6. Outlook 66
6.4. The compact support property in higher dimensions
For the white noise case the compact support result is limited to one spatial dimen-sion as otherwise solutions will not exist (as functions). However, the superprocess results on compact support go beyond one dimensional results. For the colored
6.4 The compact support property in higher dimensions 75 noise case the dimension restriction, q = 1, does not exist (except α < q), so it is natural to ask whether the compact support property holds in higher dimensions.
The main trick in our argument which relied on some spatial structure of R1 is done in (10.6), where the Itˆo-formula is applied to the function ζ(x) = (x∨0).
The functionζ is a harmonic function (w.r.t. ∆) outside the support (−∞,0] ofu0. In higher dimensions imagine that the support of u0 is contained in B(0,1)⊂Rq. Then we need a harmonic function v, which will not grow too fast:
∆v(x) = 0, x∈Rq\B(0,1),
v(x) = 0, x∈∂B(0,1),
lim sup|x|→∞v(x)|x|p = 0,
for somep >0.The last criterion is used in order to ensure finiteness of the integral R
Rqv(x)u(s, x)dx. Nontrivial solutions for this problem are well-known:
v(x) =
(log|x|, q = 2, 1− |x|q−2 q ≥3.
So we are led to the following conjecture:
Conjecture 5. Assume σ(u) = σ0uγ, γ ∈ (0,1) and σ0 > 0. Then, the compact support property holds for nonnegative solutions u ∈ C(R+, Crap) of (5.1) with b≡0 and W˙ = ˙Wk, k(x, y) =|x−y|−α, α∈(0,2∧q).
There are surely other minor modifications of equation (4.11) which might still lead to compact support property. The lower bound on σ(u) ≥ σ0uγ should be possible to be relaxed to one which only holds close to zero. This would be helpful to include cases such as Fisher-Wright noise σ(u) =p
u(1−u) or similar models.
Another minor modification should be the relaxation of the diffusion operator ∆ in a similar way as in Krylov’s work [Kry97], where he used a uniformly elliptic operatorL=a∂2+b∂+c.In fact, this would allow a (small) nonnegative drift. A very interesting other modification would be the use of this technique in the case of L´evy-noise driven SPDE.
Part II.
Proofs
76
7. An abstract result on strong solutions
We start this chapter with a lemma characterising (weak) solutions of SPDE via an approximation procedure. The proof of Lemma 5.1.1 is done after that.
Lemma 7.1.1. Let (Ω,F,Ft,P) be a filtered probability space with a noise W ,˙ which is white in time (see Definition 3.3.15). Let u : R+×Rq → R be a jointly continuous random field such that for any compact setK ⊂Rq the following holds:
E[sups≤tsupx∈K|u(s, x)|2]<∞.Let b:R→R and σ:R+×Rq×R be continuous
Proof. Assume that φ is supported within the compact set K ⊂Rq and for sim-plicity that b and σ only depend on u; the proof can also be done more general,
77
78 An abstract result on strong solutions
As I2 is the sum of two martingales we can use Doob’s inequality to obtain:
I2 ≤2E
by dominated convergence, sincekis locally integrable,φhas compact support and E[sups,z∈K|u(s, z)|2] < ∞ on the one hand and continuity of u(·,·) on the other hand.
TreatingI1 is even easier. We use the bound on band that kA∗φk∞≤C1, since
79
by the dominated convergence theorem and continuity of u.
(b)⇒ (a): By definition of the Riemann integral and by the assumption on u being continuous, we have
pointwise and hence in L2 by square-integrability of the limit. With a similar argument as that forI2, we obtain for Mcand Md defined in the following obvious manner:
80 An abstract result on strong solutions Use (b) and these two observations in the following:
E
"
sup
t≤T
|hφ, ut−u0i − Z t
0
Z
Rq
us(x)A∗φ(x) +b(us(x))φ(x)dxds
− Z t
0
Z
Rq
φ(x)σ(s, x, u)W(ds dx)|2
≤E[sup
t≤T
| Z
Rq
φ(x)(u(t, x)−u0(x))dx−Jd−Md|2+|Md−Mc|2+|Jd−Jc|2]
=o(1) (n→ ∞).
This completes the proof.
Remark 7.1.2. The assumption “white noise” in time is essential for the martingale techniques to apply. The requirement having a.s. continuous solutions might be possible to be relaxed to the case of continuity in probability.
Then we can do the proof of Lemma 5.1.1:
Proof of Proposition 5.1.1. Assume that we are given a filtered probability space (Ω,F,Ft,P) with adapted weak solutionuand noise ˙W .The latter impliesFtu,W ⊂ Ft.We want to apply Proposition 3.14 a) ⇒b) in [Kur07]. Some of his notation is required here.
LetS1 =U and S2∈C([0,∞),S0(Rq)), both Polish spaces. By Definition 3.3.15 we know that W takes values in S2.First, we check that u is compatible with W with respect to the compatibility structure C ={(FtU,FtS0) :t∈R+}. SinceB(S0) is generated by S it suffices to check compatibility for all φ∈ S(R1+q). Let t > 0 fixed and define φ1(s, x) = 1[0,t](s)φ(s, x) and φ2 = φ−φ1, both in Mα. So, by Lemma 3.3.13 we can use linearity
E[ ˙W(φ)|FtW ∨ Ftu] =E[E[ ˙W(φ1) + ˙W(φ2)|Ft]|FtW ∨ Ftu]
=E[( ˙W(φ1) + 0)|FtW ∨ Ftu]
= ˙W(φ1)
=E[ ˙W(φ)|FtW]
since ˙W is a (centred) noise white in time w.r.t. the filtration (Ft)t≥0..By the usual induction, this result extends to all bounded measurable functions on S0(R1+q).So u is compatible to ˙W .
By construction, ν = L[ ˙W] on C(R+ ×H−q(Rq). Finally, the set of convex restrictions Γ needs to be given. Let {φm ∈ D(Rq) : m ∈N} be a dense subset of
81 D(Rq) (separable!). Let for m, n∈N:
hm,n,t(u,W˙ ) =|hut−u0, φmi −
bntc
X
k=1
X
l∈Zq
n−(q+1)u(kn,nl)A∗φm(nl) +b(u(kn,nl))φm(nl)
+
bntc
X
k=1
X
l∈Zq
σ(kn,nl, u(kn,nl))φm(nl) ˙W([kn,k+1n ]×[nl,l+1n ])| ∧1 and
gn(u) =
dnte
X
k=1
1 n
X
l∈Zq,|l|∞≤n
(2n+ 1)−q
|uk/n(l
n)|+ X
j∈Zq,|j|∞≤n
(2n+ 1)−q|k(l n,j
n)uk/n(l
n)uk/n(m n)|
. Clearly the conditions lim supn∈NE[supt≤T h2m,n,t] = 0 and gn <∞ a.s. are convex constraints onM1(U × S0(R1+q)).Let
Γ ={lim sup
n→∞ gn<∞ a.s.,sup
m∈N
lim sup
n→∞ E[sup
t≤T
h2m,n,t] = 0}
be the convex constraints. Since there is a weak solution, by Lemma 7.1.1 the law µ := P◦(u,W˙ )−1 lies in SΓ,C,ν. By pointwise uniqueness (which is just pathwise uniqueness, where joint compatibitility follows as in the compatibility proof before) we know that there exists a strong compatible solution, i.e. u=F( ˙W).
8. Existence and regularity
In this chapter we want to prove the existence and regularity result Theorem 5.2.1.
Right after the theorem there was an overview of the proof. However, we quickly repeat the important steps in a more technical description.
The proof will follow closely the one given in the appendix of [MPS06] and is decomposed into several lemmas. First, we quote the classical existence result for Lipschitz coefficients. Then we derive uniform regularity results for an approximat-ing sequence of coefficients σn → σ for general σ. This regularity allows to get a certain tightness result; one of the limit points is then shown to be a mild solution.
The proofs will be less explicit in the first steps, but more explicit in the last step, the “convergence” argument from Theorem 2.6 of [Shi94].