Gaugeability with Applications to Symmetric α-Stable Processes

Takeda Masayoshi, Tohoku University, Mathematical Institute

The purpose is to give two applications of gaugeability to symmetric α-stable processes. 𝕄 = (Ω, ℙ_{x}, X_{t}) : a symmetric α-stable process on ℝ^{d} (0 < α < 2). (E^{(α)}, D(E^{(α)})) : the Dirichlet form of 𝕄^{α}. Cap(A): the capacity of a set A defined by (E^{(α)}, D(E^{(α)})). For an open set D ⊂ ℝ^{d}, 𝕄^{D} = (ℙ_{x}, X_{t}^{D}): the absorbing process on D

[figure]

Assume that 𝕄^{D} is transient, that is, Cap(ℝ^{d} \ D) > 0 if α ≥ d

G_{D}(x, y) : the Green function of 𝕄^{D}.

Definition 1.

A positive Radon measure μ on ℝ^{d} is said to be in the class S_{∞}^{D}, if for any ϵ > 0 there exists a compact set K ⊂ D and δ > 0 such that

sup_{(x,z)∈D×D\d} ∫_{K^{c}} \frac{G_{D}(x, y)G_{D}(y, z)}{G_{D}(x, z)}μ(dy) ≤ ϵ

and for all measurable sets B ⊂ K with μ(B) < δ,

sup_{(x,z)∈D×D\d} ∫_{B} \frac{G_{D}(x, y)G_{D}(y, z)}{G_{D}(x, z)}μ(dy) ≤ ϵ

μ ∈ S_{∞}^{D} ⟺ PCAF A_{t}^{μ}. (Revus Correspondence)

For a measure μ in S_{∞}^{D}, define

λ(μ; D) = inf {E^{(α)}(u, u) : u ∈ C_{0}^{∞}(D), ∫_{D} u^{2}(x)μ(dx) = 1} .

p_{t}^{μ,D}(x, y) : the integral kernel of the Feynman-Kac semigroup,

p_{t}^{μ,D}f(x) := 𝔼_{x}[exp(A_{t}^{μ})f(X_{t}); t < τ_{D}] = ∫_{D} p_{t}^{μ,D}(x, y)f(y)dy

G^{μ,D}(x, y) : the Green function, G^{μ,D}(x, y) = ∫_{0}^{∞} p_{t}^{μ,D}(x, y)dt.

\underline{Theorem 1.}(Z.Zhao, T, Z.Q.Chen)

Let μ ∈ S_{∞}^{D}. Then the following conditions are equivalent:

(i) (gaugeability) sup_{x∈D} 𝔼_{x}[e^{A_{τ_{D}}^{μ}}] < ∞

(⟺ ∃x_{0} ∈ D s.t. 𝔼_{x_{0}}[e^{A_{τ_{D}}^{μ}}] < ∞);

(ii) (subcriticality) G^{μ,D}(x, y) < ∞ for x, y ∈ D, x ≠ y;

(iii) λ(μ; D) > 1.

Applications

(i) The first is relevant with the ultracontractivity of Schrödinger semigroups p_{t}^{μ}:

p_{t}^{μ}f(x) = 𝔼_{x}[exp(A_{t}^{μ})f(X_{t})].

∥p_{t}^{μ}∥_{1,∞} :the operator norm of p_{t}^{μ} from L^{1}(ℝ^{d}) to L^{∞}(ℝ^{d}).

\underline{Theorem 2.}
Let μ ∈ S_{∞} with ∫_{ℝ^{d}} ∫_{ℝ^{d}} |x − y|^{α−d}dμ(x)dμ(y) < ∞. Then

λ(μ; ℝ^{d}) > 1 ⟺ ∥p_{t}^{μ}∥_{1,∞} ≤ \frac{c}{t^{d/α}} , t > 0.

(ii) 𝔹^{α} = (X̄_{t}, ℙ̄_{x}) : the branching α-symmetric stable process with the branching rate k, a smooth measure of 𝕄^{(α)},

ℙ̄_{x}[T > t|σ(X)] = exp(−A_{t}^{k}),

where T is the first splitting time. {p_{n}(x)}_{n≥2} : the branching mechanism Q(x) = ∑_{n≥2} np_{n}(x), μ(dx) = (Q(x) − 1)k(dx).

Assume that sup_{x∈ℝ^{d}} Q(x) < ∞.

\underline{Theorem 3.} Let K be a closed set with Cap(K) > 0. If μ ∈ S_{∞}^{ℝ^{d}\K}, then

λ(μ; ℝ^{d} \ K) > 1 ⟺ 𝔼̄_{x}[N_{K}] < ∞.

Here N_{K} is the number of branches of 𝔹^{α} ever hitting K.

References

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[CS] Z.Q. Chen and R.M. Song, General gauge and conditional gauge theorems, Ann. Probab., 30 (2002), 1313-1339.

[CZ] Z.Q. Chen and S.T. Zhang, Girsanov and Feynman-Kac type transformations for symmetric Markov processes. Ann. Inst. Henri Poincare , 38 (2002), 475-505.

[GK] A. Grigor’yan and M. Kelbert, Recurrence and transience of branching diffusion processes on Riemannian manifolds. Ann. Probab. 31 (2003), 244-284.

[TU] M. Takeda and T. Uemura, Subcriticality and Gaugeability for Symmetric α-Stable Processes, Forum Math. 16 (2004), 505-517.