Abstract
We study diffusion processes and stochastic flows which are time-changed random perturbations of a deterministic flow on a manifold. Using non-symmetric Dirichlet forms and their convergence in a sense close to the Mosco-convergence, we prove that, as the deterministic flow is accelerated, the diffusion process converges in law to a diffusion defined on a different space. This averaging principle also holds at the level of the flows. Our contributions in this article include:
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a proof of an original averaging principle for stochastic flows of kernels;
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the definition and study of a convergence of sequences of non-symmetric bilinear forms defined on different spaces;
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the study of weighted Sobolev spaces on metric graphs or “books”.
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Appendix:
Appendix:
Notation: In a topological space \(A\Subset B\) means that \(\overline {A}\subset \mathring {B}\) and \(\overline {A}\) is compact.
1.1 Weighted Sobolev Spaces
For k ≥ 1, the class of Muckenhoupt weights A2 consists all map**s \(\omega :\mathbb {R}^{k}\to [0,\infty ]\), for which there is a constant C such that for all ball B in \(\mathbb {R}^{k}\), we have
For m ≥ 2, set \(B_{m,1}:=\left (\cup _{i=1}^{m} ]0,\infty [\times \{i\}\right )\cup \{0\}\), equipped with the distance d1 defined by d1((x,i), (y,i)) = |x − y| and if i≠j, d1((x,i), (y,j)) = x + y, and d1((x,i), 0) = x. Let \(i_{1}:B_{m,1}\to \{0,\dots ,m\}\) be defined i1(x,i) = i and i1(0) = 0. Then Bm,1 is a metric graph constituted of m half lines joined at 0. Set also \(B_{m,2}:=\mathbb {R}\times B_{m,1}\), equipped with the distance d2 defined by d2((x,i), (y,i)) = ∥x − y∥, \(d_{2}((x,i),(y,j))=\|x-y^{\prime }\|\) if i≠j and where \(y^{\prime }=(y_{1},-y_{2})\), and d2((x,i),y) = ∥x − y∥ if y = (y1, 0), where ∥⋅∥ is the Euclidean norm on \(\mathbb {R}^{2}\). Then Bm,2 is a metric space constituted of m half planes joined along a line. Let also \(i_{2}: B_{m,2}\to \{0,\dots ,m\}\) be defined by i2(x1,x2,i) = i and i2(x1, 0) = 0. To simplify the notation we will simply denote d1 and d2 by d. For 1 ≤ i ≤ m, we will use the notation (0,i) = 0 ∈ Bm,1 and for \(x_{1}\in \mathbb {R}\), (x1, 0,i) = (x1, 0) ∈ Bm,2.
For m = 1 and k ∈{1, 2}, set \(B_{1,k}=\mathbb {R}^{k}\).
Let \(\omega : B_{m,k}\to [0,\infty ]\) be such that \(\omega \in L^{1}_{loc}(B_{m,k})\), i.e. such that for all \(A\Subset {\Omega }\), \(\mu (A):= {\int \limits }_{A} \omega (x) dx < \infty \), with dx the measure on Bm,k that coincides with the Lebesgue measure on Ei := {x ∈ Bm,k : ik(x) = i} for each i. For i≠j, set Ei,j := Ei ∪ Ej ∪ E0 (which is isometric to \(\mathbb {R}^{k}\), and will be thus identified to \(\mathbb {R}^{k}\)).
In the following, we fix k ∈{1, 2} and m ≥ 1 and we let Ω be an open subset of Bm,k. For i ∈{1,⋯ ,m}, set Ωi = Ω ∩ Ei and for 1 ≤ i≠j ≤ m, set Ωi,j = Ω ∩ Ei,j. Then Ωi and Ωi,j are open subsets of \(\mathbb {R}^{k}\). Denote by L2(Ω,ω) the space of all measurable functions f on Bm,k such that \({\int \limits }_{\Omega } f^{2}(x) \omega (x) dx < \infty \). For a function f ∈ L2(Ω,ω), weakly differentiable on Ωi,j for all 1 ≤ i≠j ≤ m, define the norm of f by
Let W1(Ω,ω) be the completion with respect to the norm \(\|\cdot \|_{W^{1}({\Omega },\omega )}\) of the vector space of the functions f ∈ L2(Ω,ω), that are weakly differentiable on Ωi,j for all 1 ≤ i≠j ≤ m and with \(\|f\|_{W^{1}({\Omega },\omega )}<\infty \). Suppose also that \(\frac {1}{\omega }\in L^{1}_{loc}({\Omega })\). Then, for all \(i\in \{1,\dots ,k\}\) (resp. all 1 ≤ i≠j ≤ k), the restriction of f ∈ W1(Ω,ω) to Ωi (resp. to Ωi,j) belongs to \(W^{1,1}_{loc}({\Omega }_{i})\) (resp. to \(W^{1,1}_{loc}({\Omega }_{i,j})\).
We also define H1(Ω,ω) (resp. \({H^{1}_{0}}({\Omega },\omega )\)) to be the completion of C(Ω) ∩ W1(Ω,ω) (resp. of Cc(Ω) ∩ W1(Ω,ω)), with respect to \(\|\cdot \|_{W^{1}({\Omega },\omega )}\). Define also \({W^{1}_{0}}({\Omega },\omega )\) to be the set of all f ∈ W1(Ω,ω) such that the function F = f1Ω ∈ W1(Bm,k,ω). Equipped with the inner product
W1(Ω,ω), \({W^{1}_{0}}({\Omega },\omega )\), H1(Ω,ω) and \({H^{1}_{0}}({\Omega },\omega )\) are Hilbert spaces.
Lemma A.1
Let m ≥ 1 and \(O\Subset {\Omega }\) be open subsets of Bm,k. Then there is δ > 0 such that Kδ := {x ∈ Bm,k : d(x,O) ≤ δ} is a compact set with O ⊂ Kδ ⊂Ω. Suppose that there is \(\bar {\omega }:B_{m,k}\to [0,\infty ]\) such that \(\bar {\omega }=\omega \) on Ω and such that
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when m ≥ 2, for all i≠j, the restriction \(\bar {\omega }_{i,j}\) of \(\bar {\omega }\) to Ei,j belongs to the class A2.
-
when m = 1, \(\bar {\omega }\) belongs to the class A2.
Then if \(f\in {W^{1}_{0}}(O,\omega )\), there is g ∈ L2(O,ω) such that g = 0 on O ∖ Kδ and such that for all (x,y) ∈ O2,
Moreover, there is a sequence of lipschitzian functions \(f_{n}\in {W^{1}_{0}}(O,\omega )\) such that \(\lim _{n\to \infty }\|f-f_{n}\|_{W^{1}(O,\omega )}=0\).
Proof
We only consider the case m ≥ 2, the case m = 1 being simpler.
The existence of δ and Kδ is a standard exercise.
Let \(f\in {W^{1}_{0}}(O,\omega )\) and denote by fi,j the restriction f to Ei,j, and set Oi,j := O ∩ Ei,j. Then \(f_{i,j}\in {W^{1}_{0}}(O_{i,j},\bar {\omega }_{i,j})\). Set F := f1O and Fi,j the restriction of F on Ei,j. Then \(F_{i,j}\in W^{1}(E_{i,j},\bar {\omega }_{i,j})\). Recall that Ei,j is isometric to \(\mathbb {R}^{k}\). Note that for x ∈ Oi = O ∩ Ei, we have for all j≠i, f(x) = fi,j(x) = Fi,j(x).
Since \(\bar {\omega }\in A_{2}\), we have (see [11]) \(G_{i,j}\in L^{2}(E_{i,j},\bar {\omega }_{i,j})\) such that for all \((x,y)\in E_{i,j}^{2}\),
For x ∈Ω, define \(g(x):={\sum }_{j\ne i} G_{i,j}(x)\) if x ∈Ωi. Then g ∈ L2(Ω,ω), and we have for all (x,y) ∈Ω2,
Set now \(g_{\delta }:=g 1_{K_{\delta }}+\delta ^{-1}|f|\). Then, we have that for all (x,y) ∈Ω,
Indeed, this inequality is straightforward to check if \((x,y)\in K_{\delta }^{2}\) or if (x,y) ∈ (Ω ∖ O)2. If (x,y) ∈ O × (Ω ∖ Kδ), we have d(x,y)(gδ(x) + gδ(y)) ≥ d(x,y)δ− 1|f(x)|≥|f(x) − f(y)|. This shows the first part of the lemma.
For the second part, it suffices to follow the proof of Theorem 5 of [11] (with the function gδ, and since f = fλ on Eλ and that \(E_{\lambda }\supset {\Omega }\setminus K_{\delta }\), we have fλ = 0 sur Ω ∖ Kδ). □
For an open set U ⊂Ω, define the capacity of U by
Lemma A.2
Let m ≥ 1, Ω an open subset of Bm,k and \(\omega :{\Omega }\to [0,\infty ]\) a measurable map**. For R > 0, set ΩR := {x ∈Ω : ∥x∥ < R}. Suppose that, for all R > 0, there exists a non decreasing sequence of open subsets (ΩR,n)n≥ 1 such that
-
(i)
∪n≥ 1ΩR,n = ΩR,
-
(ii)
\(\lim _{n\to \infty } \text {Cap}({\Omega }_{R}\setminus \overline {\Omega }_{R,n})=0\),
-
(iii)
for all n ≥ 1, \({W^{1}_{0}}({\Omega }_{R,n},\omega )={H^{1}_{0}}({\Omega }_{R,n},\omega )\).
Then we have \(W^{1}({\Omega },\omega )={H^{1}_{0}}({\Omega },\omega )\).
Proof
Let f ∈ W1(Ω,ω) and 𝜖 > 0. For R > 1, define \(f_{R}:{\Omega }\to \mathbb {R}\) defined by fR(x) = f(x) if ∥x∥≤ R − 1, fR(x) = (R −∥x∥)f(x) if ∥x∥∈ [R − 1,R] and fR(x) = 0 if ∥x∥ > R. Then we have that \(f_{R}\in {W^{1}_{0}}({\Omega }_{R},\omega )\) and it is easy to check that there is an R0 > 0 such that for all R > R0, \(\|f-f_{R}\|_{W^{1}({\Omega },\omega )}<\epsilon \).
From now on, we fix R > R0. Applying Theorem III.2.11 in [19], conditions (i) and (ii) ensures that there is a sequence of functions \(f_{R,n}\in {W^{1}_{0}}({\Omega }_{R,n},\omega )\) such that \(\lim _{n\to \infty }\|f_{R}-f_{R,n}\|_{W^{1}({\Omega }_{R},\omega )}=0\).
And we conclude using (iii). □
Let now G be a connected metric space, and fix k ∈{1, 2}. Suppose that there is a locally finite covering \(({\Omega }_{\ell })_{\ell \in {\mathscr{L}}}\) of G, with open sets and and with \({\mathscr{L}}\) a countable set. Suppose also that this covering is such that for each ℓ, Ωℓ is isometric to an open subset of Bm,k for some m ≥ 1. Suppose that there is a sequence of functions \((\varphi _{\ell })_{\ell \in {\mathscr{L}}}\) such that φℓ : G → [0, 1], φℓ = 0 on G ∖Ωℓ, φℓ restricted to \({\Omega }_{\ell }^{i}\) is C1, with bounded derivatives, for each \(i\in \{1,\dots , m_{\ell }\}\) and such that \({\sum }_{\ell \in {\mathscr{L}}}\varphi _{\ell }=1\).
Let \(\omega :G\to [0,\infty ]\) be a measurable map**. For each ℓ, then there is m ≥ 1 such that Ωℓ ⊂ Bm,k, we let \(\omega _{\ell }:B_{m,k}\to [0,\infty ]\) be such that ωℓ = ω on Ωℓ. Suppose that \(\omega _{\ell }\in L^{1}_{loc}(B_{m,k})\) and \(\frac {1}{\omega _{\ell }}\in L^{1}_{loc}({\Omega }_{\ell })\).
Define W1(G,ω) to be the set of all measurable functions \(f:G\to \mathbb {R}\) such that for each ℓ there is \(f_{\ell }\in W^{1}({\Omega }_{\ell },\omega _{\ell })\) with f = fℓ on Ωℓ with \(\|f\|_{W^{1}(G,\omega )}<\infty \) where
Then W1(G,ω) equipped with the innner product
is a Hilbert space. Define also \({H^{1}_{0}}(G,\omega )\) as the completion of Cc(G) ∩ W1(G,ω). Note that if f ∈ W1(G,ω), we have that for each ℓ, \(f\varphi _{\ell }\in W^{1}({\Omega }_{\ell },\omega _{\ell })\).
Lemma A.3
Suppose that for all ℓ and all R > 0 there exists a non decreasing sequence of open subsets (Ωℓ,R,n)n≥ 1 such that
-
(i)
\(\bigcup _{n\ge 1}{\Omega }_{\ell ,R,n}={\Omega }_{\ell ,R}\),
-
(ii)
\(\lim _{n\to \infty } \text {Cap}({\Omega }_{\ell ,R}\setminus \overline {\Omega }_{\ell ,R,n})=0\),
-
(iii)
for all n ≥ 1, \({W^{1}_{0}}({\Omega }_{\ell ,R,n},\omega _{\ell })={H^{1}_{0}}({\Omega }_{\ell ,R,n},\omega _{\ell })\).
Then we have \(W^{1}(G,\omega )={H^{1}_{0}}(G,\omega )\).
Proof
The proof of this lemma is almost identical to the one of Lemma A.2. We write \(f={\sum }_{\ell } f_{\ell }\), where fℓ := fφℓ. For each 𝜖 > 0 there is a finite \({\mathscr{L}}_{0}\subset \mathcal L\) such that \(\|f-{\sum }_{\ell \in {\mathscr{L}}_{0}}f_{\ell }\|_{W^{1}(G,\omega )}<\epsilon \). And then we can follow the proof of Lemma A.2, for Ω = Ωℓ and f = fℓ and to find, for all \(\ell \in {\mathscr{L}}_{0}\), an integer n ≥ 1 and a function \(g_{\ell }\in C_{c}({\Omega }_{\ell ,R,n})\cap W^{1}({\Omega }_{\ell },\omega _{\ell })\) such that \(\|f_{\ell } -g_{\ell }\|_{W^{1}(G,\omega )}<\frac {\epsilon }{|{\mathscr{L}}_{0}|}\). Then \(\|f -{\sum }_{\ell \in {\mathscr{L}}_{0}} g_{\ell }\|_{W^{1}(G,\omega )}<2 \epsilon \). □
1.2 Regularity for Section ??
In this paragraph, we complete the proofs of Proposition 7.5 and of Proposition 7.6.
Let us first recall some notation of Section ??. The space \(\widetilde M=V\cup \bigcup _{i\in I} E_{i}\) is a metric graph, with Ei =]li,ri[×{i}. There is a continuous map \(\pi :\mathbb {R}^{2}\to \widetilde M\) such that if x ∈Ωi, π(x) = (H(x),i) ∈ Ei. For v ∈ V, \(I^{+}_{v}=\{i\in I, (l_{i},i)=v\}\), \(I^{-}_{v}=\{i\in I, (r_{i},i)= v\}\) and \(I_{v}=I^{+}_{v}\cup I^{-}_{v}\). Let d(v) be the cardinal of the set Iv, which is the degree of v in the graph \(\widetilde M\). Choose i ∈ Iv and set hv = li if v = (li,i) or hv = ri if v = (ri,i). Remark that hv does not depend on the particular choice i ∈ Iv. For v ∈ V, we define \(\gamma (v):=\bigcup _{(h,i)\sim v}\gamma _{i}(h)\), the connected level set associated to the vertex v. Then for all x ∈ γ(v), H(x) = hv.
The space \(\widetilde { {\mathscr{H}}}=\{f: f\circ \pi \in H^{1}(\mathbb {R}^{2})\}\) equipped with the inner product \(\langle f,g\rangle _{\widetilde H}:=\langle f\circ \pi ,g\circ \pi \rangle _{H^{1}(\mathbb {R}^{2})}\) is a Hilbert space. Let now \(f\in \widetilde {{\mathscr{H}}}\). For i ∈ I, let \(f_{i}:]l_{i},r_{i}[\to \mathbb {R}\) be defined by fi(h) = f(h,i). Then (see Lemma 7.4), for all i ∈ I, fi is weakly differentiable and we have
where \(\displaystyle a_{i}(h)=\oint _{\gamma _{h,i}} |\nabla H| d\ell \) and \(\displaystyle T_{i}(h)=\oint _{\gamma _{h,i}} \frac {d\ell }{|\nabla H|}\). For \(\widetilde x=(h,i)\in E_{i}\), set \(a(\widetilde x)=a_{i}(h)\) and \(T(\widetilde x)=T_{i}(h)\).
In the following lemma we recall asymptotics given in chapter 8 in [9].
Lemma A.4
Let v ∈ V and i ∈ I(v). Then v = (hv,i) and as h → hv, with (h,i) ∈ Ei,
-
(1)
If d(v) = 1,
$$a_{i}(h)\sim a_{i,v} |h-h_{v}| \quad\text{ and }\quad T_{i}(h)\sim t_{i,v}$$ -
(2)
If d(v) ≥ 2,
$$a_{i}(h)\sim a_{i,v} \quad\text{ and }\quad T_{i}(h)\sim t_{i,v} |\log|h-h_{v}||$$
where ai,v and ti,v are two finite positive constants.
Proof
By definition of v, either γ(v) contains exactly one extremum of H and d(v) = 1 or γ(v) contains a saddle point of H and d(v) ≥ 3 (d(v)≠ 2 since the stationary points of H are non-degenerate).
Suppose first that d(v) = 1. Then γ(v) = {x∗}, with x∗ a local extremum of H. If H is quadratic at x∗ then, the computation of \(a(\tilde x)\) and \(T(\tilde x)\) can easily be done explicitly and gives the stated asymptotics. In the general case, one can use an asymptotic expansion or the Morse Lemma to conclude.
Suppose now that d(v) ≥ 2. Let \(\tilde x=(h,i)\to v\). Since |∇H| is continuous and that γ(v) is compact with finite length, \(a(\tilde x)\) converges to a constant \(a_{i,v}=\oint _{\gamma _{h_{v},i}}|\nabla H|d \ell \) as \(\tilde x\to v\). The main contribution for T(h) comes when the orbit γh,i is near a stationary point x∗∈ γ(v) because otherwise |∇H| is bounded away from 0. Note that there could be several such stationary points, each is a saddle. If H is quadratic in a small neighborhood \(\mathcal {V}\) of x∗ then, the computation of an asymptotic can easily be done explicitly and gives \(\oint _{\gamma _{h,i}\cap \mathcal {V}}\frac 1{|\nabla H|}d \ell \sim C^{*}|\log |h-h_{v}||\) for some C∗ > 0 which only depends on the number of times the orbit comes near x∗ (one or two times since x∗ is not degenerated) and on the Hessian matrix of H at x∗. In the general case, one can use the Morse Lemma to obtain the same asymptotics. Since each saddle point of γ(v) gives an asymptotic term of the same order, we obtain the stated result. □
Proof Proof of Proposition 7.5
Let \(f\in \widetilde {\mathcal H}\). Let \(v\in \mathcal {V}\) with d(v) ≥ 2, and i ∈ Iv. Let Ai be an open set of ]li,ri[ such that \(\bar {A}_{i}\subset ]l_{i},r_{i}[\cup \{h_{v}\}\). Note that there is c > 0 such that Ti ≥ ai ≥ c on Ai. Thus fi restricted to Ai belongs to H2(Ai) and as a consequence fi can be extended to a continuous function on ]li,ri[∪{hv}. Set Γi,v = ∂Ωi ∩ H− 1(hv). Since f ∘ π(x) = fi ∘ H(x), we have \(f_{|{\Gamma }_{i,v}}=f_{i}(h_{v})\).
Let now j ∈ Iv with j≠i. Suppose first that Γi,v ∩Γj,v≠∅, then fi(hv) = fj(hv). If Γi,v ∩Γj,v = ∅, then there is \((i_{0},i_{1},\dots ,i_{\ell })\in I_{v}^{\ell +1}\) such that i0 = i, iℓ = j and for all 1 ≤ k ≤ ℓ, \({\Gamma }_{i_{k-1},v}\cap {\Gamma }_{i_{k},v}\ne \emptyset \) and so \(f_{i}(h_{v})=f_{i_{1}}(h_{v})={\dots } =f_{i_{\ell -1}}(h_{v})=f_{j}(h_{v})\). This proves that f is continuous on \(\widetilde M\setminus \mathcal {V}_{1}\), and completes the proof of the first implication of the Proposition 7.5.
Let now \(f:M\to \mathbb {R}\) be a measurable map such that f is continuous on \(\widetilde M\setminus \mathcal {V}_{1}\), fi is weakly differentiable for all i ∈ I and \(\|f\|_{\widetilde {\mathcal H}}<\infty \). Then, for all i, \(f\circ \pi \in H^{1}({\Omega }_{i})\) and f ∘ π is continuous on M ∖Γ1, where Γ1 is the set of local extrema of H. Since Γ1 is a finite set, \(f\circ \pi \in H^{1}(\mathbb {R}^{2})\). This proves that \(f\in \widetilde {\mathcal H}\). □
Proof Proof of Proposition 7.6
To complete this proof, it remains to check that Assumption 6.9-(ii) is satisfied, i.e. that \(C_{c}(\widetilde M)\cap \widetilde {{\mathscr{H}}}\) is dense in \(\widetilde { {\mathscr{H}}}\).
Let i and j in I be such that d(Ei,Ej) = 0. Denote by vi,j the unique vertex in V such that vi,j ∈ ∂Ei ∩ ∂Ej. Set Ei,j := Ei ∪ Ej ∪{vi,j} and \(f_{i,j}=f_{|E_{i,j}}\). Let Ai,j be an open subset of Ei,j such that \(v_{i,j}\in A_{i,j}\Subset E_{i,j}\). Then (since there is c > 0 such that \(T(\widetilde x)\ge a(\widetilde x)\ge c\) in a neighborhood of v), fi,j ∈ H1(Ai,j ∩ Ei) and ui,j ∈ H1(Ai,j ∩ Ej). Since fi,j is continuous, we have that fi,j is weakly differentiable.
For each i, let ki :]li,ri[→]0,Li[ be defined such that ki(li) = 0 and \(dk_{i}(h)=\sqrt {\frac {T_{i}(h)}{a_{i}(h)}}dh\). Then ki is properly defined (in particular \(\displaystyle {\int \limits }_{l_{i}+} \sqrt {\frac {T_{i}(h)}{a_{i}(h)}}dh <\infty \) and \(\displaystyle L_{i}={\int \limits }_{l_{i}}^{r_{i}} \sqrt {\frac {T_{i}(h)}{a_{i}(h)}}dh <\infty \) only for the unique i for which Ωi is unbounded). Define a new metric graph with edges \({E^{G}_{i}}:=]0,L_{i}[\times \{i\}\) with the same adjacency rules as for \(\widetilde M\). Denote this new metric graph by G. By abuse of notation, The set of vertices of G will also be denoted by V.
For v ∈ V, let I(v) be the set of all i such that \({E^{G}_{i}}\) is an edge adjacent to v. For i ∈ I(v), set \({O^{i}_{v}}:= \{x \in {E^{G}_{i}}: d(x,v)<\frac {2L_{i}}{3}\}\) and \(O_{v}:=\{v\}\cup \cup _{i\in I(v)}{O^{i}_{v}}\). When d(v) = 1 set \({O^{0}_{v}}=O_{v}\setminus \{v\}\) and when d(v) ≥ 2, set \({O^{0}_{v}}=O_{v}\). Then \({O^{0}_{v}}\) is on open subset of Bm,1 with m = d(v) = |I(v)|.
Let G0 be the metric graph obtained out of G by taking out of G the vertices of degree 1, and denote by V0 the set of vertices of G0. Then it is easy to check that \(({O^{0}_{v}})_{v\in V^{0}}\) is a covering of G0 and that there are functions φv, v ∈ V0, such that φv : G0 → [0, 1], φv = 0 on \(G\setminus {O^{0}_{v}}\), φv restricted to \({O^{0}_{v}}\) is C1, with bounded derivatives, for each \(i\in \{1,\dots , d(v)\}\) and such that \({\sum }_{v\in V^{0}}\varphi _{v}=1\).
For \(f\in \widetilde {{\mathscr{H}}}\), let \(g:G^{0}\to \mathbb {R}\) be defined by g(v) = f(v) if v ∈ V0 and such that g(ki(h),i) = f(h,i) if (h,i) ∈ Ei.
Let i≠j such that \(d({E^{G}_{i}},{E^{G}_{j}})=0\) and set \(E^{G}_{i,j}:={E^{G}_{i}}\cup {E^{G}_{j}}\cup \{x_{i,j}\}\), we then have that \(g_{i,j}:=g_{|E^{G}_{i,j}}\) is weakly differentiable. This holds because fi,j is weakly differentiable and gi,j = fi,j ∘ hi,j, where \(h_{i,j}:E^{G}_{i,j}\to E_{i,j}\) is the continuous function, differentiable on \({E^{G}_{i}}\) and on \({E^{G}_{j}}\), defined by \(h_{i,j}(k,i)=(k_{i}^{-1}(k),i)\) if \((k,i)\in {E^{G}_{i}}\), \(h_{i,j}(k,j)=(k_{j}^{-1}(k),j)\) if \((k,j)\in {E^{G}_{j}}\) and hi,j(xi,j) = xi,j.
Then we have that \(f\in \widetilde {{\mathscr{H}}}\) if and only if g defined above belongs to W(G0,ω) with ω a measurable function on G0 such that if \((k,i)\in {E^{G}_{i}}\), \(\omega (k,i)=\omega _{i}(k)=\sqrt {a_{i}T_{i}}\circ k^{-1}_{i}(k)\).
Lemma A.5
For \(\widetilde x\in G^{0}\) and v ∈ V. We have as \(\widetilde x\to v\),
-
(1)
If d(v) = 1, \(\omega (\widetilde x)\sim c_{v} d(\widetilde x,v)\)
-
(2)
If d(v) ≥ 2, \(\omega (\widetilde x)\sim c_{v}\sqrt {|\log (d(\widetilde x,v))|} \to \infty \)
where cv > 0.
Proof
The case d(v) = 1 is straightforward. For the case d(v) ≥ 2, we let \(\widetilde x=(k,i)\in G^{0}\) and assume v = (0,i), then \(k=d(\widetilde x,v)\) and \(h:=k^{-1}_{i}(k)\) with \(k_{i}^{\prime }(h)=\sqrt {\frac {T_{i}(h)}{a_{i}(h)}}\sim C_{v}\sqrt {|\log h|}\) as h → 0. An integration by part yields that \(k(h)\sim C h\sqrt {|\log h|}\) and thus \(|\log (k)|\sim |\log h|\). Then, since \(\omega (\widetilde x)=\sqrt {a_{i}T_{i}}(h)\sim C\sqrt {|\log h|}\), we get the result. The same result holds if v = (Li,i). □
To prove that \(C_{c}(\widetilde M)\cap \widetilde {{\mathscr{H}}}\) is dense in \(\widetilde {{\mathscr{H}}}\), we will now use Lemma A.3.
Fix R > 0. For v ∈ V and n ≥ 1, set Ov,R,n = Ov,R if d(v) ≥ 2 and set Ov,R,n := {x ∈ Ov,R : d(x,v) > n− 1} if d(v) = 1. Then (i) of Lemma A.3 is satisfied.
Let us now check that (ii) is satisfied. This has only to be checked for v ∈ V with d(v) = 1. Let \({E^{G}_{i}}\) be the unique edge adjacent to v. Without loss of generality we will suppose that v = (0,i). We then have that Ov,R ∖ Ov,R,n =]0,n− 1[×{i}. To prove that Cap(Ov,R ∖ Ov,R,n) → 0. In this case, fix A > 0 such that A < R and \(A<\frac {2L_{i}}{3}\) and take (for n > A− 1), gn(k) = 1 if k ≤ n− 1 and gn(k) = 0 if k > A and \(g_{n}(k)=\frac {\log (A)-\log (k)}{\log (A)-\log (n^{-1})}\) if n− 1 ≤ k ≤ A. Then \(Cap(O_{v,R}\setminus O_{v,R,n})\le {\int \limits }_{n^{-1}}^{A} \frac {\omega _{i}(k)}{k^{2}(\log (A)+\log (n))^{2}} dk + {\int \limits }_{n^{-1}}^{A} \frac {(\log (A)-\log (k))^{2}\omega _{i}(k)}{(\log (A)+\log (n))^{2}} dk\) which converges to 0 as \(n\to \infty \). This proves (ii)
To check (iii), we shall use Lemma A.1. for the open sets Ov,R,n ⊂ Bm,1, with m = d(v). For each v,R,n, we define an extension \(\bar {\omega }\) of \(\omega _{|O_{v,R,n}}\), defined on Bm,1. By a slight abuse of notation, we set identify v as 0 in the definition of Bm,1 and keep the labels of I(v) to identify the branches of Bm,1.
If m = d(v) = 1 then recall that \(B_{1,1}=\mathbb {R}\). Ov,R,n is an open interval ]l,r[ and we let \(\bar {\omega }(k)=\omega (l)\in ]0,\infty [\) if k ≤ l and \(\bar {\omega }(k)=\omega (r)\in ]0,\infty [\) if k > r. The \(\bar {\omega }\) is continuous and positive, and it thus belongs to the class A2.
If m = d(v) ≥ 2, for i ∈ I+(v), \(O_{v,R,n}\cap {E^{G}_{i}}\) is an open interval ]0,r[ (with r = (2Li/3) ∧ R) and for k > r, we set \(\bar {\omega }(k,i)=\omega (r,i)\in ]0,\infty [\) and \(\bar {\omega }(v)=\infty \). For i ∈ I−(v), \(O_{v,R,n}\cap {E^{G}_{i}}\) is an open interval ]l,Li[ (with l = Li − (2Li/3) ∧ R) and we set
and \(\bar {\omega }(v)=\infty \). This defines \(\bar {\omega }\) on Bm,1. It remains to check that for i≠j in I(v), we have that \(\bar {\omega }_{i,j}\) belongs to the class A2. Note that \(\lim _{x\to v}\bar {\omega }_{i,j}(x)=\infty \). Since \(\omega _{i,j}(x)\sim c \sqrt {\log (k)}\) as k := d(x,v) → 0, it is a simple exercise to show that \(\bar {\omega }_{i,j}\) belongs to the class A2.
Then Lemma A.3 can be applied. This proves that \(W(G^{0},\omega )={H^{1}_{0}}(G^{0},\omega )\). Let \(f\in \widetilde {{\mathscr{H}}}\) and g ∈ W(G0,ω) defined as above by \(g(k,i)=f(k_{i}^{-1}(k),i)\) if \((k,i)\in {E^{G}_{i}}\) and g(v) = f(v) if v ∈ G0. Since \(W(G^{0},\omega )={H^{1}_{0}}(G^{0},\omega )\), for all 𝜖 > 0 there is g𝜖 ∈ Cc(G0) ∩ W(G0,ω) such that \(\|g-g_{\epsilon }\|_{W(G^{0},\omega )}<\epsilon \). Set f𝜖(h,i) = g𝜖(ki(h),i) if (h,i) ∈ Ei, f𝜖(v) = g𝜖(v) if v ∈ V0 and f𝜖(v) = 0 if v ∈ V ∖ V0. Then \(f_{\epsilon }\in C_{c}(\widetilde M)\cap \widetilde {{\mathscr{H}}}\) and \(\|f-f_{\epsilon }\|_{\widetilde {{\mathscr{H}}}}=\|g-g_{\epsilon }\|_{W(G^{0},\omega )}<\epsilon \). □
1.3 Regularity for Section ??
To prove the regularity we can apply directly Lemma A.2 since our state space will already be an open subset of B4,2. Set I := {1, 2, 3, 4}. Recall that \(\widetilde {M}=\cup _{i\in I} C_{i}\) is constituted of four cones glued along one edge. Set \(\widetilde M_{0}:=\{y\in \widetilde M: y_{1}y_{2} > 0\}\).
1.3.1 The Space \(\widetilde {\mathcal H}\)
Recall that for t ∈]0, 1[
For y ∈∪i∈IC̈i, let \(t(y)=\frac {|y_{1}|\wedge |y_{2}|}{|y_{1}|\vee |y_{2}|}\). Then \(t(y)=\frac {y_{2}}{y_{1}}\) for y ∈C̈1 ∪C̈3 and \(t(y)=\frac {y_{1}}{y_{2}}\) for y ∈C̈2 ∪C̈4.
For i ∈ I, set \(\widetilde {m}_{i}\) the measure on Ci with density hi(y)e−W(∥y∥) with respect to Lebesgue measure on Ci, where
For y ∈∪i∈IC̈i, set a(y) := νy(∇π ⊗∇π). We have that \(\nu _{y}({x_{3}^{2}})=2{y_{1}^{2}}\alpha (t)\) for y ∈C̈1, then,
The eigenvalues of a(y) are 1 and λ(t), with corresponding eigenvectors (y1,y2) and (−y2,y1). If i ∈ I and y ∈C̈i, set ai(y) = a(y).
Let us now collect some asymptotics.
Lemma A.6
As t → 0+,
As t → 1−,
Proof
Asymptotics for E and K are standard, the others are simple consequences. □
Let \(f\in \widetilde {{\mathscr{H}}}\). For i ∈ I, the map \(f_{i}:=f_{|C_{i}}\) is weakly differentiable on Ci and, using the change of variable formula Eq. ??, we have that
Set \({\Omega }:=]0,+\infty [\times B_{4,1}\subset B_{4,2}\), Ωi := Ω ∩ Ei for i ∈ I and Ωi,j := Ω ∩ Ei,j for i≠j ∈ I. Let us remark from Lemma A.6 that \(\displaystyle \mathcal {I}:={{\int \limits }_{0}^{1}} \frac {du}{(1+u^{2})\sqrt {\lambda (u)}} <+\infty \). Set \(c=\frac {\pi }{2\mathcal {I}}\) and define \(\varphi :[0,1]\to [0,\frac {\pi }2]\) by
Define \(z:\cup _{i=1}^{4} \mathring {C}_{i} \to ]0,+\infty [^{2}\) by
Note that for all i ∈ I, \(z_{i}= z_{|\mathring {C}_{i}}\) is one to one.
For \(g:{\Omega }\to \mathbb {R}\) define \(\mathcal F(g):\widetilde {M}_{0}\to \mathbb {R}\) by \(\mathcal F(g)(y)=g(z(y),i)\) if y ∈C̈i and \(\mathcal {F}(g)(y)=g(y,0)\) if y ∈ D∗. For \(f:\widetilde {M}\to \mathbb {R}\) (or for \(f:\widetilde {M}_{0}\to \mathbb {R}\)), define \(\mathcal G(f):{\Omega }\to \mathbb {R}\) such that if (z,i) ∈Ωi, then \(\mathcal {G}(f)(z,i)=f_{i}\circ z_{i}^{-1}(z)\) and if z > 0, then \(\mathcal G(f)(z,0)=f(z)\). Then, \(\mathcal G\circ \mathcal F(g)=g\) and we have that g ∈ Cc(Ω) if and only if \(\mathcal {F}(g)\in C_{c}(\widetilde {M}_{0})\).
Define \(k:[0,\frac {\pi }{2}[\to \mathbb {R}^{+}\cup \{\infty \}\) by \(k(0)=\infty \) and, for \(\phi \in ]0,\frac {\pi }{2}[\) and t = φ− 1(ϕ),
For \(z=r({\cos \limits } \phi ,{\sin \limits } \phi )\) with \((r,\phi )\in ]0,\infty [\times [0,\frac {\pi }2[\), set ω(z,i) := re−W(r)k(ϕ) and
where Rϕ is the matrix associated to the rotation of angle ϕ,
Let \(f\in \widetilde {\mathcal H}\) and \(g=\mathcal G(f)\). The change of variable z = zi(y) on each Ci yields
Recall that the norm on W1(Ω,ω) is given by
We thus get that
Lemma A.7
\(f\in \widetilde {{\mathscr{H}}}\) if and only if \(\mathcal G(f)\in W^{1}({\Omega },\omega )\).
Proof
In this proof, we fix \(f:\widetilde M\to \mathbb {R}\) and we set \(g:=\mathcal G(f)\). If \(f \in \widetilde H\), then for each i, fi is weakly differentiable on C̈i and \(g_{|{\Omega }_{i}}\) is weakly differentiable on Ωi. And Eq. A.17 show that \(g_{i}:=g_{|{\Omega }_{i}}\in W^{1}({\Omega }_{i},\omega _{i})\), where \(\omega _{i}=\omega _{|{\Omega }_{i}}\). Moreover, if \(\mathcal {O}\) be a bounded open set in Ωi,j, then \(g_{|\mathcal {O}_{i}}\in H^{1}(\mathcal {O}_{i})\), \(g_{|\mathcal {O}_{j}}\in H^{1}(\mathcal {O}_{j})\) and coincide on \(\mathcal {O}\cap E_{0}\). Since ω is bounded from below on \(\mathcal {O}\), this implies that \(g_{|\mathcal {O}}\in H^{1}(\mathcal {O})\supset W^{1}(\mathcal {O},\omega _{i,j})\). We thus have that \(g_{|{\Omega }_{i,j}}\in W^{1}({\Omega }_{i,j},\omega _{i,j})\), which proves that g ∈ W1(Ω,ω).
On the converse, if g ∈ W1(Ω,ω), then using that \(f\circ \pi =\mathcal {F}(g)\circ \pi \) and Eq. A.17, we have f ∘ π ∈ H1(m) and so \(f\in \widetilde H\). □
Lemma A.8
\({H^{1}_{0}}({\Omega },\omega )=W^{1}({\Omega },\omega )\)
We use Lemma A.2 to prove this lemma.
This lemma shows the density of \(C_{c}(\widetilde M)\cap \widetilde {{\mathscr{H}}}\) in \(\widetilde {{\mathscr{H}}}\). Indeed, \(\mathcal {F}:\widetilde H\to W^{1}({\Omega },\omega )\) and \(\mathcal {G}: W^{1}({\Omega },\omega )\) are continuous map**s and if g ∈ Cc(Ω) ∩ W1(Ω,ω) then \(\mathcal {F}(g)\in C_{c}(\widetilde M)\cap \widetilde H\). Therefore, if \(f\in \widetilde H\) and \(g=\mathcal {G}(f)\), there is a sequence gn ∈ Cc(Ω) ∩ W1(Ω,ω) approximating g in W1(Ω,ω) and so \(f_{n}:=\mathcal {F}(g_{n})\) is a sequence in \(C_{c}(\widetilde M)\cap \widetilde H\) approximating f in \(\widetilde H\).
Proof Proof of Lemma A.8
For R > 0, recall that ΩR = {z ∈Ω : ∥z∥ < R} and define, for n ≥ 1, \({\Omega }_{R,n}:=\{(r{\cos \limits } \phi , r{\sin \limits } \phi ,i) : (r,\phi ,i)\in ]0,R[\times [0,\frac \pi 2-\frac 1n[\times I\}\).
In order to apply Lemma A.2, we have to check that (i), (ii) and (iii) are satisfied. Item (i) is clearly satisfied.
To prove (ii), we construct a sequence of non negative functions (gn) on Ω, such that gn(z) = 1 in ΩR,n ∖ΩR and \(\lim _{n\to +\infty }\|g_{n}\|^{2}_{W^{1}({\Omega },\omega )}=0\). Let us first obtain asymptotics for ω.
Lemma A.9
We have, as \(\phi \to \frac {\pi }{2}^{-}\), \( k(\phi )\sim \frac {3\pi }{8c^{2}}(\frac {\pi }{2}-\phi ). \)
Proof
Using Lemma A.6 and Eq. A.12, we have that \(\frac {\pi }{2}-\varphi (t)\sim \frac {2c}{\sqrt {3}}t\) as t → 0+. Thus, since ϕ = φ(t), we have \(t\sim \frac {\sqrt {3}}{2c}(\frac {\pi }{2}-\phi )\) as \(\phi \to \frac {\pi }{2}^{-}\). From Lemma A.6 and Eq. A.14, we obtain \(k(\phi )\sim \frac {\pi \sqrt {3}}{4c}t\sim \frac {3\pi }{8c^{2}}(\frac {\pi }{2}-\phi )\) as \(\phi \to \frac {\pi }{2}^{-}\). □
Using this Lemma, we can fix \(0<\phi _{0}<\frac \pi 2\) and c0 > 0 such that \(0\leq k(\phi )\leq c_{0}(\frac {\pi }{2}-\phi )\) for all \(\phi \in [\phi _{0},\frac {\pi }{2}]\). Then we define \(f_{1}:]0,+\infty [\to \mathbb {R}\), \(f_{2,n}:[0,\frac {\pi }2[\to \mathbb {R}\) by
For i ∈ I and \(z=r(\cos \limits (\phi ),\sin \limits (\phi ))\in ]0,+\infty [^{2}\), set gn(z,i) = f1(r)f2,n(ϕ). We then have
Since \(|\nabla g_{n}(z)|^{2}=(f_{1}^{\prime })^{2}(r)f^{2}_{2,n}(\phi )+\frac 1{r^{2}}{f^{2}_{1}}(r)(f^{\prime }_{2,n})^{2}(\phi )\), we have
The integrals involving f1 are finite and does not depend on n. For f2,n, we have
Thus, \(\lim _{n\to +\infty }{\int \limits }_{\phi _{0}}^{\frac {\pi }{2}}f^{2}_{2,n}(\phi )k(\phi )d\phi =0\). We also have
Thus, \(\lim _{n\to +\infty } {\int \limits }_{\phi _{0}}^{\frac {\pi }{2}}(f^{\prime }_{2,n}(\phi ))^{2}k(\phi )d\phi =0\) and then \(\lim _{n\to +\infty }\|g_{n}\|^{2}_{W^{1}({\Omega },\omega )}=0\). Item (ii) is proven.
To prove (iii), we apply Lemma A.1. Let us first define an extension \(\bar {\omega }_{n,R}\) to B4,2 of ω which coincide with ω on ΩR,n. We define \(k_{n}:[-\pi ,\pi ]\to \mathbb {R}^{+}\cup \{\infty \}\) by
and define the weight \(\bar {\omega }_{n,R}:B_{4,2}\to \mathbb {R}^{+}\) by \(\bar {\omega }_{n,R}(z,i):=re^{-W(r\wedge R)}k_{n}(\phi )\) where \(z=(r\cos \limits \phi , r\sin \limits \phi )\) with \((r,\phi ,i)\in \mathbb {R}^{+}\times [-\pi ,\pi ]\times I\). Then, \(\bar {\omega }_{n,R}\) coincide with ω on ΩR,n. Note that for i≠j, the restriction of \(\bar {\omega }_{n,R}\) to Ei,j (identified with \(\mathbb {R}^{2}\)) does not depend on i and j. Denote this common measure ωn,R and set \(\omega _{n}:\mathbb {R}^{2}\to \mathbb {R}^{+}\cup \{+\infty \}\), defined by ωn(z) = rkn(ϕ) where \(z=r(\cos \limits \phi ,\sin \limits \phi )\in \mathbb {R}^{2}\). Since, \(0<\inf _{r>0} e^{-W(r\wedge R)} \le \sup _{r>0}e^{W(r\wedge R)}\), ωn,R belongs to the class A2 if and only if ωn belongs to the class A2.
Define for ρ > 0, \(z\in \mathbb {R}^{2}\),
where B(z,ρ) is the ball at center z and radius ρ.
Let us first obtain asymptotics for ωn.
Lemma A.10
We have, as ϕ → 0 +, \( k(\phi )\sim \frac 1{2c}\left |\log \left (\phi \right )\right |^{1/2}. \)
Proof
For ϕ > 0 close to 0 and t = φ− 1(ϕ), set δ := 1 − t. We have
As u → 0+, using Lemma A.6, we get \(\sqrt {\lambda (1-u)}\sim \frac {\sqrt {2}}{\sqrt {|\ln (u)|}}\), and thus as ϕ → 0+, by integration by parts
This entails that \(\log (\delta )\sim \log (\phi ).\) It is straightforward, using Lemma A.6 again, that \(K(t)\sqrt {\lambda (t)}\sim \frac 1{\sqrt {2}}\sqrt {|\log (\delta )|}\). Then we obtain that \(k(\phi )\sim \frac 1{2c}\sqrt {|\log (\delta )|} \sim \frac 1{2c}\sqrt {|\log (\phi )|}\). □
Let us now prove the condition A2 for ωn.
Lemma A.11
ωn satisfies the A2-Muckenhoupt condition: \(\sup _{z,\rho }C(z,\rho )<+\infty \).
Proof Proof of Lemma A.11
Note that kn is a continuous positive function on [−π,π] ∖{0}. Lemma A.10 shows that \(k_{n}(\phi )\sim \frac 1{2c}\sqrt {\left |\log |\phi |\right |}(1+o(1))\) as ϕ → 0. Thus the integrals \({\int \limits }_{-\pi }^{\pi }k_{n}(\phi )d\phi \) and \({\int \limits }_{-\pi }^{\pi }k_{n}(\phi )^{-1}d\phi \) are finite and ωn and \(\omega _{n}^{-1}\) are locally integrable on \(\mathbb {R}^{2}\). Therefore (z,ρ)↦C(z,ρ) is a continuous function. Note also that for λ > 0 and \(z\in \mathbb {R}^{2}\), ωn(λz) = λωn(z), then C(λz,λρ) = C(z,ρ). This leaves two cases to investigate: z = 0, ρ = 1 or |z| = 1, ρ > 0.
For z = 0, ρ = 1, we have
Obviously, \(C(0,\rho )=C(0,1)=C_{n}<+\infty \) does not depend on ρ.
For the second case, fix ρ > 0 and \(z=(\cos \limits \phi ,\sin \limits \phi )\), with ϕ ∈ [−π,π]. If \(\rho \geq \frac {1}{2}\), since B(z,ρ) ⊂ B(0,ρ + 1), we get \(C(z,\rho )\leq \frac {(\rho +1)^{4}}{\rho ^{4}}C_{n}\le 3^{4} C_{n}\). If \(\rho <\frac 12\), we set \(\theta =\arcsin (\rho )\in ]0,\frac {\pi }{6}[\). We have \(B(z,\rho )\subset T(z,\rho ):=\{z=r(\cos \limits \psi ,\sin \limits \psi ); (r,\psi )\in [1-\rho ,1+\rho ]\times [\phi -\theta ,\phi +\theta ]\}\). We obtain
Then we get (setting \(c_{0}=\frac {13}{3\pi ^{2}}\))
The function (ϕ,𝜃)↦D(ϕ,𝜃) is continuous on \([-\pi ,\pi ]\times ]0,\frac \pi 6]\). Let us now show that D is bounded in a neighborhood of 𝜃 = 0 +. Note that
since \(\frac {\theta ^{2}}{\sin \limits (\theta )^{2}}<2\) for all \(\theta \in [0,\frac {\pi }6]\).
Fix \(\theta _{0}\in ]0,\frac {\pi }{6}[\). We have \(\sup _{(\phi ,\theta )\in [-\pi ,\pi ]\times [\theta _{0},\frac \pi 6]} D(\phi ,\theta )<\infty \). From Lemma A.10, we get that there are 0 < c1 < C1 and 0 < c2 < C2 such that
Let 𝜃 < 𝜃0. Without loss of generality, we assume that ϕ ∈ [0,π]. If ϕ − 𝜃 ≥ 𝜃0, we get [ϕ − 𝜃,ϕ + 𝜃] ⊂ [𝜃0,π + 𝜃0] and we get, using Eq. A.37, \(D(\phi ,\theta )\leq \frac {4C_{1}}{c_{1}}\).
If ϕ − 𝜃 < 𝜃0, then [ϕ − 𝜃,ϕ + 𝜃] ⊂ [−𝜃0, 3𝜃0], and we consider again two subcases. First subcase: ϕ ≥ 2𝜃, let u := ϕ + 𝜃 ≥ 3𝜃,
In the second subcase we have ϕ < 2𝜃. We start back from Eq. A.35:
C2 is a continuous function on ]0,𝜃0] and as 𝜃 → 0+, we get (using Lemma A.10
Thus \(\lim _{\theta \to 0^{+}} C_{2}(\theta )=18.\) Finally, this gives us the result. □
With Lemma A.11, conditions of Lemma A.1 are satisfied and condition (iii) of Lemma A.2 holds. Therefore, \(W^{1}({\Omega },\omega )={H^{1}_{0}}({\Omega },\omega )\). □
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Barret, F., Raimond, O. An Averaging Principle for Stochastic Flows and Convergence of Non-Symmetric Dirichlet Forms. Potential Anal 56, 483–548 (2022). https://doi.org/10.1007/s11118-020-09893-x
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DOI: https://doi.org/10.1007/s11118-020-09893-x