  Z_{ν} = lim ⟨h_{ν}, X_{D_{n}}⟩ P_{x}-a.s. and ℕ_{x}-a.s.

for every x ∈ E and for every sequence D_{n} exhausting E.^{3} We call Z_{ν} the stochastic boundary value of h_{ν}.

The energy function for ν ∈ M(∂E) is defined by the formula

(1.6)  E_{x}(ν) = Π_{x} ∫_{0}^{τ_{E}} ψ[h_{ν}(ξ_{t})]dt.

1.3. Principal result.

Theorem 1.1. Suppose that D is a smooth open subset of a smooth domain E. If ν is a finite measure concentrated on ∂D ∩ ∂E and if E_{x}(ν) < ∞, then

(1.7) ℕ_{x}{R_{E} ⊂ D^{∗}, Z_{ν} ≠ 0} ≥ C(α)[ℕ_{x}{R_{E} ⊂ D^{∗}, Z_{ν}}]^{α/(α−1)}E_{x}(ν)^{−1/(α−1)}

where C(α) = (α − 1)^{−1}Γ(α − 1).^{4}

We prove Theorem 1.1 in Section 5 after the necessary tools have been prepared in Sections 2–4.

1Definitions of ℕ_{x} and other tools mentioned in the Introduction will be given in Section 2.

2We use the term smooth for open sets of class C^{2,λ} unless another class is indicated explicitly.

3Domains D_{n} exhaust E if D̄_{n} ⊂ D_{n+1} and if the union of D_{n} is equal to E.

4Here Γ is Euler’s gamma-function.

1.4. Applications to differential equations. We denote by U(E) the set of all positive solutions of the equation

(1.8)  Lu = ψ(u) in E.

We say that an element u of U(E) is moderate if u ≤ h for some h ∈ H(E). There exists a 1-1 correspondence between the set U_{1}(E) of all moderate solutions and a subset H_{1}(E) of H(E): h ∈ H_{1}(E) is the minimal harmonic function dominating u ∈ U_{1}(E), and u is the maximal solution dominated by h. We put ν ∈ N_{1}^{E} if h_{ν} ∈ H_{1}(E). We denote by u_{ν} the element of U_{1}(E) corresponding to h_{ν}. These elements are related by the formula

(1.9)  u_{ν}(x) + E_{x}(ν) = h_{ν}(x).

If E_{x}(ν) < ∞ for some x ∈ E, then ν ∈ N_{1}^{E}.^{5} To every closed subset K of ∂E there correspond two elements of U(E):^{6}

(1.10)  w_{K}(x) = − log P_{x}{R_{E} ∩ K = ∅} = ℕ_{x}{R_{E} ∩ K ≠ ∅}

and

(1.11)  u_{K}(x) = sup u_{ν}(x)

where the supremum is taken over all ν ∈ N_{1}^{E} concentrated on K.

We say that u ∈ U(E) is σ-moderate if there exist moderate solutions u_{n} such that u_{n} ↑ u. All solutions u_{K} are σ-moderate.

Theorem 1.1 in combination with the results presented in Chapter 11, Section 7.1 of [Dy02] and in [Dy04a], [Dy04c], [DK03], [DK04], [Ku04] makes it possible to prove the following two theorems:^{7}

Theorem 1.2. If E is a domain of class C^{4} and if L is the Laplacian Δ, then

(1.12)  u_{K} = w_{K} for all closed K ⊂ ∂E.

Theorem 1.3. Under the conditions of Theorem 1.2 all elements of U(E) are σ-moderate.

[Marcus and Véron proved in [MV04] that the equation (1.12) can be established by a purely analytical method applicable to all α > 1.]

2. Tools

2.1. h-transform and conditional diffusion. Suppose ξ is a diffusion in a domain E with the transition function p_{t}(x, dy), and let h ∈ H(E). Then

(2.1)  p_{t}^{h}(x, dy) = \frac{1}{h(x)}p_{t}(x, dy)h(y)

is the transition function of a continuous Markov process (ξ_{t}, Π̂_{x}^{h}) in E called the h-transform of ξ. We prefer to deal with measures Π_{x}^{h} = h(x)Π̂_{x}^{h} which depend linearly on h. Put Π_{x}^{ν} = Π_{x}^{h_{ν}} and Π̂_{x}^{y} = Π̂_{x}^{δ_{y}} where δ_{y} is the unit mass at a point y. The process (ξ_{t}, Π̂_{x}^{y}) can be interpreted as a diffusion starting from x ∈ E and conditioned to exit from E at y.

The following lemma is proved, for instance, in [Dy02], page 103:

5This follows, for instance, from Theorem 3.2 of Chapter 8 in [Dy02].

6w_{K} can be characterized as the maximal element of U(E) vanishing on ∂E \ K.

7Proofs of these theorems are sketched in [Dy04b]. The complete proofs are contained in the forthcoming book [Dy04d].

Lemma 2.1. For every stopping time τ and every pre-τ positive Y,

(2.2)  Π_{x}^{h}Y 1_{τ<τ_{E}} = Π_{x}Y h(ξ_{τ})1_{τ<τ_{E}}.

2.2. Measures ℕ_{x}. Denote by Z_{x} the class of all functions of the form

(2.3)  Z = ∑_{1}^{n}⟨f_{i}, X_{O_{i}}⟩

where O_{1}, …, O_{n} is a finite family of neighborhoods of x and f_{1}, …, f_{n} ∈ B(ℝ^{d}). By Theorem 1.1 in [DK04], for every x ∈ E, there exists a unique measure ℕ_{x} with the properties:

(i) For every Z ∈ Z_{x},

(2.4)  ℕ_{x}(1 − e^{−Z}) = − log P_{x}e^{−Z}.

(ii) If Ω̄ is the intersection of {X_{O} = 0} over all neighborhoods O of x, then ℕ_{x}(Ω̄) = 0.

2.3. Stochastic boundary values and range. Suppose that u ∈ B(E). A random variable Z_{u} is called a stochastic boundary value of u [we write Z_{u} = SBV(u)] if

(2.5)  Z_{u} = lim ⟨u, X_{D_{n}}⟩ P_{x}-a.s. and ℕ_{x}-a.s.

for every x ∈ E and every sequence D_{n} exhausting E.

For every u ∈ B(ℝ^{d}), we put

V_{D}(u)(x) = − log P_{x}e^{−⟨u,X_{D}⟩}.

Denote by U^{−}(E) the set of u such that V_{D}(u) ≤ u for all D ⊂ E. This condition holds for all u ≥ 0 such that Lu ≤ ψ(u) in E. In particular, it holds for u ∈ H(E). Since V_{D}(u_{1} + u_{2}) ≤ V_{D}(u_{1}) + V_{D}(u_{2}) (Theorem 2.1 of Chapter 8 in [Dy02]), the sum of two elements of U^{−}(E) belongs to U^{−}(E).

By Theorem 1.2 in [DK04], a stochastic boundary value Z_{u} exist for every u ∈ U^{−}(E) and

(2.6)  ℕ_{x}(1 − e^{−Z_{u}}) = − log P_{x}e^{−Z_{u}}.

Formula (1.5) means that Z_{ν} = SBV(h_{ν}). If ν ∈ N_{1}^{E}, then Z_{ν} is also SBV(u_{ν}).

By Theorem 1.3 in [DK04], for every domain E, there exists a random closed set R_{E} with the properties:

(a) For every open O ⊂ E and every x ∈ E, the measure X_{O} is concentrated, P_{x}-a.s. and ℕ_{x}-a.s., on R_{E}.

(b) If (a) holds for a random closed set F, then, for every x ∈ E, R ⊂ F P_{x}-a.s. and ℕ_{x}-a.s.

We call R_{E} the range of X in E. We denote by R the range of X in ℝ^{d}.

2.4. More relations between measures P_{x} and ℕ_{x}. By Theorem 1.4 in [DK04], for every u ∈ U^{−}(E) and every Borel set Γ ⊂ ∂E,

(2.7) − log P_{x}{R_{E} ∩Γ = ∅, e^{−Z_{u}}} = ℕ_{x}{R_{E} ∩Γ ≠ ∅}+ℕ_{x}{R_{E} ∩Γ = ∅, 1−e^{−Z_{u}}}.

This function is the maximal element of U(E) dominated by w_{Γ} + u.

By taking Z = 0, we get

(2.8)  − log P_{x}{R_{E} ∩ Γ = ∅} = ℕ_{x}{R_{E} ∩ Γ ≠ ∅}.

It follows from (2.7) and (2.8) that, if

P_{x}{R_{E} ∩ Γ = ∅} > 0,

then

(2.9)  ℕ_{x}{R_{E} ∩ Γ = ∅, 1 − e^{−Z}} = − log P_{x}{e^{−Z} | R_{E} ∩ Γ = ∅}.

By applying (2.7) to λZ and passing to the limit as λ → +∞, we get

(2.10) − log P_{x}{R_{E} ∩ Γ = ∅, Z = 0} = ℕ_{x}{R_{E} ∩ Γ ≠ ∅} + ℕ_{x}{R_{E} ∩ Γ = ∅, Z ≠ 0}.

By Proposition 1.1 in [DK04],

(2.11)  ℕ_{x}Z_{ν} = P_{x}Z_{ν} if P_{x}Z_{ν} < ∞.

On the other hand, for every f ∈ B(D̄),

(2.12)  P_{x}⟨f, X_{D}⟩ = Π_{x}f(ξ_{τ_{D}})

(see, e.g., [Dy02], Chapter 4, Lemma 4.1). It follows from (2.5), (2.11), (2.12), Fatou’s lemma and the mean value property of harmonic functions, that

(2.13)  ℕ_{x}Z_{ν} = P_{x}Z_{ν} ≤ h_{ν}(x) < ∞ for every ν ∈ M(∂E).

Proposition 2.1. Suppose x ∈ D, Λ is a Borel subset of ∂D and A = {R∩Λ = ∅}. We have P_{x}A > 0 and, for all Z′, Z′′ ∈ Z_{x},

(2.14) ℕ_{x}{A, (e^{−Z′} − e^{−Z′′})^{2}}

= −2 log P_{x}{e^{−Z′−Z′′} | A} + log P_{x}{e^{−2Z′} | A} + log P_{x}{e^{−2Z′′} | A}.

If Z′ = Z′′ P_{x}-a.s. on A and if P_{x}{A, Z′ < ∞} > 0, then Z′ = Z′′ ℕ_{x}-a.s. on A.

Proof. First, P_{x}A > 0 because P_{x}A = e^{−w_{Λ}(x)}. Next

(e^{−Z′} − e^{−Z′′})^{2} = 2(1 − e^{−Z′−Z′′}) − (1 − e^{−2Z′}) − (1 − e^{−2Z′′}).

Therefore (2.14) follows from (2.9). The second part of the proposition is an obvious implication of (2.14). □

2.5. Properties of superdiffusions. The following properties are often used in the theory of superdiffusions. [They are a part of the definition of branching exit Markov systems, and superdiffusions are a special case of such systems (see [Dy02], Chapters 3 and 4).]

2.5.A. (Markov property) If Y ≥ 0 is measurable with respect to the σ-algebra generated by X_{D′}, D′ ⊂ D and Z ≥ 0 is measurable with respect to the σ-algebra generated by X_{D′′}, D′′ ⊃ D, then

(2.15)  P_{μ}(Y Z) = P_{μ}(Y P_{X_{D}}Z).

2.5.B. If μ(E) = 0, then P_{μ}{X_{E} = μ} = 1.

We use 2.5.A, 2.5.B and Proposition 2.1 to prove the next proposition.

Proposition 2.2. Let D ⊂ E be two open sets. Then, for every x ∈ D, X_{D} and X_{E} coincide P_{x}-a.s. and ℕ_{x}-a.s. on the set A = {R_{D} ⊂ D^{∗}}.

[Note that

(2.16)  D^{∗} = {x ∈ D̄ : d(x, Λ) > 0}

where Λ = ∂D ∩ E.] 3. Relations between superdiffusions and conditional diffusions

in two open sets

3.1. Now we consider two bounded smooth open sets D ⊂ E. We denote by Z̃_{ν} the stochastic boundary value of h̃_{ν}(x) = ∫_{∂D} k_{D}(x, y)ν(dy) in D; Π̃_{x}^{y} refers to the diffusion in D conditioned to exit at y ∈ ∂D.

Theorem 3.1. Put A = {R_{D} ⊂ D^{∗}}. For every x ∈ D,

(3.1)  R_{E} = R_{D} P_{x}-a.s. and ℕ_{x}-a.s.

and

(3.2)  Z_{ν} = Z̃_{ν} P_{x}-a.s. and ℕ_{x}-a.s. on A

for all ν ∈ N_{1}^{E} concentrated on ∂D ∩ ∂E.

Proof. 1^{∘}. First, we prove (3.1). Clearly, R_{D} ⊂ R_{E} P_{x}-a.s. and ℕ_{x}-a.s. for all x ∈ D. We get (3.1) if we show that, if O is an open subset of E, then, for every x ∈ D, X_{O} = X_{O∩D} P_{x}-a.s. on A and, for every x ∈ O ∩ D, X_{O} = X_{O∩D} ℕ_{x}-a.s. on A. For x ∈ O∩D this follows from Proposition 2.2 applied to O∩D ⊂ O because {R_{D} ⊂ D^{∗}} ⊂ {R_{O∩D} ⊂ (O ∩ D)^{∗}}. For x ∈ D \ O, P_{x}{X_{O} = X_{D∩O} = δ_{x}} = 1.

2^{∘}. Put

(3.3)  D_{m}^{∗} = {x ∈ D̄ : d(x, E \ D) > 1/m}.

To prove (3.2), it is sufficient to prove that it holds on A_{m} = {R_{D} ⊂ D_{m}^{∗}} for all sufficiently large m. First we prove that, for all x ∈ D,

(3.4)  Z_{ν} = Z̃_{ν} P_{x}-a.s. on A_{m}.

We get (3.4) by proving that both Z_{ν} and Z̃_{ν} coincide P_{x}-a.s. on A_{m} with the stochastic boundary value Z^{∗} of h_{ν} in D.

Let

E_{n} = {x ∈ E : d(x, ∂E) > 1/n}, D_{n} = {x ∈ D : d(x, ∂D) > 1/n}.

If n > m, then

A_{m} ⊂ A_{n} ⊂ {R_{D} ⊂ D_{n}^{∗}} ⊂ {R_{D_{n}} ⊂ D_{n}^{∗}}.

We apply Proposition 2.2 to D_{n} ⊂ E_{n} and we get that, P_{x}-a.s. on {R_{D_{n}} ⊂ D_{n}^{∗}} ⊃ A_{m}, X_{D_{n}} = X_{E_{n}} for all n > m, which implies Z^{∗} = Z_{ν}.

3^{∘}. Now we prove that

(3.5)  Z^{∗} = Z̃_{ν} P_{x}-a.s. on A_{m}.

Consider h^{0} = h_{ν} − h̃_{ν} and Z^{0} = Z_{ν}^{∗} − Z̃_{ν}. If y ∈ ∂D ∩ ∂E, then

(3.6)  k_{E}(x, y) = k_{D}(x, y) + Π_{x}{τ_{D} < τ_{E}, k_{E}(ξ_{τ_{D}}, y)}.

Therefore

(3.7)  h^{0}(x) = Π_{x}{ξ_{τ_{D}} ∈ ∂D ∩ E, h_{ν}(ξ_{τ_{D}})}.

This is a harmonic function in D. It vanishes on Γ_{m} = ∂D ∩ D_{m}^{∗} = ∂E ∩ D_{m}^{∗}.

We claim that, for every ε > 0 and every m, h^{0} < ε on Γ_{m,n} = ∂E_{n} ∩ D_{m}^{∗} for all sufficiently large n. [If this is not true, then there exists a sequence n_{i} → ∞ such that z_{n_{i}} ∈ Γ_{m,n_{i}} and h^{0}(z_{n_{i}}) ≥ ε. If z is a limit point of z_{n_{i}}, then z ∈ Γ_{m} and h^{0}(z) ≥ ε.]

All measures X_{D_{n}} are concentrated, P_{x}-a.s., on R_{D}. Therefore A_{m} implies that they are concentrated, P_{x}-a.s., on D_{m}^{∗}. Since Γ_{m,n} ⊂ D_{m}^{∗}, we conclude that, for all sufficiently large n, ⟨h^{0}, X_{D_{n}}⟩ < ε⟨1, X_{D_{n}}⟩ P_{x}-a.s. on A_{m}. This implies (3.5).

4^{∘}. If ν ∈ M(∂E) and Z_{ν} = SBV(h_{ν}), then

(3.8)  ℕ_{x}Z_{ν} = P_{x}Z_{ν} ≤ h_{ν}(x) < ∞.

Note that P_{x}A > 0. It follows from (3.8) that Z_{ν} < ∞ P_{x}-a.s. and therefore P_{x}{A, Z_{ν} < ∞} > 0. By Proposition 2.1, (3.2) follows from (3.4) . □

3.2. We also need the following result (see [Dy04a], Lemma 3.2).

Theorem 3.2. Suppose that D ⊂ E are smooth open sets. Denote by F̃ the σalgebra in Ω generated by the sets {s < τ_{D}, ξ_{s} ∈ B} where s ≥ 0, B ∈ B(E). We have

(3.9)  Π̃_{x}^{y}Y = Π_{x}^{y}{τ_{D} = τ_{E}, Y }

for all x ∈ D, y ∈ ∂E ∩ ∂D and for all Y ∈ F̃.

Corollary 3.1. If

(3.10)  F_{t} = exp [ − ∫_{0}^{t} a(ξ_{s}) ds]

where a is a positive continuous function on [0, ∞), then, for y ∈ ∂D ∩ ∂E,

(3.11)  Π̃_{x}^{y}F_{τ_{D}} = Π_{x}^{y}{τ_{D} = τ_{E}, F_{τ_{E}}}.

Indeed, it is easy to see that Fτ̃ ∈ F̃.

4. Equations connecting P_{x} and ℕ_{x} with Π_{x}^{ν}

4.1.

Theorem 4.1. Let Z_{ν} = SBV(h_{ν}), Z_{u} = SBV(u) where ν ∈ N_{1}^{E} and u ∈ U(E). Then

(4.1)  P_{x}Z_{ν}e^{−Z_{u}} = e^{−u(x)}Π_{x}^{ν}e^{−Φ(u)}

and

(4.2)  ℕ_{x}Z_{ν}e^{−Z_{u}} = Π_{x}^{ν}e^{−Φ(u)}

where

(4.3)  Φ(u) = ∫_{0}^{τ_{E}} ψ′[u(ξ_{t})]dt.

Proof. Formula (4.1) follows from Theorem 3.1 in Chapter 9 of [Dy02]. To prove (4.2), we observe that, for every λ ≥ 0, λZ_{ν} + Z_{u} = SBV(v) where v = λh_{ν} + u ∈ U^{−}(E) and therefore, by (2.6),

(4.4)  ℕ_{x}(1 − e^{−λZ_{ν}−Z_{u}}) = − log P_{x}e^{−λZ_{ν}−Z_{u}}.

By taking the derivatives with respect to λ at λ = 0,^{8} we get

(4.5)  ℕ_{x}Z_{ν}e^{−Z_{u}} = P_{x}Z_{ν}e^{−Z_{u}}/P_{x}e^{−Z_{u}}.

By Theorem 1.1 of Chapter 9 in [Dy02],

(4.6)  P_{x}e^{−Z_{u}} = e^{−u(x)}.

8The differentiation under the integral signs is justified by (2.13).

Therefore (4.4) follows from (4.1), (4.5) and (4.6). □

Theorem 4.2. Suppose that D ⊂ E are bounded smooth open sets and Λ, L, D^{∗} are the sets introduced in Theorem 1.1. Let ν be a finite measure on ∂D ∩ ∂E, x ∈ E and E_{x}(ν) < ∞. Put

\begin{aligned}w_{Λ}(x) & = ℕ_{x}\{R_{D} ∩ Λ ≠ ∅\}, \\ v_{s}(x) & = w_{Λ}(x) + ℕ_{x}\{R_{D} ∩ Λ = ∅, 1 − e^{−sZ_{ν}} \} \end{aligned} ; (4.7)

for x ∈ D and let w_{Λ}(x) = v_{s}(x) = 0 for x ∈ E \ D. For every x ∈ E, we have

(4.8)  ℕ_{x}{R_{E} ⊂ D^{∗}, Z_{ν}} = Π_{x}^{ν}{A, e^{−Φ(w_{Λ})}},
(4.9)  ℕ_{x}{R_{E} ⊂ D^{∗}, Z_{ν} ≠ 0} = ∫_{0}^{∞} Π_{x}^{ν}{A, e^{−Φ(v_{s})}}ds

where Φ is defined by (4.3) and

(4.10)  A = {τ_{E} = τ_{D}} = {ξ_{t} ∈ D for all t < τ_{E}}.

Remark. Since E_{x}(ν) < ∞, ν belongs to N_{x}^{E} and to N_{x}^{D}.

Proof. 1^{∘}. If x ∈ E \ D, then, ℕ_{x}-a.s., R_{E} is not a subset of D^{∗} because R_{E} contains supports of X_{O} for all neighborhoods O of x and we can choose O such that Ō ∩ D^{∗} = ∅. On the other hand, Π_{x}^{ν}(A) = 0. Therefore (4.8) and (4.9) hold independently of values of w_{Λ} and v_{s}.

2^{∘}. Now we assume that x ∈ D. Put A = {R_{D} ⊂ D^{∗}}. We claim that

A = {R_{E} ⊂ D^{∗}} ℕ_{x}-a.s.

Indeed, {R_{E} ⊂ D^{∗}} ⊂ A because R_{D} ⊂ R_{E}. By Theorem 3.1, A ⊂ {R_{D} = R_{E}} ℕ_{x}-a.s. Hence, A ⊂ {R_{E} ⊂ D^{∗}}.

By Theorem 3.1, R_{D} = R_{E} and Z_{ν} = Z̃_{ν} ℕ_{x}-a.s. on A. Therefore

\begin{aligned}ℕ_{x}\{R_{E} ⊂ D^{∗}, Z_{ν}\} & = ℕ_{x}\{A, Z_{ν}\} = ℕ_{x}\{A, Z̃_{ν}\}, \\ ℕ_{x}\{R_{E} ⊂ D^{∗}, Z_{ν}e^{−sZ_{ν}} \} & = ℕ_{x}\{A, Z_{ν}e^{−sZ_{ν}} \} = ℕ_{x}\{A, Z̃_{ν}e^{−sZ̃_{ν}} \}. \end{aligned} ; (4.11)

Formula (4.7) defines two elements of U(D). The stochastic boundary value Z_{Λ} of w_{Λ} in D is equal to ∞1_{A^{c}} (Remark 1.2 on p. 133 in [Dy02]) and therefore

(4.12)  e^{−Z_{Λ}} = 1_{A}.

By (2.7) and (2.8), v_{s}(x) = − log P_{x}{R_{D} ∩ Λ = ∅, e^{−sZ_{ν}} } and, by Remark 2.1 on p. 137 in [Dy02], the stochastic boundary value Z^{s} of v_{s} in D is equal to Z_{Λ} + sZ̃_{ν}. Hence,

(4.13)  e^{−Z^{s}} = 1_{A}e^{−sZ̃_{ν}} .

By (4.11), (4.12) and (4.13),

(4.14)  ℕ_{x}{A, Z_{ν}} = ℕ_{x}{Z̃_{ν}e^{−Z_{Λ}}}

and

(4.15)  ℕ_{x}{A, Z_{ν}e^{−sZ_{ν}} } = ℕ_{x}{Z̃_{ν}e^{−Z^{s}}}.

By applying formula (4.2) to Z̃_{ν} and to the restriction of w_{Λ} to D, we conclude from (4.14) that

(4.16)  ℕ_{x}{A, Z_{ν}} = Π̃_{x}^{ν} exp [ − ∫_{0}^{τ_{D}} ψ′[w_{Λ}(ξ_{s})]ds]

and, by Corollary 3.1,

(4.17)  ℕ_{x}{A, Z_{ν}} = Π_{x}^{ν}{A, e^{−Φ(w_{Λ})}}.

Analogously, (4.2) applied to the restriction of v_{s} to D, in combination with (4.15) and (3.11), yields

(4.18)  ℕ_{x}{A, Z_{ν}e^{−sZ_{ν}} } = Π_{x}^{ν}{A, e^{−Φ(v_{s})}}.

Formula (4.8) follows from (4.17) and formula (4.9) follows from (4.18) because

(4.19)  ℕ_{x}{A, Z_{ν} ≠ 0} = lim_{t→∞} ℕ_{x}{A, 1 − e^{−tZ_{ν}} }

and

(4.20)  1 − e^{−tZ_{ν}} = ∫_{0}^{t} Z_{ν}e^{−sZ_{ν}} ds.
□

5. Proof of Theorem 1.1

We use the following two elementary inequalities:

5.A. For all a, b ≥ 0 and 0 < β < 1,

(5.1)  (a + b)^{β} ≤ a^{β} + b^{β}.

Proof. It is sufficient to prove (5.1) for a = 1. Put f(t) = (1 + t)^{β} − t^{β}. Note that f(0) = 1 and f′(t) ≤ 0 for t > 0. Hence f(t) ≤ 1 for t ≥ 0. □

5.B. For every finite measure M, every positive measurable function Y and every β > 0,

M(Y^{−β}) ≥ M(1)^{1+β}(MY)^{−β}.

Indeed f(y) = y^{−β} is a convex function on ℝ_{+}, and we get 5.B by applying Jensen’s inequality to the probability measure M/M(1).

Proof of Theorem 1.1. 1^{∘}. If x ∈ E \ D, then, ℕ_{x}-a.s., R_{E} is not a subset of D^{∗} (see the proof of Theorem 4.2). Hence, both sides of (1.7) vanish.

2^{∘}. Suppose x ∈ D. By (2.6), ℕ_{x}(1 − e^{−sZ_{ν}}) = u_{sν}. Thus (4.7) implies v_{s} ≤ w_{Λ} + u_{sν}. Therefore, by 5.A, v_{s}^{α−1} ≤ w_{Λ}^{α−1} + u_{sν}^{α−1} and, since u_{sν} ≤ h_{sν} = sh_{ν}, Φ(v_{s}) ≤ Φ(w_{Λ}) + s^{α−1}Φ(h_{ν}).

Put A = {R_{E} ⊂ D^{∗}}. It follows from (4.9) that

(5.2)  ℕ_{x}{A, Z_{ν} ≠ 0} ≥ Π_{x}^{ν}{A, ∫_{0}^{∞} e^{−Φ(w_{Λ})−s^{α−1}Φ(h_{ν})}ds}.

Note that ∫_{0}^{∞} e^{−as^{β}}ds = Ca^{−1/β} where C = ∫_{0}^{∞} e^{−t^{β}}dt. Therefore (5.2) implies

(5.3)  ℕ_{x}{A, Z_{ν} ≠ 0} ≥ CΠ_{x}^{ν}{A, e^{−Φ(w_{Λ})}Φ(h_{ν})^{−1/(α−1)}}.

The right side in (5.3) is equal to CM(Y^{−β}) where β = 1/(α − 1), Y = Φ(h_{ν}) and M is the measure with the density 1_{A}e^{−Φ(w_{Λ})} with respect to Π_{x}^{ν}. We get from (5.3) and 5.B that

ℕ_{x}{A, Z_{ν} ≠ 0} ≥ CM(1)^{1+β}(MY)^{−β}

= C[Π_{x}^{ν}{A, e^{−Φ(w_{Λ})}}]^{α/(α−1)}[Π_{x}^{ν}{A, e^{−Φ(w_{Λ})}Φ(h_{ν})}]^{−1/(α−1)}.

By (4.8), Π_{x}^{ν}{A, e^{−Φ(w_{Λ})}} = ℕ_{x}{R_{E} ⊂ D^{∗}, Z_{ν}} and, since Π_{x}^{ν}{A, e^{−Φ(w_{Λ})}Φ(h_{ν})} ≤ Π_{x}^{ν}Φ(h_{ν}), we have

(5.4)  ℕ_{x}{A, Z_{ν} ≠ 0} ≥ C[ℕ_{x}{R_{E} ⊂ D^{∗}, Z_{ν}}]^{α/(α−1)}[Π_{x}^{ν}Φ(h_{ν})]^{−1/(α−1)}.

3^{∘}. By the definition of h-transform, for every f ∈ B(E) and every h ∈ H(E),

Π_{x}^{h} ∫_{0}^{τ_{E}} f(ξ_{t})dt = ∫_{0}^{∞} Π_{x}^{h}{t < τ_{E}, f(ξ_{t})}dt = ∫_{0}^{∞} Π_{x}{t < τ_{E}, f(ξ_{t})h(ξ_{t})}dt.

By taking f = αh_{ν}^{α−1} and h = h_{ν} we get

(5.5)  Π_{x}^{ν}Φ(h_{ν}) = αE_{x}(ν).

Formula (1.7) follows from (5.4) and (5.5). □

References

[Dy91] E. B. Dynkin, A probabilistic approach to one class of nonlinear differential equations, Probab. Th. Rel. Fields 89 (1991), 89–115. MR 1109476 (92d:35090)

[Dy02] , Diffusions, superdiffusions and partial differential equations, American Mathematical Society, Providence, RI, 2002. MR 1883198 (2003c:60001)

[Dy04a] , On upper bounds for positive solutions of semilinear equations, J. Functional Analysis 210 (2004), 73–100. MR 2051633

[Dy04b] , Superdiffusions and positive solutions of nonlinear partial differential equations, Uspekhi Matem. Nauk 59 (2004), to appear.

[Dy04c] , Absolute continuity results for superdiffusions with applications to differential equations, C. R. Acad. Sc. Paris, Série I, 338 (2004), 605–610. MR 2056468

[Dy04d] , Superdiffusions and positive solutions of nonlinear partial differential equations, American Mathematical Society, Providence, RI, 2004, to appear.

[DK03] E. B. Dynkin and S. E. Kuznetsov, Poisson capacities, Math. Research Letters 10 (2003), 85–95. MR 1960126 (2003k:31005)

[DK04] , ℕ-measures for branching exit Markov systems and their applications to differential equations, Probab. Theory and Related Fields, to appear.

[Ku04] S. E. Kuznetsov, An upper bound for positive solutions of the equation Δu = u^{α}, Amer. Math. Soc., Electronic Research Announcements, to appear.

[MV04] M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion, J. European Math. Soc., to appear.

[Ms02] B. Mselati, Classification et représentation probabiliste des solutions positives de Δu = u^{2} dans un domaine, Thése de Doctorat de l’Université Paris 6, 2002.

[Ms04] B. Mselati, Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation, Memoirs of the American Mathematical Society 168 (2004), no. 798, to appear.

Department of Mathematics, Cornell University, Ithaca, NY 14853

E-mail address: ebd1@cornell.edu