Upper bounds getdocc6b9. 250KB Jun 04 2011 12:04:57 AM

3.1 Upper bounds

Let Y λ be the process associated with the following Dirichlet form: E Y λ f , f = 1 2 Z R d d X i, j=1 a i j x ∂ f x ∂ x i ∂ f x ∂ x j d x + Z Z |x− y|≤λ f y − f x 2 J x, yd x d y, 3.3 so that Y λ has jumps of size less than λ only. Let N λ be the exceptional set corresponding to the Dirichlet form defined by 3.3. Let P Y λ t be the semigroup associated with E Y λ . We will use the arguments in [FOT94] and [CKS87] as indicated in the proof of Lemma 2.3 to obtain the existence of the heat kernel p Y λ t, x, y as well as some upper bounds. For any v, ψ ∈ F , we can define Γ λ [v]x = 1 2 ∇v · a∇v + Z |x− y|≤λ vx − v y 2 J x, yd y, D λ ψ 2 = ke −2ψ Γ λ [e ψ ]k ∞ ∨ ke 2 ψ Γ λ [e −ψ ]k ∞ , and provided that D λ ψ ∞, we set E λ t, x, y = sup{|ψ y − ψx| − t D λ ψ 2 ; D λ ψ ∞}. Proposition 3.4. There exists a constant c 1 such that the following holds. p Y λ t, x, y ≤ c 1 t −d2 exp[ −E λ 2t, x, y], ∀ x, y ∈ R d \N λ, and t ∈ 0, ∞, where p Y λ t, x, y is the transition density function for the process Y λ associated with the Dirichlet form E Y λ . Proof. Similarly to Proposition 3.1, we write E Y λ f , f = E Y λ c f , f + E Y λ d f , f , Since J x, y ≥ 0 for all x, y ∈ R d , we have E Y λ f , f ≥ E Y λ c f , f . 3.4 We have the following Nash inequality; see Section VII.2 of [Bas97]: k f k 21+ 2 d 2 ≤ c 2 E Y λ c f , f k f k 4 d 1 . This, together with 3.4 yields k f k 21+ 2 d 2 ≤ c 2 E Y λ f , f k f k 4 d 1 . Now applying Theorem 3.25 from [CKS87], we get the required result. ƒ 322 We now estimate E λ t, x, y to obtain our first main result. Proof of Theorem 2.4. Let us write Γ λ as Γ λ [v] = Γ c λ [v] + Γ d λ [v], where Γ c λ [v] = 1 2 ∇v · a∇v, and Γ d λ [v] = Z |x− y|≤λ vx − v y 2 J x, yd y. Fix x , y ∈ € R d \N λ Š × € R d \N λ Š . Let µ 0 be constant to be chosen later. Choose ψx ∈ F such that |ψx − ψ y| ≤ µ|x − y| for all x, y ∈ R d . We therefore have the following: ¯ ¯ ¯e −2ψx Γ d λ [e ψ ]x ¯ ¯ ¯ = e −2ψx Z |x− y|≤λ e ψx − e ψ y 2 J x, yd y = Z |x− y|≤λ e ψ y−ψx − 1 2 J x, yd y ≤ c 1 Z |x− y|≤λ |ψx − ψ y| 2 e 2 |ψx−ψ y| J x, yd y ≤ c 1 µ 2 e 2 µλ Z |x− y|≤λ |x − y| 2 J x, yd y = c 1 µ 2 K λe 2 µλ , where K λ = sup x ∈R d Z |x− y|≤λ |x − y| 2 J x, yd y. Some calculus together with the ellipticity condition yields: ¯ ¯ ¯e −2ψx Γ c λ [e ψ ]x ¯ ¯ ¯ = 1 2 ¯ ¯ ¯e −2ψx ∇e ψx · a∇e ψx ¯ ¯ ¯ = 1 2 ¯ ¯ ∇ψx · a∇ψx ¯ ¯ ≤ 1 2 Λk∇ψk 2 ∞ ≤ 2µ 2 Λ. Combining the above we obtain ¯ ¯ ¯e −2ψx Γ λ [e ψ ]x ¯ ¯ ¯ ≤ c 1 µ 2 K λe 2 µλ + 2µ 2 Λ. Since we have similar bounds for ¯ ¯ e 2 ψx Γ λ [e −ψ ]x ¯ ¯ , we have − E λ 2t; x, y ≤ 2t D λ ψ 2 − |ψ y − ψx| ≤ 2tµ 2 € c 1 K λe 2 µλ + 2Λ Š − ¯ ¯ ¯ ¯ µx − y · x − y |x − y | ¯ ¯ ¯ ¯ . 3.5 323 Taking x = x , y = y and µ = λ = 1 in the above and using Proposition 3.4 together with the fact that t ≤ 1, we obtain p Y t, x , y ≤ c 2 t − d 2 e −|x − y | , Since x and y were taken arbitrarily, we obtain the required result. ƒ The following is a consequence of Proposition 3.4 and an application of Meyer’s construction. Proposition 3.5. Let r ∈ 0, 1]. Then for x ∈ R d \N , P x sup s ≤t r 2 |X s − x| r ≤ 1 2 , where t is a small constant. Proof. The proof is a follow up of that of the Theorem 2.4, so we refer the reader to some of the notations there. Let λ be a small positive constant to be chosen later. Let Y λ be the subprocess of X having jumps of size less or equal to λ. Let E Y λ and p Y λ t, x, y be the corresponding Dirichlet form and probability density function respectively. According to Proposition 3.4, we have p Y λ t, x, y ≤ c 1 t −d2 exp[ −E λ 2t, x, y]. 3.6 Taking x = x and y = y in 3.5 yields − E λ 2t; x , y ≤ 2tµΛ + 2c 2 t µ 2 K λe 2 µλ − µ|x − y | 3.7 Taking λ small enough so that Kλ ≤ 1 2c 2 , the above reduces to −E λ 2t; x , y ≤ 2tµ 2 Λ + tλ 2 µλ 2 e 2 µλ − µ|x − y | ≤ 2tµ 2 Λ + tλ 2 e 3 µλ − µ|x − y |. Upon setting µ = 1 3 λ log 1 t 1 2 and choosing t such that t 1 2 ≤ λ 2 , we obtain −E λ 2t; x , y ≤ c 3 t 1 2 log t 2 + t λ 2 1 t 1 2 − |x − y | 3 λ log 1 t 1 2 ≤ c 3 t 1 2 log t 2 + 1 + log[t |x − y |6λ ]. Applying the above to 3.6 and simplifying p Y λ t, x , y ≤ c 4 e c 3 t 1 2 log t 2 t |x − y |6λ t −d2 = c 4 e c 3 t 1 2 log t 2 t |x − y |12λ−d2 t |x − y |12λ = c 4 e c 3 t 1 2 log t 2 t |x − y |12λ−d2 e |x0−y0| 12 λ log t . For small t, the above reduces to p Y λ t, x , y ≤ c 5 t |x − y |12λ−d2 e −c 6 |x − y |12λ 3.8 324 Let us choose λ = c 7 r d with c 7 124 so that for |x − y | r2, we have |x − y |12λ − d2 d 2. Since t is smallless than one, we obtain P x |Y λ t − x | r2 ≤ Z |x − y|r2 c 5 t |x − y|12λ−d2 e −c 3 |x − y|12λ d y ≤ c 5 t d 2 Z |x − y|r2 e −c 3 |x − y|12λ d y. We bound the integral on the right hand side to obtain P x |Y λ t − x | r2 ≤ c 7 t d 2 e −c 8 r . Therefore there exists t 1 0 small enough such that for 0 ≤ t ≤ t 1 , we have P x |Y λ t − x | r2 ≤ 1 8 . We now apply Lemma 3.8 of [BBCK] to obtain P x sup s ≤t 1 |Y λ s − Y λ | ≥ r ≤ 1 4 ∀ s ∈ 0, t 1 ]. 3.9 We can now use Meyer’s argumentRemark 3.3 to recover the process X from Y λ . Recall that in our case J x, y = J x, y1 |x− y|≤λ so that after using Assumptions 2.2a and choosing c 7 smaller if necessary, we obtain sup x N x ≤ c 9 r −2 , where c 9 depends on the K i s and N x = Z R d J x, z − J x, zdz. Set t 2 = t r 2 with t small enough so that t 2 ≤ t 1 . Recall that U 1 is the first time at which we introduce the big jump. We thus have P x sup s ≤t 2 |X s − x | ≥ r ≤ P x sup s ≤t 2 |X s − x | ≥ r, U 1 t 2 + P x sup s ≤t 2 |X s − x | ≥ r, U 1 ≤ t 2 ≤ P x sup s ≤t 2 |Y λ s − x| ≥ r + P x U 1 ≤ t 2 = 1 4 + 1 − e −sup Nt 2 = 1 4 + 1 − e −c 9 t . By choosing t smaller if necessary, we get the desired result. ƒ Remark 3.6. It can be shown that the process Y λ is conservative. This fact has been used above through Lemma 3.8 of [BBCK]. 325

3.2 Lower bounds