non-linearities fulfil certain Lipschitz conditions. Since in this case, existence of solutions has not been established yet, we first have to extend the main existence result of the aforementioned Lp-theory to this class of equations. This will be done in Theorem 5.13. Afterwards, we can apply Theorem 5.1 and obtain spatial regularity in the scale (∗) of Besov spaces, see Theorem 5.15.
The examples and remarks presented in Section 5.1 have been partially worked out in col-laboration with F. Lindner, S. Dahlke, S. Kinzel, T. Raasch, K. Ritter, and R.L. Schilling [25].
integer γ + 2∈ N are considered. Secondly, we obtain Lp-integrability in time of the Bτ,τα (O)-valued process for arbitrary p≥2 fulfilling the assumptions (i) or (ii) from Theorem 3.13. With the techniques used in [25] just Lτ-integrability in time can be established. Thirdly, we do not need the extra assumption u∈Lp([0, T]×Ω;Bp,ps (O)) for some s >0. Due to Corollary 4.2, it suffices thatu∈Hγ+2p,θ−p(O, T).
Next, we give some examples of applications of Theorem 5.2 and interpret our result from the point of view of the question whether adaptivity pays, cf. our motivation for studying topic (T1) from Section 1.1. We are mainly interested in the Hilbert space case p = 2 since, as already pointed out in Section 1.1, it provides a natural setting for numerical discretization techniques like adaptive wavelet methods, see also the expositions in [105, 125] for more details. We begin with an application of Theorem 5.2 for particular parametersγ, θ ∈Rand p= 2.
Example 5.4. Assume that we have given coefficientsaij,bi,c,σik, andµk, withi, j∈ {1, . . . , d}
andk∈N, fulfilling Assumption 3.1 withγ = 0. Furthermore, fix arbitraryf ∈H02,d+2(O, T),g∈ H12,d(O, T;`2) and u0 ∈U2,d2 =L2(Ω,F0,P;H2,d1 (O)). Then, by an application of Theorem 3.13 withγ = 0,p= 2 andθ=d, Eq. (3.1) has a unique solutionu∈H22,d(O, T). Due to Theorem 5.2,
u∈L2(ΩT;Bτ,τα (O)), 1 τ = α
d +1
2, for all 0< α < d
d−1. (5.2) In the two-dimensional case, this means that
u∈L2(ΩT;Bτ,τα (O)), 1 τ = α
2 +1
2, for all 0< α <2.
Note that if we assume slightly more regularity on the coefficients, the initial condition u0 and the free terms f and g, we can included the border case α = d/(d−1) in (5.2). To this end, assume that the coefficients aij,bi, c, σik, and µk, with i, j ∈ {1, . . . , d} and k∈ N, fulfil Assumption 3.1 for some arbitrary positiveγ >0. Furthermore, fix an arbitraryε >0 and assume thatf ∈Hγ2,d−ε+2(O, T),g∈Hγ+12,d−ε(O, T;`2) andu0 ∈U2,d−εγ+2 =L2(Ω,F0,P;H2,d−εγ+2 (O)). Then, there exists anε1 ∈(0, κ1) with κ1 >0 from Theorem 3.13(ii), such that Eq. (3.1) has a unique solution u∈Hγ+22,d−ε
1(O, T). Due to Theorem 5.2, u∈L2(ΩT;Bτ,τα (O)), 1
τ = α d +1
2, for all 0< α <min
γ+ 2,
1 +ε1 2
d d−1
, and therefore, sinceγ and ε1 are strictly positive,
u∈L2(ΩT;Bτ,τα (O)), 1 τ = α
d +1
2, for all 0< α≤ d d−1, which in the two-dimensional case yields
u∈L2(ΩT;Bτ,τα (O)), 1 τ = α
d +1
2, for all 0< α≤2.
The example above shows that equations of the type (3.1) on general bounded Lipschitz domains have spatial Besov regularity in the scale (∗) up to order α = 2. In order to answer the question whether this is enough for justifying the development of spatially adaptive wavelet methods, we have to compare this result with the spatial Sobolev regularity of the solution under consideration. We give now a concrete example of an SPDE of the type (3.1) with solution u ∈H22,d(O, T) whose spatial Besov regularity in the scale (∗) is strictly higher than its spatial Sobolev regularity.
V1 V8
V7
V5
V4
V3
V2
β1 β2
β3
β4
β5 β6 β7
β8
V6
Figure 5.1: Polygon in R2 withβmax=β6 = 5π/4.
Example 5.5.We consider an equation of the type (3.1) on a polygonal domain O ⊂R2 and show that, under natural conditions on the data of the equation, if the underlying domain is not convex, the spatial Besov regularity of the solution in the scale (∗) is strictly higher than its spatial Sobolev smoothness. In particular, this shows that, generically, solutions to linear SPDEs on bounded Lipschitz domains behave as described in (1.11), so that the use of spatially adaptive methods is recommended. For more details on the link between regularity theory and the convergence rates of numerical methods we refer to Section 1.1.
Let O ⊂R2 be a simply connected bounded domain in R2 with a polygonal boundary ∂O such that O lies on one side of ∂O. It can be described by a finite set {Vn :n = 1, . . . , N} of vertices of the boundary numbered, e.g., according to their order in ∂O in counter-clockwise orientation. Forn∈ {1, . . . , N}, we writeβn∈(0,2π) for the interior angle at the vertexVnand denote by βmax the maximal interior angle ofO, i.e.,
βmax:= max
βn:n= 1, . . . , N .
An example of such a domain with βmax = 5π/4 is shown in Figure 5.1. Assume that we have an initial condition u0 ∈U2,22 (O) additionally satisfying
u0 ∈L2(Ω,F0,P; ˚W21(O))∩Lq(Ω,F0,P;L2(O))
for some q > 2. Furthermore, let f ∈ L2(ΩT;L2(O)) ,→ H02,4(O, T) and let g ∈ H2,21 (O;`2).
Typically, we make slight abuse of notation and write galso for the constant stochastic process g∈H12,d(O, T;`2) withg(ω, t) :=gfor all (ω, t)∈ΩT. Then, due to Theorem 3.13, the stochastic heat equation
du= ∆u+f
dt+gkdwtk on ΩT × O, u(0) =u0 on Ω× O,
)
(5.3) has a unique solution u∈H22,2(O, T).
We want to compare the spatial Besov regularity of the solution to Eq. (5.3) in the scale (∗) with its spatial Sobolev regularity. Regarding the spatial regularity in the non-linear approxi-mation scale (∗), an application of Theorem 5.2 yields
u∈L2(ΩT;Bτ,τα (O)), 1 τ = α
2 +1
2 = α+ 1
2 , for all 0< α <2. (5.4)
Concerning the spatial Sobolev regularity of the solution, by our analysis so far, we can only guarantee that
u∈L2(ΩT; ˚W21(O)),
which is a consequence of Proposition 4.1. Together with (5.4), this suggests that the Besov regularity of the solution to Eq. (5.3) in the scale (∗) is generically higher than its spatial Sobolev regularity in the following sense: There exist polygonal domainsO ⊂R2 and free terms f andg fulfilling the assumptions from above, such that
˜
sSobmax(u)<2, (5.5)
with ˜sSobmax(u) as introduced in (1.12). We can confirm this statement by exploiting the recent results from [92]. Therefore, let us denote by ∆D2,w : D(∆D2,w) ⊆ L2(O) → L2(O) the weak Dirichlet-Laplacian on L2(O), i.e.,
D(∆D2,w) :=
u∈W˚21(O) : ∆u∈L2(O) ,
∆D2,wu:= ∆u, u∈D(∆D2,w).
From Proposition 3.18 we already know that our solutionu∈H22,2(O, T) is also the unique weak solution (in the sense of Da Prato and Zabczyk [32]) of the L2(O)-valued ordinary SDE
du(t)−∆D2,wu(t) dt=f(t) dt+ dWQ(t), t∈[0, T], u(0) =u0,
)
(5.6) driven by the H2,21 (O)-valued Q-Wiener process (WQ(t))t∈[0,T] := P
k∈Ngkwkt
t∈[0,T] with co-variance operatorQ:=P
k∈Nhgk,·iH1
2,2(O)gk ∈ L1(H2,21 (O)). Moreover, due to Theorem 3.8(ii), sup
t∈[0,T]
E h
ku(t)k2L
2(O)
i
≤E h
sup
t∈[0,T]
ku(t)k2L
2(O)
i
≤Ckuk2H2
2,2(O,T)<∞, and by [32, Theorem 5.4], for all t∈[0, T],
u(t) =S2(t)u0+ Z t
0
S2(t−s)f(s) ds+ Z t
0
S2(t−s) dWQ(s) P-a.s., where
S2(t) t≥0 denotes the contraction semigroup on L2(O) generated by (∆D2,w, D(∆D2,w)).
Thus, u is the unique (up to modifications) mild solution of Eq. (5.6) which is studied in [92], see also [91, Chapter 4]. Therein, techniques from [57, 58] have been adapted to the stochastic setting, and it has been shown that this solution can be divided into a spatially regular and a spatially irregular part, regularity being measured by means of Sobolev spaces. In particular, if we assume that the range of the covariance operator Qis dense in H2,21 (O),→L2(O), it follows from [92, Example 3.6] that
u /∈L2(ΩT;W2s(O)) for any s >1 + π βmax
. (5.7)
Thus, ifO is not convex, we have
˜
sSobmax(u)≤1 + π βmax
<2,
with ˜sSubmax(u) as defined in (1.12). Together with (5.4), this shows that the solution to Eq. (5.3) generically behaves as described in (1.11). Therefore, the development of suitable spatially adap-tive numerical methods is completely justified.
r
1 q
3/2
1
1 2
W21(O) 2
3 2
B22/3,2/3(O)
1 +π/βmax= 9/5
1
Figure 5.2: Spatial Besov regularity in the scale Bτ,τα (O), 1/τ = (α+ 1)/2, versus spatial Sobolev regularity of the solution of Eq. (5.3), illustrated in a DeVore/Triebel diagram.
Figure 5.2 shows a DeVore/Triebel diagram illustrating the situation described above (see Remark 2.63 for details on the visualisation of Besov spaces using this type of diagrams). The fact that (5.4) holds, is represented by the solid segment {(1/τ, α) : 1/τ = (α + 1)/2,0 ≤ α < 2} of the L2(O)-non-linear approximation line and the annulus at (3/2,2), which stands for the Besov space B2/3,2/32 (O). The point at (1/2,1) shows that u ∈L2(ΩT;W21(O)). In this situation, by Theorem 2.61 and standard interpolation results, see, e.g. [117, Corollary 1.111], u ∈ L2(ΩT;Bq,qr (O)) for all (1/q, r) in the interior of the polygon with vertices at the points (1/2,0), (1/2,1), (3/2,2), (2,2), and (1,0). This is indicated by the shaded area. The border at (1/2,3/2) illustrates the following consequence of (5.7): For anyε >0, there exists a polygonal domain O ⊂ R2, such that u /∈ L2(ΩT;W23/2+ε(O)). The concrete border for the example in Figure 5.1 is indicated by the annulus at (1/2,1 +π/βmax) = (1/2,9/5), which stands for the Sobolev space W29/5(O).
In the following example we are concerned with equations of the form (3.1) driven by a specific type of noise.
Example 5.6. We consider an equation of the type (3.1) driven by a time-dependent version of the stochastic wavelet expansion introduced in [1] in the context of Bayesian non-parametric regression and generalized in [14] and [24]. This noise model is formulated in terms of a wavelet basis expansion on the domainO ⊂Rd with random coefficients of prescribed sparsity and thus tailor-made for applying adaptive techniques with regard to the numerical approximation of the corresponding SPDEs. Via the choice of certain parameters specifying the distributions of the wavelet coefficients it also allows for an explicit control of the spatial Besov regularity of the noise. We first describe the general noise model and then deduce a further example for the application of Theorem 5.2.
Let {ψλ : λ ∈ ∇} be a multiscale Riesz basis of L2(O) consisting of scaling functions at a fixed scale level j0 ∈ Z and of wavelets at level j0 and all finer levels. We follow [27] and use the same notation as in Section 1.1. Information like scale level, spatial location and type of the wavelets or scaling functions are encoded in the indices λ ∈ ∇. In particular, we write
∇=S
j≥j0−1∇j, where forj≥j0 the set∇j ⊂ ∇contains the indices of all waveletsψλ at scale level j and where ∇j0−1 ⊂ ∇ is the index set referring to the scaling functions at scale level j0
which we denote by ψλ,λ∈ ∇j0−1, for the sake of notational simplicity;|λ|:=j for all λ∈ ∇j. We make the following assumptions concerning our basis. Firstly, the cardinalities of the index sets∇j,j ≥j0−1, satisfy
C−12jd ≤ |∇j| ≤C2jd, j ≥j0−1, (5.8) with a constantC which does not depend onj. Secondly, we assume that the basis admits norm equivalences similar to those described in Theorem 4.4. That is, there exists ans0 ∈N(depending on the smoothness of the scaling functions ψλ, λ ∈ ∇j0−1, and on the degree of polynomial exactness of their linear span), such that, given p, q > 0, max{0, d(1/p−1)} < s < s0, and a real valued distribution v∈ D0(O), we have v∈Bp,qs (O) if, and only if, v can be represented as v=P
λ∈∇cλψλ, (cλ)λ∈∇ ⊂R(convergence inD0(O)), such that
∞
X
j=j0−1
2jq s+d 12−1p X
λ∈∇j
|cλ|ppq
!1q
<∞. (5.9)
Furthermore,kvkBs
p,q(O) is equivalent to the (quasi-)norm (5.9). Concrete constructions of bases satisfying these assumptions can be found e.g. in [42–44] or [19, 20], see also [27, Section 2.12 together with Section 3.9] for a detailed discussion. Concerning the family of independent stan-dard Brownian motions (wkt)t∈[0,T],k∈N, in (3.1), we modify our notation and write (wtλ)t∈[0,T], λ∈ ∇, instead. The description of the noise model involves parametersa1 ≥0 ,a2∈[0,1],b∈R, with a1 +a2 > 1. For every j ≥j0 −1 we set ςj := (j−(j0−2))bd22−a1(j−(j20−1))d and let Yλ, λ ∈ ∇j, be {0,1}-valued Bernoulli distributed random variables on (Ω,F0,P) with parameter pj = 2−a2(j−(j0−1))d, such that the random variables and processes Yλ, (wλt)t∈[0,T], λ∈ ∇, are stochastically independent. The noise in our equation will be described by the L2(O)-valued stochastic process (Mt)t∈[0,T]defined by
Mt:=
∞
X
j=j0−1
X
λ∈∇j
ςjYλψλ·wtλ, t∈[0, T]. (5.10)
Using (5.9), (5.8) and a1 + a2 > 1, it is easy to check that the infinite sum converges in L2(ΩT;L2(O)) as well as in L2(Ω;C([0, T];L2(O))). Moreover, by the choice of the parame-ters a1, a2 and b one has an explicit control of the convergence of the infinite sum in (5.10) in the (quasi-)Banach spacesLp2(ΩT;Bps1,q(O)),s < s0,p1, q >0,p2 ≤q. (Compare [24] which can easily be adapted to our setting.)
For simplicity, let us consider the two-dimensional case, i.e., d = 2. Assume that we have a given f ∈ H02,2(O, T), an initial condition u0 ∈ U2,22 (O), and coefficients aij, bi and c, with i, j∈ {1, . . . , d}, fulfilling Assumption 3.1 with σ= 0 and µ= 0. We consider the equation
du= aijuxixj+biuxi+cu+f
dt+ς|λ|Yλψλdwλt on ΩT × O, u(0) =u0 on Ω× O,
)
(5.11) where we sum over all λ ∈ ∇ instead of k ∈ N. That is, we understand this equation similar to equations of the type (3.1), where the role of k ∈ N in the required definitions is taken by λ∈ ∇. In this setting, letg:= (gλ)λ∈∇:= (ς|λ|Yλψλ)λ∈∇. Sincea1+a2>1 and kgk
H02,2(O,T;`2)= p2/TkMkL2(ΩT;L2(O)) we have g ∈ H02,2(O, T;`2). Let us impose a bit more smoothness on g
and assume that a1+a2>2. This is sufficient to ensure that g∈H12,2(O, T;`2), since kgk2
H12,2(O,T;`2) =E Z T
0
kgk2H1
2,2(O;`2)dt
=EhZ T 0
X
λ∈∇
kgλ(t,·)k2H1
2,2(O)dti
=TE h X∞
j=j0−1
X
λ∈∇j
ςj2Yλ2kψλk2H1 2,2(O)
i ,
so that by (2.28), kgk2
H12,2(O,T;`2)≤C
∞
X
j=j0−1
X
λ∈∇j
ςj2pj
X
|α|≤1
kρ|α|Dαψλk2L
2(O)
≤C
∞
X
j=j0−1
X
λ∈∇j
ςj2pjkψλk2W1 2(O).
Since W21(O) = B2,21 (O), see Theorem 2.60(ii), we can use the equivalence (5.9) with v = ψλ followed by (5.8) with d= 2 to obtain
kgk2
H12,2(O,T;`2)≤C
∞
X
j=j0−1
X
λ∈∇j
ςj2pj22j
=C
∞
X
j=j0−1
|∇j|(j−(j0−2))2b2−2a1(j−(j0−1))2−2a2(j−(j0−1))22j
≤C
∞
X
j=j0−1
(j−(j0−2))2b2−2j(a1+a2−2). Thus g∈H12,2(O, T;`2) and for any ϕ∈ C0∞(O),
X
λ∈∇
Z · 0
(gλ, ϕ) dwtλ= (M·, ϕ) P-a.s.
in C([0, T];R), see also Proposition 3.6 and the definition of stochastic integrals from Subsec-tion 2.2.3 for details. As in the examples above, by Theorem 3.13, there exists a unique soluSubsec-tion of Eq. (5.11) in the class H22,2(O, T). As shown in Examples 5.5, in general, the solution pro-cess is not in L2(ΩT;W2s(O)) for all s < 2, but, by Theorem 5.2, it belongs to every space L2(ΩT;Bτ,τα (O)) withα <2 andτ = 2/(α+ 1).
We make the following note regarding adaptive versus uniform methods in Sobolev spaces.
Remark 5.7.As already mentioned in the introduction, in different deterministic settings, there exist adaptive wavelet-based schemes realising the convergence rate of the best m-term approximation error in the energy norm. This norm is determined by the equation and is usually equivalent to anL2(O)-Sobolev norm and not to theL2(O) norm itself. Thus, the question arises whether our regularity results underpin the use of adaptivity also in the case that the error is measured in a suitable Sobolev norm. Again this question can be decided after a rigorous regularity analysis of the target function, since the results on the link between regularity theory and the convergence rate of approximation methods discussed in Section 1.1 can be generalised to the case where the error is measured in a Sobolev spacesW2r(O) withr >0 instead ofLp(O).
r
1 q
1 +π/βmax
W2sSobmax(O) 1
1 2
W21(O)
3/2
2
3
1 2
Bατ∗∗,τ∗(O) B2/3,2/32 (O)
1 q 7→ 2q
Figure 5.3: Spatial Besov regularity in the scaleBτ,τα (O), 1/τ =α/2, versus spatial Sobolev regularity of the solution to Eq. (5.3),
illustrated in a DeVore/Triebel diagram.
Let us denote by {ηλ :λ∈ ∇} a wavelet basis of W2r(O) for somer > 0. Such a basis can be obtained by rescaling a wavelet basis {ψλ :λ∈ ∇} of L2(O) as the one used in Example 5.6 and by using the norm equivalence (5.9), see, e.g., [27] or [41]. For the error of the bestm-term wavelet approximation error in this Sobolev norm, it is well-known that
u∈Bτ,τα (O), 1
τ = α−r d +1
2 implies σm,Wr
2(O)(u)≤C m−(α−r)/d, (5.12) where
σm,Wr
2(O)(u) := inf n
ku−umkWr
2(O):um ∈Σem,Wr
2(O)
o
with
Σem,Wr
2(O):=
X
λ∈Λ
cληλ : Λ⊂ ∇, Λ
=m, cλ ∈R, λ∈Λ
,
see, e.g., [125, Corollary 3.2] and the references therein, in particular, [27]. Therefore, similar to the L2(O)-setting, on the one hand, the decay rate of the best m-term wavelet approximation error inE =W2r(O) depends on the Besov regularity of the target function. On the other hand, the convergence rate of uniform numerical methods is determined by the Sobolev regularity of the solution to be approximated. It is well-known that, under fairly natural conditions, if um, m∈N, is a uniform approximation scheme of u, then,
ku−umkWr
2(O)≤C m−(s−r)/dkukWs
2(O), m∈N;
see, e.g., [37], [46] or [61] for details. If we consider uniform wavelet approximation, the following converse assertion also holds: Ifu /∈W2s(O), then the convergence rate of the uniform method in W2r(O) is limited by (s−r)/d, see, e.g., [125, Proposition 3.2] and the references therein. This means that, if the error is measured in W2r(O), adaptivity pays if the spatial smoothness of the solution in the Besov spaces from (5.12) is strictly higher than its spatial Sobolev regularity.
Let us consider the setting from Example 5.5 and discuss the relationship between the spa-tial Sobolev and Besov regularity in view of approximation in W21(O), i.e., r = 1. We use a DeVore/Triebel diagram to visualise our explanations, see Figure 5.3. Due to (5.7),
˜
sSobmax(u)≤1 + π βmax
, (5.13)
with ˜sSobmax(u) as defined in (1.12). Thus, supn
(s−1)/2 :u∈L2(ΩT;W2s(O))o
≤ π 2βmax
. (5.14)
Let us assume that the spatial Sobolev regularity of the solution u reaches its maximum, i.e., that u ∈ L2(ΩT;W21+π/βmax(O)), cf. (5.7). Then, due to (5.4), by Theorem 2.61 and standard interpolation results, see, e.g. [117, Corollary 1.111],
u∈L2(ΩT;Bατ,τ(O)), 1
τ = α−1 2 +1
2 = α
2 for all 0< α < α∗ := βmax+ 3π βmax+π . This is illustrated in Figure 5.3 by the solid segment of the line 1/q 7→2/qdelimited by the origin and the annulus at (1/τ∗, α∗) = (α∗/2, α∗). Thus, the decay rate of the best m-term wavelet approximation error in W21(O) with respect to the space coordinates goes up to π/(βmax+π), which is greater than π/(2βmax) whenever βmax> π, i.e., whenever the polygonal domainO is not convex. Therefore, also in this setting, the implementation of adaptive wavelet methods is justified.
In all the other examples from above we consider general bounded Lipschitz domains. In this case, we do not have an explicit bound for the spatial Sobolev regularity of the solution. Thus, we can only assume the limit case βmax = 2π. Inserting this into the calculations from above, we can say that, in the worst case,
sup
(s−1)/2 :u∈L2(ΩT;W2s(O)) ≤ 1
4. (5.15)
Simultaneously,
u∈L2(ΩT;Bτ,τα (O)), 1
τ = α−1 2 +1
2 = α
2 for all 0< α < 5 3.
Since (5/3−1)/2 = 2/6 >1/4, the development of optimal adaptive algorithms with respect to the space coordinates, where the error is measured inW21(O), is recommended. We illustrate this limiting case in Figure 5.4 by using again a DeVore/Triebel diagram.
We conclude this section with an example showing that, in contrast to what is known to hold for deterministic equations, adaptive wavelet methods for SPDEs may pay even if the underlying domain is smooth.
Example 5.8. Let O be a bounded Cu1-domain (and, therefore, a bounded Lipschitz domain) in Rd. Furthermore, letaij,bi,c, andµk, withi, j ∈ {1, . . . , d} andk∈N, be given coefficients satisfying Assumption 3.1 withγ = 0,σ = 0 and suitable constantsδ0andK. Fixp∈[2,∞) and let f ∈ H1p,d−1+p(O, T), g ∈ H2p,d−1(O, T;`2) and u0 ∈ Up,d−13 (O). Then, by [72, Theorem 2.9]
there exists a unique solution u of Eq. (3.1), which is in the class H3p,d−ε(O, T) for any ε > 0;
see also Remark 3.14(ii). Due to Theorem 5.1 this yields u∈Lp(ΩT;Bτ,τα (O)), 1
τ = α d +1
p, for all 0< α <
1 +ε
p d
d−1, ε∈(0,1).
r
1 q
1
1 2
W21(O)
3/2
W23/2(O) 2
3
1 2
5/3
6 5
B5/6,5/65/3 (O) B2/3,2/32 (O)
1 q 7→ 2q
Figure 5.4: Spatial Besov regularity in the scaleBτ,τα (O), 1/τ =α/2, versus spatial Sobolev regularity of the solution to equations of type (3.1), illustrated in a DeVore/Triebel diagram.
Thus, in the two-dimensional case, if p= 2, we have u∈Lp(ΩT;Bτ,τα (O)), 1
τ = α+ 1
2 , for all 0< α <3.
What about the spatial Sobolev regularity of this solution? It is known from [78, Example 1.2]
that if we considerR+ instead ofO, there exists a non-random g, continuously differentiable on [0,∞)×[0,∞) such that the second partial derivatives with respect to the space coordinates of the solution to the heat equation
du= ∆udt+gkdwkt, u
∂R+ = 0, u(0) = 0,
on R+, do not lie in L2(ΩT;L2(R+)). This is due to the incompatibility of the noise with the zero Dirichlet boundary condition. Exploiting the compatibility results from [55], it is reasonable to expect that we can construct similar examples on smooth bounded domains, with maximal spatial Sobolev regularity strictly less than the spatial Besov regularity in the non-linear ap-proximation scale (∗). If this is indeed the case, it shows that in the stochastic setting, adaptive methods are a serious alternative to uniform methods even if the underlying domain is smooth.
It is worth noting that this would be completely different from what is known to hold in the deterministic setting, where adaptivity does not pay on smooth domains.