Elliptic operators generating stochastic semigroups

Hendrik Vogt

Fachrichtung Mathematik, Technische Universität Dresden, D-01062 Dresden, Germany

Abstract

We use an intrinsic metric type approach to investigate when C_{0}-semigroups generated by second order elliptic differential operators are stochastic. We give a new condition for stochasticity that encompasses the volume growth conditions by Karp and Li and by Perelmuter and Semenov.

MSC 2000: 47D07, 35J15, 47B44

1 Introduction

Let Ω ⊆ ℝ^{n} be an open set, and let a: Ω → ℝ^{n×n} be a measurable function with values in the positive semidefinite matrices. We assume that a is sectorial, i.e., there exists α ⩾ 1 such that

(aξ · η)^{2} ⩽ α(aξ · ξ)(aη · η) a.e.  (ξ, η ∈ ℝ^{n}).  (1.1)

Let m ∈ L_{1,loc}(Ω) such that m > 0 a.e. and ma is locally integrable, and let μ := mλ^{n}, where λ^{n} is the Lebesgue measure on Ω. We are going to study conditions under which certain realisations of the operator \frac{1}{m}∇ · (ma∇) generate stochastic C_{0}-semigroups on L_{1}(μ). Throughout the paper, the function spaces are assumed to be real vector spaces.

In order to give a precise meaning to the operator in question, we define the sectorial form τ_{0,max} in L_{2}(μ) by

D(τ_{0,max}) := {u ∈ L_{2}(μ) ∩ W_{1,loc}^{1}(Ω); a∇u · ∇u ∈ L_{1}(μ)},
τ_{0,max}(u, v) := ∫_{Ω} a∇u · ∇v dμ.

Note that C_{c}^{∞}(Ω) ⊆ D(τ_{0,max}) since ma is locally integrable. Let τ_{0} be a restriction of τ_{0,max} that is densely defined and satisfies F ∘ u ∈ D(τ_{0}) for all u ∈ D(τ_{0}) and all normal contractions F ∈ C^{∞}(ℝ). Examples for D(τ_{0}) are D(τ_{0,max}) itself, C_{c}^{∞}(Ω), and the intersection of D(τ_{0,max}) with C^{∞}(Ω), {u|_{Ω}; u ∈ C^{∞}(ℝ^{n})} or {u|_{Ω}; u ∈ C_{c}^{∞}(ℝ^{n})}.

We are particularly interested in the case that the form τ_{0} is Neumann type, by which we mean that

D(τ_{0,max}) ∩ {u|_{Ω}; u ∈ C_{c}^{∞}(ℝ^{n})} ⊆ D(τ_{0}).  (1.2)

This condition will be needed (in combination with the assumption μ = λ^{n}) in Examples 3.2 and 3.5 and in Corollary 4.4.

We assume that τ_{0} is closable; then τ := \overline{τ_{0}} is a (non-symmetric) Dirichlet form (see, e.g., [MaRö92; Prop. I.4.10]). For conditions ensuring closability of τ_{0} we refer to [RöWi85]; see also [VoVo03; Prop. A.1] for the case τ_{0} = τ_{0,max}. Let A be the m-sectorial operator in L_{2}(μ) associated with τ. Then −A generates a C_{0}-semigroup T_{2} on L_{2}(μ), and the Beurling-Deny criteria imply that T_{2} extrapolates to a substochastic C_{0}-semigroup T_{1} on L_{1}(μ) and to a sub-Markovian weak^{∗}-continuous semigroup T_{∞} on L_{∞}(μ).

Within the above framework, we are going to study when the semigroup T_{1} is stochastic, i.e., when ||T_{1}(t)f||_{1} = ||f||_{1} for all t ⩾ 0, 0 ⩽ f ∈ L_{1}(μ). Related problems studied in the literature are conservativeness of the L_{∞}-semigroup and L_{1}-uniqueness. In the symmetric case it is clear that T_{1} is stochastic if and only if T_{∞} is conservative, i.e., T_{∞}11_{Ω} = 11_{Ω} for all t ⩾ 0.

If C_{c}^{∞}(Ω) ⊆ D(A_{1}), where A_{1} is the generator of T_{1}, then A_{1}|_{C_{c}^{∞}(Ω)} is said to be L_{1}-unique if C_{c}^{∞}(Ω) is a core for A_{1}. For Ω = ℝ^{n}, D(τ_{0}) = C_{c}^{∞}(ℝ^{n}) and sufficiently regular coefficients, stochasticity of T_{1} is equivalent to L_{1}-uniqueness of A_{1}|_{C_{c}^{∞}(Ω)}. It follows from [Sta99; Cor. 2.2] that this equivalence holds if m = φ^{2} for some φ ∈ W_{2,loc}^{1}(ℝ^{n}), a ∈ L_{∞,loc}(ℝ^{n})^{n×n}, ∂_{j}a ∈ L_{2,loc}(μ)^{n×n} (j = 1, …, n) and a is locally Hölder continuous and locally uniformly elliptic. (These assumptions imply that A_{1} is an extension of the operator −\frac{1}{m}∇·(ma∇) with domain C_{c}^{∞}(Ω).)

There are two important means to approaching the above problems: Lyapunov functions and the intrinsic metric of τ. In [Dav85; Sec. 2] a Lyapunov function approach going back to [Kha60] is used to study conservativeness in the case of smooth symmetric coefficients (and manifolds, not just subdomains of ℝ^{n}). This approach has been generalised in [Sta99; Sec. I.1] to non-symmetric operators with non-smooth coefficients; there, stochasticity is studied for non-divergence form operators, but the operators can be written in divergence form.

1.1 Example. Let Ω = ℝ^{n}, D(τ_{0}) = C_{c}^{∞}(ℝ^{n}), m = 11_{ℝ^{n}} and a(x) = a_{0}(|x|)I_{n} (x ∈ ℝ^{n}), where a_{0}: ℝ → (0, ∞) is an even C^{∞}-function satisfying ∫_{0}^{∞} \frac{r}{a_{0}(r)} dr = ∞ and I_{n} is the identity matrix. Then from [Dav85; Thm. 2.4] it follows that T_{1} is stochastic. Indeed, for the Lyapunov function

u(x) := n + ∫_{0}^{|x|} \frac{r}{a_{0}(r)} dr (x ∈ ℝ^{n})

we obtain that u(x) → ∞ as |x| → ∞ and a∇u(x) = a_{0}(|x|) · \frac{|x|}{a_{0}(|x|)} \frac{x}{|x|} = x for all x ∈ ℝ^{n}. Thus, u ∈ C^{∞}(ℝ^{n}) and ∇ · (a∇u) = n ⩽ u, so T_{∞} is conservative by [Dav85; Thm. 2.4].

The above might inspire the conjecture that T_{1} is stochastic if instead of a(x) = a_{0}(|x|)I_{n} one only assumes

||a(x)|| ⩽ a_{0}(|x|) (x ∈ ℝ^{n}).

This conjecture is known to be true in dimension n = 1 (see Proposition 2.2), but in higher dimensions we can only prove it under the additional assumption that r ↦ r^{−2}a_{0}(r) is increasing on (R, ∞) for some R > 0; see Corollary 4.4.

As pointed out in [PeSe87], Lyapunov functions are particularly useful for proving conservativeness of semigroups generated by non-divergence form operators. In the case of divergence form operators (or stochasticity and non-divergence form operators), the method has the disadvantage that it leads to assumptions on the derivative of the coefficient matrix a; see, e.g., [Dav85; Thm. 2.5] and [Sta99; Prop. 1.10(c)]. These assumptions can be avoided by working with the intrinsic metric (or related functions) as in [PeSe87] and [Stu94; Sec. 3]. We would like to point out, however, that with an intrinsic metric approach one can recover neither Example 1.1 nor the well-known criterion for stochasticity in the one-dimensional case; see Proposition 2.2 and Example 2.4 below.

In [Stu94], the question of conservativeness is studied in the general abstract context of strongly local symmetric regular Dirichlet forms, under the assumption that the intrinsic metric induces the topology of Ω. Sturm gives a condition for conservativeness in terms of the volume growth of intrinsic balls, which in the context of the canonical Dirichlet form on Riemannian manifolds is due to [KaLi83] and [Gri87]. (Note that in our framework, the form τ need neither be symmetric nor regular. Moreover, our assumptions do not imply that the intrinsic metric of τ induces the topology of Ω.)

In Theorem 4.6, the main result of the paper, we will present a new condition for stochasticity that generalises the above mentioned volume growth condition. We will follow an intrinsic metric approach and work with a function ψ satisfying the condition

0 ⩽ ψ ∈ W_{1,loc}^{1}(Ω),  a∇ψ · ∇ψ ⩽ 1 a.e.,  ρ ∘ ψ ∈ D(τ_{0}) (0 ⩽ ρ ∈ C_{c}^{∞}[0, ∞)). (1.3)

This is parallel to the use of the function U in [PeSe87].

1.2 Remarks. (a) Assume that μ(Ω) < ∞. Then ψ = 11_{Ω} satisfies condition (1.3) if and only if 11_{Ω} ∈ D(τ_{0}). Observe that the latter already implies that ∫ Au = τ(u, 11_{Ω}) = 0 for all u ∈ D(A) and hence that T_{1} is stochastic.

(b) Condition (1.3) clearly depends on the choice of the form τ_{0}. Let 0 ⩽ ψ ∈ W_{1,loc}^{1}(Ω) such that a∇ψ · ∇ψ ⩽ 1 a.e. For τ_{0} = τ_{0,max} it is straightforward that then condition (1.3) is satisfied if ψ(x) → ∞ as |x| → ∞ and μ(Ω ∩ B(0, r)) < ∞ for all r > 0. If one only assumes τ_{0} to be Neumann type (see (1.2)), then one needs in addition that ψ is the restriction of a C^{∞}-function on ℝ^{n}. In the case D(τ_{0}) = C_{c}^{∞}(Ω), condition (1.3) is satisfied if and only if ψ ∈ C^{∞}(Ω) and ψ(x) → ∞ as x → ∂Ω.

(c) If ψ satisfies condition (1.3) then one has ρ ∘ ψ ∈ D(τ) ∩ D(τ_{0,max}) for all 0 ⩽ ρ ∈ W_{∞,c}^{1}[0, ∞). Indeed, let R > 0 such that spt ρ ⊆ [0, R), and let c := ||ρ′||_{∞}. Choose 0 ⩽ ρ̃ ∈ C_{c}^{∞}[0, ∞) such that ρ̃(r) = c(R +1−r) for all r ∈ [0, R]. Then ρ ⩽ ρ̃, so ρ̃∘ψ ∈ L_{2}(μ) implies ρ ∘ ψ ∈ L_{2}(μ). Moreover,

a∇(ρ ∘ ψ) · ∇(ρ ∘ ψ) ⩽ c^{2}11_{[ψ⩽R]}a∇ψ · ∇ψ ⩽ a∇(ρ̃ ∘ ψ) · ∇(ρ̃ ∘ ψ) ∈ L_{1}(μ)

and hence ρ∘ψ ∈ D(τ_{0,max}). Finally, let (ρ_{k}) be a sequence in C_{c}^{∞}[0, ∞) converging uniformly to ρ that satisfies spt ρ_{k} ⊆ [0, R) and ||ρ′_{k}||_{∞} ⩽ c for all k ∈ ℕ. Then as above we obtain that ρ_{k} ∘ ψ ⩽ ρ̃ ∘ ψ and τ_{0}(ρ_{k} ∘ ψ) ⩽ τ_{0}(ρ̃ ∘ ψ) for all k ∈ ℕ. Therefore, (ρ_{k} ∘ ψ) is a bounded sequence in D(τ) with ρ_{k} ∘ ψ → ρ ∘ ψ in L_{2}(μ) as k → ∞, and from the closedness of τ we infer that ρ ∘ ψ ∈ D(τ).

(d) We note the following important consequence of part (c): If D(τ_{0}) = D(τ) ∩ D(τ_{0,max}) and ψ satisfies (1.3) then ρ ∘ ψ satisfies (1.3) for all 0 ⩽ ρ ∈ W_{∞}^{1}[0, ∞) with ||ρ′||_{∞} ⩽ 1 and ρ(r) → ∞ as r → ∞.

The paper is organised as follows. In Section 2 we recall the well-known criterion for stochasticity in the one-dimensional case and give an example that illustrates the limitations of intrinsic metric approaches. In Section 3 we generalise the result of [PeSe87] to our framework and give a new proof. In Section 4 we present new conditions for stochasticity in terms of growth of the coefficient matrix a (in the case μ = λ^{n}) and in terms of volume growth.

2 The one-dimensional case

In this section we assume that n = 1, Ω = (c, d) for some −∞ ⩽ c < d ⩽ ∞ and C_{c}^{∞}(Ω) ⊆ D(τ_{0}). We first show that then D(τ) contains all compactly supported elements of D(τ_{0,max}).

2.1 Lemma. Under the above assumptions one has D(τ_{0,max}) ∩ W_{1,c}^{1}(Ω) ⊆ D(τ). If, in addition, μ((c, x_{0})) = μ((x_{0}, d)) = ∞ for some (and hence all) x_{0} ∈ (c, d), then D(τ_{0,max}) ⊆ D(τ), i.e., C_{c}^{∞}(Ω) is a core for τ_{0,max}.

Proof. First observe that W_{∞,c}^{1}(Ω) ⊆ D(τ) by a standard convolution argument (recall m, ma ∈ L_{1,loc}(Ω)). Let u ∈ D(τ_{0,max}) ∩ W_{1,c}^{1}(Ω). Let c_{0} := min spt u, d_{0} := max spt u and ε > 0 such that d_{1} := d_{0} + ε < d. For k ∈ ℕ define u_{k} ∈ W_{∞,c}^{1}(Ω) by u_{k}(x) := ∫_{c}^{x}(u′(y) ∧ k) ∨ (−k) dy for x ⩽ d_{0}, u_{k}(x) := u_{k}(d_{0})(1 − ε^{−1}(x − d_{0})) for d_{0} < x < d_{1} and u_{k}(x) := 0 for x ⩾ d_{1}. Then spt u_{k} ⊆ [c_{0}, d_{1}] for all k ∈ ℕ and u_{k} → u uniformly, hence u_{k} → u in L_{2}(μ) as k → ∞. Moreover, |u′_{k}| ⩽ |u′| + |u_{k}(d_{0})|ε^{−1}11_{[d_{0},d_{1}]} for all k ∈ ℕ and u′_{k} → u′ a.e. as k → ∞. We conclude that u_{k} → u in D(τ_{0,max}), and the first assertion follows.

Assume now that μ((c, x_{0})) = μ((x_{0}, d)) = ∞, and let u ∈ D(τ_{0,max}). Then u ∈ C(Ω), and lim inf_{x→c} |u(x)| = lim inf_{x→d} |u(x)| = 0 since u ∈ L_{2}(μ). It follows that u_{ε} := (|u| − ε)^{+} sgn u can be approximated in D(τ_{0,max}) by compactly supported elements of D(τ_{0,max}). Moreover, u_{ε} → u in D(τ_{0,max}) as ε → 0, so the first assertion implies the second one.

For the remainder of the section we assume that (ma)^{−1} ∈ L_{1,loc}(Ω). Then the form τ_{0,max} is closed: For a Cauchy sequence (u_{k}) in D(τ_{0,max}) we obtain that u_{k} → u in L_{2}(μ) and (ma)^{1/2}u′_{k} → f in L_{2}(Ω) for some u ∈ L_{2}(μ) and f ∈ L_{2}(Ω), hence u′_{k} → (ma)^{−1/2}f in L_{1,loc}(Ω). It follows that u ∈ W_{1,loc}^{1}(Ω), (ma)^{1/2}u′ = f ∈ L_{2}(Ω), and thus u ∈ D(τ_{0,max}), u_{k} → u in D(τ_{0,max}).

It is well-known that in the case D(τ_{0}) = C_{c}^{∞}(Ω), the semigroup T_{1} is stochastic if and only if the boundary points c and d are non-exit in the classification of Feller; see, e.g., [Osh92; Lemma 3.1] or [Aze74; Prop. 4.6]. The latter condition is still sufficient for stochasticity if as above we only assume C_{c}^{∞}(Ω) ⊆ D(τ_{0}). For the convenience of the reader we include a proof that is based on the same ansatz as the proofs in the next two sections.

2.2 Proposition. Let x_{0} ∈ (c, d), M(x) := ∫_{x_{0}}^{x} m(y) dy (c < x < d) and assume that

∫_{x_{0}}^{c} \frac{M(x)}{m(x)a(x)} dx = ∫_{x_{0}}^{d} \frac{M(x)}{m(x)a(x)} dx = ∞.

Then T_{1} is stochastic.

Proof. Define v ∈ W_{1,loc}^{1}(Ω) by v(x) := ∫_{x_{0}}^{x} \frac{M(y)}{m(y)a(y)} dy. Then v ⩾ 0 and v(x) → ∞ as x → c or x → d. Thus, for ε > 0 we have v_{ε} := (1 − εv)^{+} ∈ W_{1,c}^{1}(Ω) and 0 ⩽ v_{ε} ⩽ 1. Moreover, v′_{ε} = −ε\frac{M}{ma}11_{[εv⩽1]}, hence ma(v′_{ε})^{2} = \frac{ε^{2}M^{2}}{ma} 11_{[εv⩽1]} is integrable, and therefore v_{ε} ∈ D(τ_{0,max}). By Lemma 2.1 we obtain that v_{ε} ∈ D(τ) for all ε > 0.

Let now 0 ⩽ f ∈ L_{1}∩L_{2}(μ) and u := (1+A)^{−1}f. For ε > 0 let c_{ε} ∈ (c, x_{0}) and d_{ε} ∈ (x_{0}, d) such that spt v_{ε} = [c_{ε}, d_{ε}]. Then

τ(u, v_{ε}) = ∫_{c}^{d} mau′v′_{ε} = −ε ∫_{c_{ε}}^{d_{ε}} u′M = −εuM|_{c_{ε}}^{d_{ε}} + ε ∫_{c_{ε}}^{d_{ε}} um
= −εu(d_{ε})M(d_{ε}) + εu(c_{ε})M(c_{ε}) + ε ∫_{c_{ε}}^{d_{ε}} u dμ ⩽ ε||u||_{1}

since u ⩾ 0, M(d_{ε}) > 0 and M(c_{ε}) < 0. Using v_{ε} ↑ 1 as ε → 0, we conclude that

0 ⩽ ||f||_{1} − ||u||_{1} = ∫ (f − u) dμ = ∫ Au dμ = lim_{ε→0} τ(u, v_{ε}) ⩽ 0,

which implies the assertion.

2.3 Remark. It is easy to see that the condition ∫_{x_{0}}^{c} \frac{M(x)}{m(x)a(x)} dx = ∞ can be omitted in the above result if μ((c, x_{0})) < ∞ and the Neumann boundary condition is posed at c. An analogous observation holds for the right boundary point d.

The following example indicates that Proposition 2.2 cannot be proved by an intrinsic metric approach.

2.4 Example. Let m := 11_{ℝ} and define a ∈ L_{1,loc}(ℝ) by

a(x) := \begin{cases}1 & if 2^{k} ⩽ |x| ⩽ 2^{k} + 2^{−k}, k ∈ ℕ, \\ e^{|x|} & otherwise. \end{cases}

Then M(x) = x for all x ∈ ℝ and

∫_{0}^{−∞} \frac{x}{a(x)} dx = ∫_{0}^{∞} \frac{x}{a(x)} dx ⩾ ∑_{k=1}^{∞} ∫_{2^{k}}^{2^{k}+2^{−k}} x dx = ∞,

so T_{1} is stochastic by Proposition 2.2.

Let now ψ ∈ W_{1,loc}^{1}(ℝ) with a(ψ′)^{2} ⩽ 1 a.e. Then

∫_{0}^{∞} |ψ′(x)| dx ⩽ ∫_{0}^{∞} a(x)^{−1/2} dx < ∫_{0}^{∞} e^{−|x|/2} dx + ∑_{k=1}^{∞} 2^{−k} = 3,

in the same way ∫_{−∞}^{0} |ψ′(x)| dx < 3, and thus the intrinsic distance between 0 and ±∞ is less than 3. It also follows that there is no function ψ satisfying condition (1.3).

On the other hand, for m = 11_{ℝ} and a(x) = ((1 + |x|) ln(2 + |x|))^{2} (x ∈ ℝ) one obtains that the intrinsic distance between 0 and ±∞ is ∞, yet T_{1} is not stochastic (cf. [Dav85; Example B]). Therefore, the intrinsic distance between 0 and ±∞ yields no information about whether T_{1} is stochastic or not.

3 A condition of Perelmuter and Semenov

We start with a simple condition for T_{1} being stochastic, which is needed for the proof of Theorem 3.3.

3.1 Lemma. Assume that there exists a sequence (v_{k}) ⊆ D(τ_{0}) such that

sup_{k∈ℕ}τ(v_{k}) < ∞,  sup_{k∈ℕ}||v_{k}||_{∞} < ∞

and v_{k}|_{B_{k}} = 1 a.e. for some sequence (B_{k}) of measurable subsets of Ω with B_{k} ↑ Ω as k → ∞. Then T_{1} is stochastic.

Proof. For u ∈ D(τ_{0}) we have (a∇u · ∇v_{k})^{2} ⩽ α(a∇u · ∇u)(a∇v_{k} · ∇v_{k}) by (1.1) and hence

τ(u, v_{k})^{2} ⩽ ατ(v_{k}) ∫_{Ω\B_{k}} a∇u · ∇u dμ → 0 (k → ∞).

Using sup_{k} τ(v_{k}) < ∞ again, we infer that τ(u, v_{k}) → 0 as k → ∞ for all u ∈ D(τ). For u ∈ (1 + A)^{−1}(L_{1} ∩ L_{2}(μ)) we conclude that

∫_{Ω} Au dμ = lim_{k→∞} ∫_{Ω} Au · v_{k} dμ = lim_{k→∞} τ(u, v_{k}) = 0,

which implies the assertion.

3.2 Example. Assume that n ⩽ 2, μ = λ^{n}, τ_{0} is Neumann type (i.e., (1.2) holds) and that a is bounded. Then the assumptions of Lemma 3.1 are satisfied for v_{k} = v(\frac{·}{k})|_{Ω}, where v ∈ C_{c}^{∞}(ℝ^{n}), v = 1 on B(0, 1). Indeed,

τ(v_{k}) = \frac{1}{k^{2}} ∫_{Ω} a∇v(\frac{·}{k}) · ∇v(\frac{·}{k}) dλ^{n} ⩽ k^{n−2}||a||_{∞}||∇v||_{∞}^{2} · λ^{n}(spt v),

and v_{k} = 1 on B_{k} = B(0, k) ∩ Ω.

In the next theorem we give a much better (but not quite sharp) condition for T_{1} being stochastic; see also Theorem 4.6 below. The result is essentially due to [PeSe87], except for the more restrictive framework of that paper. (Perelmuter and Semenov assumed that Ω = ℝ^{n}, a ∈ C^{∞} and μ = λ^{n}; the case of more general measures μ was studied in [Lis99; Thm. 4], for a the identity matrix. Our main contribution is a new and simpler proof.

3.3 Theorem. Assume that there exists ψ satisfying condition (1.3) such that

∫_{[r_{k}⩽ψ⩽r_{k}+d_{k}]} a∇ψ · ∇ψ dμ ⩽ e^{cd_{k}^{2}} − 1  (k ∈ ℕ)  (3.1)

for some c > 0 and (r_{k}), (d_{k}) ⊆ (0, ∞) with r_{k} → ∞ as k → ∞. Then T_{1} is stochastic.

3.4 Remarks. (a) Because of a∇ψ · ∇ψ ⩽ 1, assumption (3.1) is in particular satisfied if

μ([r_{k} ⩽ ψ ⩽ r_{k} + d_{k}]) ⩽ e^{cd_{k}^{2}} − 1  (k ∈ ℕ).  (3.2)

It is straightforward that the latter holds, e.g., if lim inf_{r→∞} μ([r ⩽ ψ ⩽ 2r])^{1/r^{2}} < ∞.

(b) In [PeSe87], the (equivalent, see introduction) question of L_{1}-uniqueness is studied, and instead of condition (3.1) the following is assumed: There exist sequences (t_{k}), (ε_{k}), (β_{k}) ⊆ (0, ∞) such that t_{k} → ∞, ε_{k} → 0 as k → ∞ and

M := sup_{k∈ℕ} \frac{ε_{k}}{β_{k}} \sqrt{t_{k}} μ([1 ⩽ ε_{k}ψ ⩽ 1 + β_{k}])^{1/(t_{k}+1)} < ∞  (3.3)

(cf. [PeSe87; Rem. 3]). We now show: If (3.3) is satisfied then condition (3.2) holds for r_{k} = 1/ε_{k}, d_{k} = β_{k}/ε_{k}, c = M^{2} and all k with t_{k} ⩾ 1. Observe that (3.3) is equivalent to

μ([r_{k} ⩽ ψ ⩽ r_{k} + d_{k}]) = μ([1 ⩽ ε_{k}ψ ⩽ 1 + β_{k}]) ⩽ (\frac{Md_{k}}{\sqrt{t_{k}}} )^{t_{k}+1} (k ∈ ℕ),

so we only have to show that (Md_{k}/\sqrt{t})^{t+1} ⩽ e^{M^{2}d_{k}^{2}} − 1 for all t ⩾ 1. With x = M^{2}d_{k}^{2}/t this inequality reads x^{(t+1)/2} ⩽ e^{tx} − 1. The latter is clear for 0 ⩽ x ⩽ 1: then x^{(t+1)/2} ⩽ x ⩽ e^{x} − 1 ⩽ e^{tx} − 1. For x > 1 we can estimate

x^{(t+1)/2} ⩽ x^{t} = e^{t ln x} ⩽ e^{t(x−1)} ⩽ e^{−1}e^{tx} ⩽ e^{tx} − 1

since e^{tx} ⩾ e.

Conversely, it is easy to see that (3.2) implies (3.3) with t_{k} = d_{k}^{2}, ε_{k} = 1/r_{k} and β_{k} = d_{k}/r_{k}, so the assumption in [PeSe87; Rem. 3] is satisfied if (3.2) holds with d_{k} → ∞.

3.5 Example (cf. the corollary in [PeSe87; p. 719]). Assume that μ = λ^{n} and that τ_{0} is Neumann type (i.e., (1.2) holds), and let R > 1. Define ψ ∈ C^{∞}(Ω) by ψ(x) := \sqrt{ρ(|x|)}, where ρ ∈ C^{∞}[0, ∞) is increasing and satisfies ρ(r) = ln R for r ⩽ R and ρ(r) = ln r for r ⩾ R + 1. Then ∇ψ = 0 on B(0, R)∩Ω and ∇ψ(x) = \frac{1}{2}(ln |x|)^{−1/2}|x|^{−1} \frac{x}{|x|} for x ∈ Ω\B(0, R+1). Thus, if there exists c > 0 such that

a_{rr}(x) := a(x)\frac{x}{|x|} · \frac{x}{|x|} ⩽ c|x|^{2} ln |x|  (x ∈ Ω, |x| > R)  (3.4)

then δψ satisfies condition (1.3) for some δ > 0 (cf. Remark 1.2(b)). Moreover, for x ∈ Ω and r > R + 1 we have ψ(x) ⩽ 2r if and only if |x| ⩽ e^{4r^{2}}, so

λ^{n}([r ⩽ δψ ⩽ 2r]) ⩽ ω_{n}e^{4nr^{2}/δ^{2}} (r > δ(R + 1)),

where ω_{n} = λ^{n}(B(0, 1)). Therefore, Theorem 3.3 is applicable if (3.4) is satisfied.

It is shown in [Dav85; Example B] that T_{1} is not stochastic if (Ω, μ) = (ℝ^{n}, λ^{n}), D(τ_{0}) = C_{c}^{∞}(ℝ^{n}), ε > 0 and a(x) = (1 + |x|)^{2}(ln(1 + |x|))^{1+ε}I_{n} for all x ∈ ℝ^{n}, where I_{n} is the identity matrix. In this sense condition (3.4) is rather sharp; however, in Example 4.5 below we will show that the condition can be slightly relaxed.

Since we do not assume τ_{0} to be closed, we need the following auxiliary lemma for the proof of Theorem 3.3. We note that the assertion of part (a) is a rather strong result; for our purposes weak convergence would suffice.

3.6 Lemma. (a) Let φ: ℝ → ℝ be Lipschitz continuous, φ(0) = 0. Let u ∈ D(τ), (u_{k}) a sequence in D(τ) such that u_{k} → u in D(τ). Then φ ∘ u_{k} → φ ∘ u in D(τ).

(b) Let 0 ⩽ u ∈ D(τ) ∩ L_{∞}(μ). Then there exists a sequence (u_{k}) ⊆ D(τ_{0}) such that u_{k} → u in D(τ) and 0 ⩽ u_{k} ⩽ ||u||_{∞} a.e. for all k ∈ ℕ.

Proof. (a) follows from [Anc76; Théorème 10].

(b) Let (v_{j}) be a sequence in D(τ_{0}) with v_{j} → u in D(τ), and let (ρ_{k}) ⊆ C^{∞}(ℝ) be a sequence of normal contractions satisfying 0 ⩽ ρ_{k}(t) ↑ t^{+} ∧ ||u||_{∞} =: ρ(t) as k → ∞, for all t ∈ ℝ. Then by part (a) we have D(τ_{0}) ∋ ρ_{k} ∘ v_{j} → ρ_{k} ∘ u in D(τ) as j → ∞, for all k ∈ ℕ. Moreover, ρ_{k} ∘u → ρ∘u = u in D(τ) (see, e.g., [Anc76; Prop. 11]), so the assertion follows.

We point out that in the case that (d_{k}) is not bounded, Theorem 3.3 can be deduced from our main result, Theorem 4.6 below (see Remark 4.8(c)). Here we give an independent proof that is much more elementary.

Proof of Theorem 3.3. Let ρ ∈ C^{∞}(ℝ) be a decreasing function such that ρ = 1 on (−∞, 0) and ρ = 0 on (1, ∞). For k ∈ ℕ define ρ_{k} ∈ C_{c}^{∞}[0, ∞) by ρ_{k}(t) := ρ(\frac{t−r_{k}}{d_{k}} ); then v_{k} := ρ_{k} ∘ ψ ∈ D(τ_{0}) by condition (1.3) and v_{k} = 1 on [ψ ⩽ r_{k}]. Moreover, with C := ||ρ′||_{∞}^{2} and w := a∇ψ · ∇ψ ( ⩽ 1) we obtain that

a∇v_{k} · ∇v_{k} = (ρ′_{k} ∘ ψ)^{2}a∇ψ · ∇ψ ⩽ Cd_{k}^{−2}11_{[r_{k}⩽ψ⩽r_{k}+d_{k}]}w  (k ∈ ℕ).  (3.5)

Suppose first that (d_{k}) is bounded. Then

τ(v_{k}) ⩽ Cd_{k}^{−2} ∫_{[r_{k}⩽ψ⩽r_{k}+d_{k}]} w dμ ⩽ Cd_{k}^{−2}(e^{cd_{k}^{2}} − 1) (k ∈ ℕ)

implies sup_{k} τ(v_{k}) < ∞, and the assertion follows from Lemma 3.1.

If (d_{k}) is not bounded then we can assume without loss of generality that d_{k} → ∞ as k → ∞. Let δ > 0 and v ∈ D(τ_{0}). For 0 ⩽ u ∈ D(τ_{0}) ∩ L_{∞}(μ) we have ln(1 + u/δ) ∈ D(τ_{0}), ∇ ln(1 + u/δ) = (u + δ)^{−1}∇u and hence

(a∇u · ∇v)^{2} ⩽ α(a∇u · ∇ ln(1 + u/δ)) · (u + δ)(a∇v · ∇v),

so we obtain the estimate

τ(u, v)^{2} ⩽ ατ(u, ln(1 + u/δ)) ∫ (u + δ)(a∇v · ∇v) dμ.

By Lemma 3.6 it follows that this estimate is valid for all 0 ⩽ u ∈ D(τ) ∩ L_{∞}(μ).

Let now 0 ⩽ f ∈ L_{1}(μ) ∩ L_{∞}(μ), u := (1 + A)^{−1}f. Then 0 ⩽ u ∈ D(τ) ∩ L_{1}(μ) ∩ L_{∞}(μ), so due to (3.5) we infer for all k ∈ ℕ that

⟨Au, v_{k}⟩^{2} = τ(u, v_{k})^{2} ⩽ α⟨Au, ln(1 + u/δ)⟩ · Cd_{k}^{−2} ∫_{[r_{k}⩽ψ⩽r_{k}+d_{k}]}(u + δ)w dμ
⩽ α||Au||_{1} ln(1 + ||u||_{∞}/δ) · Cd_{k}^{−2}(∫_{[r_{k}⩽ψ⩽r_{k}+d_{k}]} u dμ + δ(e^{cd_{k}^{2}} − 1)).

Choosing δ = e^{−2cd_{k}^{2}} we conclude that ⟨Au, v_{k}⟩^{2} → 0 as k → ∞, and hence ∫ Au dμ = 0. Thus we have shown ∫ Au dμ = 0 for all u ∈ (1 + A)^{−1}(L_{1} ∩ L_{∞}(μ)), and the assertion follows.

4 New conditions for stochasticity

In this section we present new conditions in terms of volume growth and of growth of the coefficient matrix a that imply T_{1} being stochastic. All the results are based on the following abstract condition.

4.1 Proposition. Assume that there exist an integrable function φ: [0, 1] → [1, ∞] and a sequence (v_{k}) ⊆ D(τ_{0}) such that sup_{k} ||v_{k}||_{∞} < ∞, v_{k} → 1 a.e. and

∫_{Ω} \frac{a∇v_{k} · ∇v_{k}}{φ ∘ u} dμ → 0  (k → ∞)  (4.1)

for all u = (1 + A)^{−1}f, where 0 ⩽ f ∈ L_{∞,c}(Ω), ||f||_{∞} ⩽ 1. Then T_{1} is stochastic.

Proof. Let 0 ⩽ f ∈ L_{∞,c}(Ω), ||f||_{∞} ⩽ 1 and u := (1 + A)^{−1}f. We are going to show that

τ(u, v)^{2} ⩽ α||φ||_{1}||Au||_{1} ∫_{Ω} \frac{a∇v · ∇v}{φ ∘ u} dμ  (v ∈ D(τ_{0})).  (4.2)

Then by (4.1) we obtain that

∫_{Ω} Au dμ = lim_{k→∞} ∫_{Ω} Au · v_{k} dμ = lim_{k→∞} τ(u, v_{k}) = 0,

hence ∫ Au dμ = 0 for all u ∈ (1 + A)^{−1}(L_{∞,c}(Ω)), and the assertion follows.

Let ε > 0. Then there exists an increasing sequence (φ_{j}) ⊆ C^{∞}[0, 1] such that φ_{j} ⩾ 1, ||φ_{j}||_{1} ⩽ (1+ε)||φ||_{1} for all j ∈ ℕ and lim_{j→∞} φ_{j} ⩾ φ pointwise. Let j ∈ ℕ, define h_{j} ∈ C^{∞}[0, 1] by h_{j}(t) := ∫_{0}^{t} φ_{j}(s) ds, and fix v ∈ D(τ_{0}). For 0 ⩽ u ∈ D(τ_{0}) with ||u||_{∞} ⩽ 1 we obtain that h_{j} ∘ u ∈ D(τ_{0}), ∇(h_{j} ∘ u) = (φ_{j} ∘ u)∇u and thus

(a∇u · ∇v)^{2} ⩽ α(a∇u · ∇(h_{j} ∘ u)) · (φ_{j} ∘ u)^{−1}(a∇v · ∇v),

so we can estimate

τ(u, v)^{2} ⩽ ατ(u, h_{j} ∘ u) ∫_{Ω}(φ_{j} ∘ u)^{−1}(a∇v · ∇v) dμ.  (4.3)

Let now 0 ⩽ f ∈ L_{∞,c}(Ω), ||f||_{∞} ⩽ 1 and u := (1 + A)^{−1}f. Then 0 ⩽ u ∈ D(τ) and ||u||_{∞} ⩽ 1. By Lemma 3.6 there exists a sequence (u_{k}) ⊆ D(τ_{0}) such that 0 ⩽ u_{k} ⩽ 1 for all k ∈ ℕ, u_{k} → u a.e. and in D(τ), and h_{j} ∘ u_{k} → h_{j} ∘ u in D(τ) as k → ∞. Moreover, (φ_{j} ∘ u_{k})^{−1} → (φ_{j} ∘ u)^{−1} a.e. as k → ∞ and ||(φ_{j} ∘ u_{k})^{−1}||_{∞} ⩽ 1 for all k ∈ ℕ, so from (4.3) it follows that

τ(u, v)^{2} ⩽ ατ(u, h_{j} ∘ u) ∫_{Ω}(φ_{j} ∘ u)^{−1}(a∇v · ∇v) dμ ⩽ α||Au||_{1}||h_{j} ∘ u||_{∞} ∫_{Ω} \frac{a∇v · ∇v}{φ_{j} ∘ u} dμ.

Finally, observe that ||h_{j} ∘ u||_{∞} ⩽ ||h_{j}||_{∞} = ||φ_{j}||_{1} ⩽ (1 + ε)||φ||_{1}. Therefore, letting first j → ∞ and then ε → 0, we conclude that (4.2) holds, and the proof is complete.

Remark. The above proof can be simplified if the function φ is continuous. In the applications below, φ will even be continuous and decreasing.

Proposition 4.1 will be applied via the following result, which is a preliminary version of our main result, Theorem 4.6 below.

4.2 Proposition. Assume that there exist ψ satisfying condition (1.3) and a decreasing integrable function φ: [0, 1] → [0, ∞] such that

exp(−φ ∘ (a∇ψ · ∇ψ)) ∈ L_{1}(μ).

Then T_{1} is stochastic.

Proof. First observe that we can assume without loss of generality that φ is continuous and that φ(1) = 0. Moreover, by replacing φ with φ − ln we can assume that φ(0) = ∞, φ(t) ⩾ ln t^{−1} for all t ∈ (0, 1) and that φ is strictly decreasing and hence bijective.

We are going to apply Proposition 4.1 with φ_{1}: [0, 1] → [1, ∞] defined by φ_{1}(0) := ∞ and φ_{1}(t) := \frac{1}{t}φ^{−1}(\frac{1}{2} ln \frac{1}{t}) for 0 < t ⩽ 1. Substituting s = \frac{1}{2} ln \frac{1}{t} we compute

∫_{0}^{1} φ_{1}(t) dt = ∫_{0}^{1} φ^{−1}(\frac{1}{2} ln \frac{1}{t}) \frac{dt}{t} = 2 ∫_{0}^{∞} φ^{−1}(s) ds = 2 ∫_{0}^{1} φ(r) dr < ∞,

so φ_{1} is integrable. Next we show that

\frac{x}{φ_{1}(t)} ⩽ t + exp(−φ(x))  (t, x ∈ [0, 1]).  (4.4)

This inequality is trivial in the cases t = 0 and x ⩽ tφ_{1}(t) = φ^{−1}(\frac{1}{2} ln \frac{1}{t}), so let 0 < t ⩽ 1 and φ^{−1}(\frac{1}{2} ln \frac{1}{t}) < x ⩽ 1. Then φ(x) < \frac{1}{2} ln \frac{1}{t} and hence exp(−φ(x)) > t^{1/2}. Moreover, φ(t^{1/2}) ⩾ ln t^{−1/2} (see the beginning of the proof), thus φ^{−1}(ln t^{−1/2}) ⩾ t^{1/2}, and we conclude that

\frac{x}{φ_{1}(t)} = \frac{tx}{φ^{−1}(ln t^{−1/2})} ⩽ t^{1/2} < exp(−φ(x)).

Let now ρ ∈ C_{c}^{∞}[0, ∞) be a decreasing function with ρ(0) = 1. For k ∈ ℕ let v_{k} := ρ ∘ (ψ/k). Then v_{k} ∈ D(τ_{0}) by condition (1.3), ||v_{k}||_{∞} ⩽ 1 for all k ∈ ℕ and v_{k} → 1 a.e. as k → ∞. Moreover, by (4.4) we obtain for 0 ⩽ f ∈ L_{∞,c}(Ω), ||f||_{∞} ⩽ 1 and u := (1 + A)^{−1}f that \frac{a∇ψ·∇ψ}{φ_{1}∘u} ⩽ u + exp(−φ ∘ (a∇ψ · ∇ψ)) ∈ L_{1}(μ) and hence

∫_{Ω} \frac{a∇v_{k} · ∇v_{k}}{φ_{1} ∘ u} dμ ⩽ ||ρ′||_{∞}^{2}k^{−2} ∫_{Ω} \frac{a∇ψ · ∇ψ}{φ_{1} ∘ u} dμ → 0 (k → ∞),

so the assertion follows from Proposition 4.1.

Remark. The function φ_{1} used in the above proof is strictly decreasing: Without loss of generality it was assumed that φ + ln is decreasing; thus 2φ + ln = 2(φ + ln) − ln and hence r ↦ re^{2φ(r)} is strictly decreasing. Since φ is strictly decreasing, this implies that s ↦ φ^{−1}(s)e^{2s} is strictly increasing on (0, ∞), so t ↦ φ_{1}(t) = φ^{−1}(\frac{1}{2} ln \frac{1}{t}) exp(2 · \frac{1}{2} ln \frac{1}{t}) is strictly decreasing on (0, 1).

In the next two results, which deal with the special case μ = λ^{n}, we give conditions for T_{1} being stochastic in terms of the growth of the coefficient matrix a at ∞.

4.3 Corollary. Let μ = λ^{n} and assume that there exist ψ satisfying condition (1.3) and a decreasing function h: [0, ∞) → (0, 1] such that

∫_{1}^{∞} \frac{h(r)}{r} dr < ∞, (a∇ψ · ∇ψ)(x) ⩽ h(|x|) (x ∈ Ω).

Then T_{1} is stochastic.

Proof. Assume without loss of generality that h|_{[0,1]} = 1 and that h: [1, ∞) → (0, 1] is bijective. Then ln ∘ h^{−1}: (0, 1] → [0, ∞) is decreasing, and

∫_{0}^{1} ln h^{−1}(t) dt = ∫_{0}^{∞} h(e^{s}) ds = ∫_{1}^{∞} h(r) \frac{dr}{r} < ∞.

Thus, φ := (n + 1) ln ∘ h^{−1} (with φ(0) := ∞) is decreasing and integrable. Moreover,

exp(−φ ∘ (a∇ψ · ∇ψ))(x) ⩽ exp(−φ(h(|x|))) = |x|^{−n−1} ∧ 1 (x ∈ Ω),

so the assertion follows from Proposition 4.2.

The following result generalises Example 3.5; its proof illustrates the possible interplay between the functions ψ and h in Corollary 4.3.

4.4 Corollary. Assume that μ = λ^{n}, τ_{0} is Neumann type (i.e., (1.2) holds) and that there exist an open set X ⊆ (0, ∞) and an increasing function g: (0, ∞) → [1, ∞) such that

∫_{1}^{∞} \frac{dr}{rg(r)} = ∞, a_{rr}(x) = a(x)\frac{x}{|x|} · \frac{x}{|x|} ⩽ |x|^{2}g(|x|) (x ∈ Ω, |x| ∈ X)

and lim inf_{r→∞} λ(X ∩ (0, r))/r > 0. Then T_{1} is stochastic.

If τ_{0} = τ_{0,max} then the assumption on X to be open can be omitted.

Proof. Without loss of generality we can assume that g ∈ C^{∞}(0, ∞) (convolution with a C_{c}^{∞}(−1, 0)-function) and that (0, 1) ⊆ (0, ∞) \ X. Let ε, R > 0 such that λ(X ∩ (0, r)) ⩾ 2εr for all r > R. Since X is open, there exists a C^{∞}-function χ: (0, ∞) → [0, 1] such that χ = 0 on (0, ∞) \ X and ∫_{0}^{r} χ(s) ds ⩾ εr for all r > R.

Define f: [0, ∞) → [1, ∞) by f(r) = 1 + ∫_{0}^{r} \frac{χ(s)}{sg(s)} ds and 0 ⩽ ψ ∈ C^{∞}(Ω) by ψ(x) := ln f(|x|). With c := 2/ε we obtain that

∫_{r}^{cr} χ(s) ds ⩾ ∫_{0}^{cr} χ(s) ds − r ⩾ εcr − r = r = \frac{1}{c^{2} − c} ∫_{cr}^{c^{2}r} ds  (r > R).  (4.5)

Since g is increasing, we deduce that

∫_{R}^{∞} \frac{χ(s)}{sg(s)} ds ⩾ \frac{1}{c^{2} − c} ∫_{cR}^{∞} \frac{1}{sg(s)} ds = ∞,

so f(r) → ∞ as r → ∞. Moreover,

(a∇ψ · ∇ψ)(x) = (\frac{χ(|x|)}{f(|x|) · |x|g(|x|)})^{2} a_{rr}(x) ⩽ \frac{1}{f(|x|)^{2}g(|x|)} ⩽ 1

for all x ∈ Ω, hence by Remark 1.2(b) it follows that ψ satisfies condition (1.3). Finally, h := \frac{1}{f^{2}g} is decreasing and as above we deduce from (4.5) that

\frac{1}{c^{2} − c} ∫_{cR}^{∞} \frac{h(r)}{r} dr ⩽ ∫_{R}^{∞} \frac{χ(r)h(r)}{r} dr = ∫_{R}^{∞} \frac{f′(r)}{f(r)^{2}} dr = −\frac{1}{f(r)}|_{R}^{∞} = \frac{1}{f(R)} < ∞,

so the assertion follows from Corollary 4.3.

In the case τ_{0} = τ_{0,max}, the above proof works with χ = 11_{X}. Then we only have ψ ∈ W_{∞,c}^{1}(Ω), but ψ still satisfies condition (1.3).

4.5 Example. The condition of Corollary 4.4 is satisfied, e.g., if there exists c > 0 such that

a_{rr}(x) ⩽ c|x|^{2} ln |x| ln ln |x| (x ∈ Ω, |x| > 10).

The following is the main result of the paper; the proof will be given at the end of the section.

4.6 Theorem. Assume that there exist ψ satisfying condition (1.3), a decreasing integrable function φ: [0, 1] → [0, ∞] and an increasing function v: (0, ∞) → (3, ∞) such that

∫_{0}^{∞} \frac{r}{ln v(r)} dr = ∞,  ∫_{[ψ⩽r]} exp(−φ ∘ (a∇ψ · ∇ψ)) dμ ⩽ v(r) (r > 0).  (4.6)

Then T_{1} is stochastic.

Setting φ = 0 in this theorem, we obtain the following volume growth condition for stochasticity.

4.7 Corollary. Assume that there exist ψ satisfying condition (1.3) and an increasing function v: (0, ∞) → (3, ∞) such that

∫_{0}^{∞} \frac{r}{ln v(r)} dr = ∞, μ([ψ ⩽ r]) ⩽ v(r) (r > 0).

Then T_{1} is stochastic.

4.8 Remarks. (a) In the context of stochastical completeness of Riemannian manifolds, the volume growth condition in Corollary 4.7 is due to [KaLi83] and [Gri87]. In [Gri89] it is shown that the condition is sharp in the following sense: Given a smooth increasing function v: (0, ∞) → (3, ∞) such that ∫_{0}^{∞} \frac{r}{ln v(r)} dr < ∞, there exists a complete Riemannian manifold M such that μ(B(x, r)) ⩽ v(r) for all r > 0 and some x ∈ M, and M is not stochastically complete, i.e., the L_{1}-semigroup associated with the canonical Dirichlet form on M is not stochastic. (Here, μ(B(x, r)) denotes the Riemannian volume of the geodesic ball with centre x and radius r.) This does not mean, however, that the condition of Corollary 4.7 is necessary; see Example 4.9 below.

(b) The condition ∫_{1}^{∞} \frac{r}{ln v(r)} dr = ∞ is satisfied, e.g., if v(r) = e^{cr^{2}} or v(r) = e^{cr^{2} ln r} for some c > 0. Observe that in the former case, Corollary 4.7 is already covered by Theorem 3.3 (see Remark 3.4(a)), but it is not hard to see that in the latter case it is not.

(c) As pointed out in Section 3, Theorem 3.3 can be deduced from Theorem 4.6 if (d_{k}) is not bounded. Note that in this case we can assume without loss of generality that (d_{k}) is increasing and that r_{k} +d_{k} < r_{k+1} for all k ∈ ℕ. Moreover, D(τ_{0}) = D(τ)∩D(τ_{0,max}) without loss of generality.

Let ρ: [0, ∞) → [0, ∞) be such that ρ(0) = 0, and ρ has slope 1 on the intervals [r_{k}, r_{k} +d_{k}] and slope 0 otherwise. Then ρ(r) → ∞ as r → ∞, so ψ̃ := ρ ∘ ψ satisfies condition (1.3) by Remark 1.2(d). Moreover, for k ∈ ℕ and r̃_{k} := ∑_{j=1}^{k} d_{j} we obtain that

∫_{[ψ⩽r̃_{k}]} a∇ψ̃ · ∇ψ̃ dμ = ∑_{j=1}^{k} ∫_{[r_{j}⩽ψ⩽r_{j}+d_{j}]} a∇ψ · ∇ψ dμ ⩽ ∑_{j=1}^{k} e^{cd_{j}^{2}} ⩽ ke^{cd_{k}^{2}}.

We now define the functions φ and v by φ(0) := ∞, φ(t) := ln \frac{1}{t} for 0 < t ⩽ 1 and v(r) := ke^{cd_{k}^{2}} for r̃_{k−1} < r ⩽ r̃_{k}, k ∈ ℕ (where r̃_{0} := 0). Then

∫_{[ψ⩽r]} exp(−φ ∘ (a∇ψ · ∇ψ)) dμ = ∫_{[ψ⩽r]} a∇ψ · ∇ψ dμ ⩽ v(r) (r > 0)

and

∫_{0}^{∞} \frac{r}{ln v(r)} dr = ∑_{k=1}^{∞} ∫ r̃_{k} ; r̃_{k−1} \frac{r}{ln k + cd_{k}^{2}} dr ⩾ ∑_{k=1}^{∞} \frac{d_{k}^{2}/2}{ln k + cd_{k}^{2}} = ∞.

Thus, we can apply Theorem 4.6 to obtain that T_{1} is stochastic.

We now present two examples for situations in which Theorem 4.6 is applicable, but Corollary 4.7 is not. These examples illustrate that if one has good control of the coefficients a and m on a suitable small portion of Ω then no control on the remainder of Ω is needed.

4.9 Example. We are going to apply Theorem 4.6 with φ|_{(0,1]} = 0, φ(0) = ∞. Let (r_{k}) ⊆ (0, ∞) be a sequence with r_{k+1} ⩾ r_{k} + 1 for all k ∈ ℕ.

(a) Let Ω := ℝ×(0, 1) ⊆ ℝ^{2}, and for k ∈ ℕ let Ω_{k} := {x ∈ Ω; r_{k} < |x_{1}| < r_{k} +1}. Assume that m ⩽ 1 and a_{11} ⩽ k^{2} on Ω_{k}, for all k ∈ ℕ, and that

D(τ_{0}) ⊇ {u|_{Ω}; u ∈ C^{∞}(ℝ^{2}), u(x_{1}, 0) = u(x_{1}, 1) (x_{1} ∈ ℝ)}.

(This includes Neumann and periodic boundary conditions.) There exists 0 ⩽ ψ_{0} ∈ C^{∞}(ℝ) such that ψ_{0}(x) ⩾ \frac{1}{2} ln k and |ψ′_{0}(x)| ⩽ \frac{1}{k} for all k ∈ ℕ and all x ∈ ℝ with r_{k} ⩽ |x| ⩽ r_{k} + 1, ψ′_{0}(x) = 0 otherwise. Then for ψ ∈ C^{∞}(Ω) defined by ψ(x) := ψ_{0}(x_{1}) we obtain that ψ satisfies condition (1.3) ((a∇ψ · ∇ψ)(x) = a_{11}(x)ψ′_{0}(x_{1})^{2} ⩽ 1 for all x ∈ Ω.)

Let now r > 0 and choose k ∈ ℕ such that \frac{1}{2} ln k ⩽ r < \frac{1}{2} ln(k + 1). Then [ψ ⩽ r] is contained in (−r_{k+1}, r_{k+1}) × (0, 1), hence

∫_{[ψ⩽r]} exp(−φ ∘ (a∇ψ · ∇ψ)) dμ ⩽ ∑_{j=1}^{k} μ(Ω_{j}) ⩽ 2k ⩽ 2e^{2r},

so from Theorem 4.6 it follows that T_{1} is stochastic. Observe that we have no control of μ([ψ ⩽ r]) in this example, so we cannot apply Corollary 4.7.

(b) Let Ω := ℝ^{n}, and for k ∈ ℕ let Ω_{k} := \{x ∈ ℝ^{n}; r_{k} < |x| < r_{k} + \frac{1}{k}\}. Assume that μ(Ω_{k}) ⩽ exp((ln k)^{2}) and a_{rr} ⩽ 1 on Ω_{k}, for all k ∈ ℕ, and that D(τ_{0}) ⊇ C_{c}^{∞}(ℝ^{n}). There exists a radially symmetric function 0 ⩽ ψ ∈ C^{∞}(ℝ^{n}) such that ψ ⩾ \frac{1}{2} ln k and |∇ψ| ⩽ 1 on Ω_{k} for all k ∈ ℕ, ∇ψ = 0 otherwise. Then ψ satisfies condition (1.3), and with the same argument as in part (a) one shows that

∫_{[ψ⩽r]} exp(−φ ∘ (a∇ψ · ∇ψ)) dμ ⩽ e^{2r+4r^{2}} (r > 0),

so T_{1} is stochastic by Theorem 4.6.

For the proof of Theorem 4.6 we need the following technical result.

4.10 Lemma. Let γ: (0, ∞) → (1, ∞) be a measurable function with ∫_{0}^{∞} \frac{r}{γ(r)} dr = ∞. (a) If γ is increasing then ∫_{0}^{∞} \frac{r}{γ(r)+r^{2}} dr = ∞.

(b) If γ(r) ⩾ r^{2} for all r > 0 then there exists a decreasing and bijective function g: (0, ∞) → (0, 1) such that g ∉ L_{1}(0, ∞) and γ ∘ (g^{2})^{−1} ∈ L_{1}(0, 1).

Proof. (a) Let γ̃(r) := γ(r) + r^{2} for all r > 0. If there exists r_{0} > 0 such that γ(r) ⩾ r^{2} for all r ⩾ r_{0} then ∫_{0}^{∞} \frac{r}{γ̃(r)} dr ⩾ ∫_{r_{0}}^{∞} \frac{r}{2γ(r)} dr = ∞. If there exists no such r_{0} then there exists a sequence (r_{k}) ⊆ (1, ∞) such that γ(r_{k}) < r_{k}^{2} and r_{k+1} ⩾ 2r_{k} for all k ∈ ℕ. We conclude that γ̃(r) ⩽ γ̃(r_{k}) < 2r_{k}^{2} for all k ∈ ℕ and all r ⩽ r_{k}, and therefore

∫_{0}^{∞} \frac{r}{γ̃(r)} dr ⩾ ∑_{k=1}^{∞} \frac{1}{2r_{k}^{2}} ∫_{r_{k}/2}^{r_{k}} r dr = ∑_{k=1}^{∞} \frac{1}{2r_{k}^{2}} · \frac{1}{2}(r_{k}^{2} − r_{k}^{2}/4) = ∞.

(b) We define f, F, g̃: (0, ∞) → (0, ∞) by

f(r) := \frac{r}{γ(r)},  F (r) := 2 + ∫_{0}^{r} f(s) ds,  g̃(r) := ∫_{r}^{∞} \frac{ds}{γ(s)F (s)}.

(The function g̃ is defined since γ(s) ⩾ s^{2} for all s > 0.) Then g̃ is strictly decreasing and continuous, and

∫_{0}^{∞} g̃(r) dr = ∫_{0}^{∞} \frac{1}{γ(s)F (s)} ∫_{0}^{s} dr ds = ∫_{0}^{∞} \frac{f(s)}{F (s)} ds = ln F (s)|_{0}^{∞} = ∞

since F (r) → ∞ as r → ∞ by the assumption.

Let now c := 1/g̃(0+) and g := cg̃. Then g: (0, ∞) → (0, 1) is decreasing and bijective. It remains to show that γ ∘ (g^{2})^{−1} ∈ L_{1}(0, 1). Because of F (r) ⩾ 2 and γ(r) ⩾ r^{2} we have

\frac{2rf(r)}{F (r)^{2}} = \frac{2r^{2}}{γ(r)F (r)^{2}} ⩽ \frac{1}{F (r)}

and hence

\frac{1}{F (r)} ⩽ \frac{2}{F (r)} − \frac{2rf(r)}{F (r)^{2}} = \frac{d}{dr} \frac{2r}{F (r)}

for all r > 0. Thus, ∫_{0}^{s} \frac{dr}{F (r)} ⩽ \frac{2r}{F (r)} |_{0}^{s} = \frac{2s}{F (s)} for all s > 0. We conclude that

∫_{0}^{1} γ ∘ (g^{2})^{−1}(r) dr = ∫_{∞}^{0} γ(r)(g^{2})′(r) dr = −2c^{2} ∫_{0}^{∞} γ(r)g̃(r)g̃′(r) dr

= 2c^{2} ∫_{0}^{∞} \frac{g̃(r)}{F (r)} dr = 2c^{2} ∫_{0}^{∞} \frac{1}{γ(s)F (s)} ∫_{0}^{s} \frac{dr}{F (r)} ds

⩽ 2c^{2} ∫_{0}^{∞} \frac{2s}{γ(s)F (s)^{2}} ds = −\frac{4c^{2}}{F (s)}|_{0}^{∞} = \frac{4c^{2}}{F (0+)} = 2c^{2}.

Proof of Theorem 4.6. Define γ: (0, ∞) → (1, ∞) by γ(r) := ln v(r + 1) + (r + 1)^{2}. Then

∫_{0}^{∞} \frac{r}{γ(r)} dr = ∫_{1}^{∞} \frac{r − 1}{ln v(r) + r^{2}} dr = ∞

by Lemma 4.10(a) since v is increasing, and γ(r) ⩾ r^{2} for all r > 0. By Lemma 4.10(b) there exists a decreasing and bijective function g: (0, ∞) → (0, 1) such that g ∉ L_{1}(0, ∞) and φ_{0} := γ ∘ (g^{2})^{−1} ∈ L_{1}(0, 1).

Without loss of generality we assume that D(τ_{0}) = D(τ)∩D(τ_{0,max}). Let ρ(r) := ∫_{0}^{r} g(s) ds for all r > 0. Then ||ρ′||_{∞} ⩽ 1 and ρ(r) → ∞ as r → ∞, so ψ̃ := ρ ∘ ψ satisfies condition (1.3) by Remark 1.2(d). Moreover,

a∇ψ̃ · ∇ψ̃ = (g ∘ ψ)^{2}a∇ψ · ∇ψ.

Let now w := exp(−φ ∘ (a∇ψ · ∇ψ)) and φ̃ := φ_{0} + φ. Since both φ_{0} and φ are decreasing and (g ∘ ψ)^{2} ⩽ 1, a∇ψ · ∇ψ ⩽ 1 a.e., we infer that

φ̃ ∘ (a∇ψ̃ · ∇ψ̃) ⩾ φ_{0} ∘ (g ∘ ψ)^{2} + φ ∘ (a∇ψ · ∇ψ) = γ ∘ ψ − ln w.

We conclude that

∫_{Ω} exp(−φ̃ ∘ (a∇ψ̃ · ∇ψ̃)) dμ ⩽ ∑_{k=1}^{∞} ∫_{[k−1⩽ψ<k]} exp(−γ ∘ ψ + ln w) dμ

⩽ ∑_{k=1}^{∞} exp(−γ(k − 1)) ∫_{[ψ⩽k]} w dμ ⩽ ∑_{k=1}^{∞} exp(−k^{2}) < ∞,

where we have used that exp(−γ(k −1)) = exp(−k^{2})/v(k) for all k ∈ ℕ. Since φ̃ is decreasing and integrable, the assertion follows from Proposition 4.2.

4.11 Remark. In the above proof, only the estimate

∫_{[k−1⩽ψ<k]} exp(−φ ∘ (a∇ψ · ∇ψ)) dμ ⩽ v(k) (k ∈ ℕ)

was needed. This observation, however, is not useful for weakening the assumptions of Theorem 4.6. Indeed, if the above estimate is satisfied for some increasing function v: (0, ∞) → (3, ∞) such that ∫_{0}^{∞} \frac{r}{ln v(r)} dr = ∞ then a straightforward computation shows that (4.6) holds with ṽ(r) := (r + 1)v(r + 1) in place of v(r) (take into account Lemma 4.10(a)).

Acknowledgement. The author thanks Jürgen Voigt for valuable discussions.

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