Characterizing limit points getdoc28f9. 373KB Jun 04 2011 12:04:11 AM

ON −1 between the time-grid points in N −2 N . We obtain that P ‚ sup t ≤T k ˜ A ξ N t − ˆ A ξ N t k −λ ≥ 7N −14 Œ ≤ P € ∃t ∈ [0, T ] ∩ N −2 N , s ∈ [0, T ], |s − t| ≤ N −2 such that kAξ N t − Aξ N s k −λ + 〈ν N , ψ · t − ψ · s 〉 −λ ≥ 7N −14 Š ≤ P € ∃t ∈ [0, T ] ∩ N −2 N , s ∈ [0, T ], |s − t| ≤ N −2 such that kAξ N t − Aξ N s k −λ ≥ 6N −14 Š for N big enough. Next note that the value of A ξ N t x changes only at jump times of P t x; y or Q m,i, j.k t x; y 1 , . . . , y m ; z, i, j, k = 0, 1, m ≥ 2 for some y ∼ x respectively for some y 1 , . . . , y m ∼ x and arbitrary z ∈ N −1 Z and that each jump of A ξ N t is by definition of A ξ N t bounded by N −12 . Then, writing P a for a Poisson variable with mean a, we get as a further bound on the above X l ∈Z P € ∃z ∈ N −1 Z ∩ l, l + 1], ∃t ∈ [0, T ] ∩ N −2 N , ∃s ∈ [t, t + N −2 ] with § A ξ N t z − Aξ N s z ∧ Aξ N t+N −2 z − Aξ N s z ª ≥ N −14 e λ|l|−1 ‹ ≤ X l ∈Z N N 2 T P   C N −12   X y ∼0 P N −2 0; y + X i, j,k=0,1,m ≥2 X y 1 ,..., y m ∼0 X u Q m,i, j,k N −2 0; y 1 , . . . , y m ; u    ≥ N −14 e λ|l|−1    ≤ X l ∈Z CT N 3 P € C N −12 P € N −2 €€ N − θ N Š + C Q ŠŠ ≥ N −14 e λ|l|−1 Š ≤ X l ∈Z CT N 3 P € P € N −2 € N + C Q ŠŠŠ p ≥ C N p 4 e λp|l|−1 for some p 0. Now apply Chebyshev’s inequality. Choose p 0 such that 3 − p4 0. Then the resulting sum is finite and goes to zero for N → ∞. b The proof of part b follows as the proof of Lemma 7b of [13].

3.7 Characterizing limit points

To conclude the proof of Theorem 2.10 we can proceed as in Section 4 in [13], except for the proof of weak uniqueness of 21. We shall give a short overview in what follows. The interested reader is referred to [13] for complete explanations. In short, Lemma 3.6b implies for all φ ∈ C c that sup t 〈ν N t , φ〉 − 〈Aξ N t , φ〉 ≤ Cλ k Dφ, N −12 k λ N →∞ → 0. 59 The C-tightness of A ξ N t : t ≥ 0 in C 1 follows from the results of Subsection 3.6. 649 This in turn implies the C-tightness of ν N t : t ≥ 0 as cadlag Radon measure valued processes with the vague topology. Indeed, let ϕ k , k ∈ N be a sequence of smooth functions from R to [0, 1] such that ϕ k x is 1 for |x| ≤ k and 0 for |x| ≥ k + 1. Then a diagonalization argument shows that C-tightness of ν N t : t ≥ 0 as cadlag Radon measure valued processes with the vague topology holds if and only if C-tightness of ϕ k d ν N t : t ≥ 0 as cadlag M F [−k + 1, k + 1]-valued processes with the weak topology holds. Here, M F [−k + 1, k + 1] denotes the space of finite measures on [ −k + 1, k + 1]. Now use Theorem II.4.1 in Perkins [15] also see the Remark following it and the proof of sufficiency on pages 157-159 to obtain C-tightness of ϕ k d ν N t : t ≥ 0 in D M F [−k + 1, k + 1]. The compact containment condition i in [15] is obvious. The second condition ii in [15] follows from 59 and the C-tightness of A ξ N t : t ≥ 0 in C 1 together with Lemma 3.7a. Observe in particular, that 59 implies the existence of a subsequence A ξ N k t , ν N k t that converges to u t , ν t . Hence, we can define variables with the same distributions on a different probability space such that with probability one, for all T ∞, λ 0, φ ∈ C c , sup t ≤T Aξ N k t − u t −λ → 0 as k → ∞, sup t ≤T 〈φ, ν N k t 〉 − 〈φ, ν t 〉 → 0 as k → ∞, where we used 0 ≤ Aξ N k t ≤ 1 and thus 0 ≤ u t x ≤ 1 a.s. for the first limit. We obtain in particular ν t d x = u t xd x for all t ≥ 0. It remains to investigate u t in the special case with no short-range competition, i.e. where q k,m,N 0 j = q k,m,N 1 j , j = 0, 1. Take φ t ≡ φ ∈ C 3 c in 37. We get M N t φ =〈ν N t , φ〉 − 〈ν N , φ〉 − Z t 〈ν N s − , ∆ φ 〉ds − E 1 t φ 60 − X k=0,1 1 − 2k X m ≥2, j=0,1 q k,m,N 0 j Z t 〈 € F j ◦ A € ξ N s − ŠŠ × € F 1 −k ◦ A € ξ N s − ŠŠ m −1 1 € ξ N s − · = k Š , φ〉ds − E 3 t φ. From 30 and 36 and the Burkholder-Davis-Gundy inequality 49 we obtain that the error terms converge to zero for all 0 ≤ t ≤ T almost surely. Taylor’s theorem further shows that replace N k by N for notational ease ∆ φ x N = N − θ N cN N p N 2 X y ∼x N φ y − φx N → ∆ φ 6 x as x N → x and N → ∞ on the support of φ. Using this in 60 we can show that M N t φ converges to a continuous martingale M t φ satisfying M t φ = Z φxu t xd x − Z φxu xd x − Z t Z ∆ φx 6 u s xd x ds 61 − X k=0,1 1 − 2k X m ≥2, j=0,1 q k,m 0 j Z t Z F j u s x F 1 −k u s x m −1 F k u s x φxd x ds. 650 To exchange the limit in N → ∞ with the infinite sum we used [16], Proposition 11.18 together with Hypothesis 2.7. Recall in particular, that 0 ≤ F l u s x ≤ 1 for l = 0, 1. To show that M t φ is indeed a martingale we used in particular 33 to see that 〈M N φ〉 t ≤ Cλt kφ k 2 λ 〈1, e −2λ 〉 is uniformly bounded. Therefore, M N t φ : N ≥ N and all its moments are uniformly integrable, using the Burkholder-Davis-Gundy inequality of the form 49 once more. We can further calculate its quadratic variation by making use of 32 for N → ∞ together with the uniform integrability of M N t φ 2 : N ≥ N . Use our results for φ ∈ C 3 c , note that C 3 c is dense in C 2 c with respect to the norm k f k≡k f k ∞ + k f ′ k ∞ + k ∆ f k ∞ , and use 61 to see that u t solves the martingale problem associated with the SPDE 21. It is now straightforward to show that, with respect to some white noise, u t is actually a solution to 21 see Rogers and Williams [17], V.20 for the similar argument in the case of SDEs.

3.8 Uniqueness in law