Existence and space-time regularity for stochastic heat equations on p.c.f. fractals

We define linear stochastic heat equations (SHE) on p.c.f.s.s. sets equipped with regular harmonic structures. We show that if the spectral dimension of the set is less than two, then function-valued"random-field"solutions to these SPDEs exist and are jointly H\"older continuous in space and time. We calculate the respective H\"older exponents, which extend the well-known results on the H\"older exponents of the solution to SHE on the unit interval. This shows that the"curse of dimensionality"of the SHE on $\mathbb{R}^n$ depends not on the geometric dimension of the ambient space but on the analytic properties of the operator through the spectral dimension. To prove these results we establish generic continuity theorems for stochastic processes indexed by these p.c.f.s.s. sets that are analogous to Kolmogorov's continuity theorem. We also investigate the long-time behaviour of the solutions to the fractal SHEs.


Introduction
The stochastic heat equation (or SHE) on R n , n ∈ N is a stochastic partial differential equation which can be expressed formally as ∂u ∂t (t, x) = Lu(t, x) +Ẇ (t, x), u(0, ·) = u 0 for (t, x) ∈ [0, ∞) × R n , where L is the Laplacian on R n , u 0 is a (sufficiently regular) function on R n andẆ is a space-time white noise on R × R n . Written in the differential notation of stochastic calculus this is equivalent to du(t) = Lu(t)dt + dW (t), where W is a cylindrical Wiener process on L 2 (R n ). A solution to this SPDE is a process u = (u(t) : t ∈ [0, T ]) taking values in some space containing L 2 (R n ) that satisfies the above equations in some weak sense; see [4] for details. The SHE on R n is one of the prototypical examples of an SPDE and has been widely studied, see for example [5], [8] and [21]. It has two notable properties that are relevant to the present paper. The first is its so-called "curse of dimensionality". Solutions to the SHE on R n are function-valued only in the case n = 1; in dimension n ≥ 2 solutions are forced to take values in a wider space of distributions on R n , see [21]. Secondly if n = 1 and u 0 = 0 then the solution is unique and jointly Hölder continuous in space and time, see again [21]. One of the aims of the present paper is to investigate what happens regarding these two properties in the setting of finitely ramified fractals, which behave in many ways like spaces with dimension between one and two.
The family of spaces that we will be considering is the class of connected postcritically finite self-similar (or p.c.f.s.s.) sets endowed with regular harmonic structures. This family includes many well-known fractals such as the Sierpinski gasket and the Vicsek fractal but not the Sierpinski carpet. The unit interval [0, 1] also has several formulations in the language of p.c.f.s.s. sets that belong to this family. Analysis on these sets is a relatively young field which started with the construction of a "Brownian motion" on the Sierpinski gasket in [9], [17] and [3]. This broader theory was then developed and provides a concrete framework where reasonably explicit results can be obtained, see [15] and [2]. Associated with a regular harmonic structure on a p.c.f.s.s. set (F, (ψ i ) M i=1 ) is an operator, called the Laplacian on F , which is the generator of a "Brownian motion" on F by analogy with the Laplacian on R n as the generator of Brownian motion in R n . We will see that there exists a constant d s > 0 associated with the harmonic structure known as the spectral dimension, and it will turn out that the assumption that the harmonic structure is regular implies that d s ∈ [1,2). The existence of a Laplacian allows us to define certain PDEs and SPDEs on F , such as a heat equation and a stochastic heat equation. The former has been studied extensively, see [15,Chapter 5] and further references. The latter is the subject of the present paper.
For examples of some previous work in this area, in [6] it is shown that on certain fractals a stochastic heat equation can be defined which yields a random-field solution, that is, a solution which is a random map [0, T ] × F → R. We extend this result in the main theorem of Section 4 of the present paper. In [14] (see also [12]) it is shown that solutions to some nonlinear stochastic heat equations on more general metric measure spaces have Hölder continuous paths when considered as a random map from a "time" set to some space of functions. However in that paper the authors do not consider the Hölder exponents of the solution when considered as a random field, which is what we will do.
The structure of the present paper is as follows: In the following subsection we describe the precise set-up of the problem and the specific SPDE that we will be studying, and state a theorem which is an important corollary of our main result. In Section 2 we recall some useful spectral theory for Laplacians on p.c.f.s.s. sets from [15] and show that (unique) solutions to the SPDE exist as L 2 (F )-valued stochastic processes.
In Section 3 we prove generic results analogous to Kolmogorov's continuity theorem for families of random variables indexed by F and by [0, 1] × F . In Section 4 we show that the resolvent densities associated with the Laplacian are Lipschitz continuous with respect to the resistance metric on F . More importantly we also show that evaluations of solutions to the SPDE at points (t, x) ∈ [0, ∞) × F can be done in a well-defined way, which is necessary for us to talk about continuity of these solutions. Section 5 contains the main results of the paper, which use our continuity theorems to establish space-time Hölder continuity of solutions to the SPDE and compute the respective Hölder exponents. Section 6 serves as a "coda" of the paper, where we prove results on the invariant measures and long-time behaviour of the solutions to the SPDE.

Description of the problem
Let M ≥ 2 be an integer. Let S = (F, (ψ i ) M i=1 ) be a connected p.c.f.s.s. set (see [15]) such that F is a compact metric space and the ψ i : F → F are injective strict contractions on F . Let I = {1, . . . , M } and for each n ≥ 0 let W n = I n . Let W * = n≥0 W n and let W = I N . We call the sets W n , W * and W word spaces and we call their elements words. Note that W 0 is a singleton containing an element known as the empty word. Words w ∈ W n or w ∈ W will be written in the form w = w 1 w 2 w 3 . . . with w i ∈ I for each i. For a word w = w 1 , . . . , w n ∈ W * , let ψ w = ψ w1 • · · · • ψ wn and let F w = ψ w (F ).
If W is endowed with the standard product topology then there is a canonical continuous surjection π : W → F given in [2,Lemma 5.10]. Let P ⊂ W be the postcritical set of S (see [15,Definition 1.3.4]), which is finite by assumption. Then let F 0 = π(P ), and for each n ≥ 1 let F n = w∈Wn ψ w (F 0 ). Let F * = ∞ n=0 F n . It is easily shown that (F n ) n≥0 is an increasing sequence of finite subsets and that F * is dense in F .
Let the pair (A 0 , r) be a regular irreducible harmonic structure on S such that r = (r 1 , . . . , r M ) ∈ R M for some constants r i > 0, i ∈ I (harmonic structures are defined in [15,Section 3.1]). Here regular means that r i ∈ (0, 1) for all i. Let r min = min i∈I r i and r max = max i∈I r i . If n ≥ 0, w = w 1 , . . . w n ∈ W * then write r w : Then let µ be the self-similar Borel probability measure on F such that for any n ≥ 0, if w ∈ W n then µ(F w ) = r d H w . In other words, µ is the self-similar measure on F in the sense of [15,Section 1.4] associated with the weights r d H i on I. Let (E, D) be the regular local Dirichlet form on L 2 (F, µ) associated with this harmonic structure, as given by [15,Theorem 3.4.6]. This Dirichlet form is associated with a resistance metric R on F , defined by which generates the original topology on F , by [15,Theorem 3.3.4]. Additionally, let Then by [15,Corollary 3.4.7], (E, D 0 ) is a regular local Dirichlet form on L 2 (F \ F 0 , µ). By [2,Chapter 4], associated with the Dirichlet form (E, D) on L 2 (F, µ) is a µsymmetric diffusion X N = (X N t ) t≥0 which itself is associated with a C 0 -semigroup of contractions S N = (S N t ) t≥0 . Let L N be the generator of this diffusion. Likewise associated with (E, D 0 ) we have a µ-symmetric diffusion X D with C 0 -semigroup of contractions S D and generator L D . The process X D is similar to X N , except for the fact that it is absorbed at the points F 0 , whereas X N is reflected. The letters N and D indicate Then all the conditions given above are satisfied. We have D = H 1 [0, 1] and E(f, g) = 1 0 f g . The associated generators L N and L D are respectively the standard Neumann and Dirichlet Laplacians on [0, 1]. In particular, the induced resistance metric R is none other than the standard Euclidean metric. This interpretation of the unit interval as a p.c.f.s.s. set that fits into our set-up will be useful to us later on.
The object of study in the present paper is the following SPDE on F : for all s, t ∈ [0, ∞) and f, g ∈ L 2 (F, µ). Note that W is not an L 2 (F, µ)-valued process; to be precise, it takes values in some separable Hilbert space in which L 2 (F, µ) can be continuously embedded (see [4]). The vast majority of results in this paper hold regardless of the value of b; whenever this is not the case it will be explicitly stated.
The SPDE (1.1) in the case α = 0 will be called the stochastic heat equation or SHE for (A 0 , r) on F . It is well known (see for example [21]) that the solution to the standard SHE on [0, 1] with initial condition u 0 = 0 is jointly continuous with Hölder exponents of essentially 1 2 in space and essentially 1 4 in time (the meaning of "essentially" is given in Definition 2.10). The following extension of this result is a simple consequence of our main result Theorems 5.6 and 5.7 and was the original motivation for the writing of the present paper: Theorem 1.2. Equip F with the resistance metric R. Then for each b ∈ {N, D}, the SHE for (A 0 , r) on F with u 0 = 0 has a unique solution u = (u(t, x)) (t,x)∈[0,∞)×F which is jointly continuous, essentially 1 2 -Hölder continuous in space (i.e. in (F, R)) and essentially is the spectral dimension of (F, R).
Note that many p.c.f.s.s. sets F can be embedded into Euclidean space in such a way that R is equivalent to the Euclidean metric up to some exponent. Therefore, for such sets, we can also make sense of the above result with respect to a spatial Euclidean metric, see Remark 5.8.
, and r i = n+1 n+3 for all i ∈ I. In fact for n = 1 we have the binary decomposition of the unit interval and recover the usual case. For n = 2 the diffusion X N is known as Brownian motion on the Sierpinski gasket and is ubiquitous in the field of analysis on fractals ( [9], [17], [3]). We can compute d H = log(n+1) log(n+3)−log(n+1) and d s = 2 log(n+1) log(n+3) . This gives us a family of examples which live naturally in R n for any geometric dimension n and where the spectral dimension can be made arbitrarily close to 2 by taking n large. Using the properties of the resistance metric we can have solutions that have arbitrarily small spatial (with respect to the Euclidean metric) and temporal Hölder exponents. See Remark 5.8 for further discussion.

Existence (and uniqueness)
Definition 2.1. Henceforth we let H = L 2 (F, µ). Denote the inner product on H by ·, · µ . Let T > 0. Following [4], an H-valued predictable process u = (u(t) : Notice that for any f ∈ H, u is a solution to (1.1) with u 0 = 0 if and only if u + S b f is a solution to (1.1) with u 0 = f . Thus we can safely assume that u 0 = 0, and so we are interested in the properties of the stochastic convolution Observe that if a solution exists for u 0 = 0, then it must equal W b α up to versions.
The first thing to investigate is the validity of the operator . S b can be extended to an analytic semigroup (which we will identify with S b ), 2 is known as a Bessel potential, see [14], [20].

Spectral theory of Laplacians
Now that we have established the close relationship between L b and E, we may make use of the spectral theory of these Laplacians developed in [15,Chapters 4 and 5]. We summarise the useful definitions and results below: so that i∈I r i ≥ 1 = i∈I r d H i and thus d H ≥ 1. It follows that d s ∈ [1, 2). lim k→∞ λ b k = ∞. We assume that they are given in ascending order: There exist constants c 1 , c 2 , c 3 > 0 such that if k ≥ 2 then Proof. This is a simple corollary of results in [15,Chapters 4,5], in particular Theorem 4.5.4 and Lemma 5.1.3.

Remark 2.8.
Note that all functions f ∈ D must be at least 1 2 -Hölder continuous with respect to the resistance metric since for all x, y ∈ F (see [2,Proposition 7.18]). Thus it makes sense to consider ϕ b k (x) for x ∈ F . The above proposition then implies that |ϕ b k (x)| ≤ c 3 |λ b k | ds 4 for all x ∈ F , k ≥ 2.
Remark 2.9. The reason why we require k ≥ 2 in the above proposition is that we may have λ b 1 = 0. In this case it follows that E(ϕ b 1 , ϕ b 1 ) = 0. By the properties of the resistance metric R, for any distinct x 1 , x 2 ∈ F we have that It follows that ϕ b 1 is constant. Since ϕ b 1 µ = 1 and µ is a probability measure we conclude that ϕ b 1 ≡ 1. This confirms that if 0 is an eigenvalue it must necessarily have multiplicity 1, so we always have λ b 2 > 0. It also implies that we have λ b 1 = 0 if and only if b = N , since the non-zero constant functions are elements of D \ D 0 . In the case that λ b 1 > 0, we will assume that c 1 , c 2 , c 3 are chosen such that the estimates in the above proposition hold for k ≥ 1.
The existence of a complete orthonormal basis of eigenfunctions of L b allows us to write down series representations of elements of H and operators defined on subspaces of H in a way analogous to the Fourier series representations of elements of L 2 (0, 1). For example, an element f ∈ H has a series representation and the domain of Ξ(−L b ) is exactly those f ∈ H for which the above expression is in H.
In particular k for all f ∈ H.

Existence of solution
Recall the expression (2.1). If we can show that W b α (t) ∈ H almost surely for every t > 0, then we have a unique global solution of (1.1) for u 0 = 0, and thus by the discussion after Definition 2.1 we have a unique global solution for any initial value u 0 ∈ H. In fact we can do better than that: In particular for any α ≥ 0, b ∈ {N, D} and any initial condition u 0 ∈ H there exists a unique (up to versions) global solution to (1.1). There exists an H-continuous version of this solution. Moreover if u 0 = 0 then this version is essentially 1 Hölder continuous on compact intervals.
Proof. We refer to the proof of [10,Theorem 5.13]. By Itō's isometry for Hilbert spaces we have that for some constant C > 0, and the last expression is square-integrable on the interval [0, T ]. Therefore finding such a β is sufficient for W b α (t) to be square-integrable. We see from Proposition 2.7 that ds and the final expression is finite for β > ds 4 . Since we know that d s < 2 we can pick any β ∈ ( ds The continuity statements then directly follow from [10, Theorems 5.10 and 5.17].

Some Kolmogorov-type continuity theorems
It is well-known that solutions to the one-dimensional stochastic heat equation are essentially 1 4 -Hölder continuous in time and essentially 1 2 -Hölder continuous in space, so we would like to prove analogous results for our SPDE. It will become clear that the natural "spatial" metric to use on F is the resistance metric R.
The usual method of proving continuity of processes indexed by R is to use Kolmogorov's continuity theorem. Our aim in this section is to prove versions of this theorem for the spaces F and [0, 1] × F .

Partitions and neighbourhoods
We introduce some more theory and notation from [15] and develop it further for our purposes.
Definition 3.1. If n ≥ 1 and w = w 1 . . . w n ∈ W n then let If n = 0 and w ∈ W 0 then w is the empty word and we set Σ w := W.
Stochastic heat equations on p.c.f. fractals which is a partition, see [15,Definition 1.5.6]. Notice that if w ∈ Λ(a) then r min a < r w ≤ a.
For n ≥ 1 let Λ n = Λ(2 −n ). Let Λ 0 be the singleton containing the empty word; this is also a partition.
In particular v is not the empty word, so w is not the empty word (since n 1 ≥ n 2 ), so it follows that m 2 ≥ 2 and m 1 ≥ 1. But then n 1 , n 2 ≥ 1 so Proof. Let n ≥ 0 and x ∈ F n . Recall the canonical continuous surjection π : W → F and the post-critical set P . By assumption x ∈ F n = w∈Wn ψ w (F 0 ) = w∈Wn ψ w (π(P )), so there exists w ∈ W n and v ∈ Σ w such that v n+1 v n+2 . . . ∈ P and π(v) = x. By the definition of P it follows that for all integer i ≥ 0 we must have that v n+i+1 v n+i+2 . . . ∈ P . Now consider the sequence w i : Definition 3.7. For n ≥ 0 and x, y ∈ F n Λ let x ∼ n y if there exists w ∈ Λ n such that x, y ∈ F w . Then (F n Λ , ∼ n ) can be interpreted as a graph.
Proof. By the refinement property (Lemma 3.4) we have that Σ v ⊂ Σ w and so there exist There exists a constant c g > 0 such that if n ≥ 0 and w ∈ Λ n , then ) is a connected graph and its graph diameter is at most c g .
Consider by the refinement property (Lemma 3.4) that we must have By [15,Lemma 4.2.3] it must be the case that the quantities |{w ∈ Λ n : F w x}| and |{w ∈ Λ n : F w ∩ D 0 n (x) = ∅}| are bounded over all n ≥ 0 and all x ∈ F . Let In particular, observe that D 0 n (x) ⊆ D 1 n (x), and that if x, y ∈ F n Λ with x ∼ n y then y ∈ D 0 n (x). Definition 3.11. For x ∈ F and ε > 0 let B(x, ε) be the closed ball in (F, R) with centre x and radius ε.
The next result shows that the resistance metric R is topologically well-behaved with respect to the structure of the p.c.f.s.s. set F and the partitions Λ n . Compare similar results obtained in [11,Lemmas 3.2,3.4].
For the first inclusion, let D h ⊆ D be the set of harmonic functions (see [15, We now take g to be the harmonic extension to H of the indicator function 1 D 0 n (x) | F n Λ : F n Λ → R. Then by self-similarity, if w ∈ Λ n then the function g • ψ w on F agrees exactly with an element of D h . Evidently g(x) = 1, and if y / ∈ D 1 n (x) then g(y) = 0. Therefore it follows by the definition of the resistance metric and the comment in Definition 3.3 that where c 4 is defined in (3.1), and this completes the proof.
The next result gives bounds on the growth of the cardinality of the sets Λ n in terms of the Hausdorff dimension d H .
Proof. For n ≥ 0 and v ∈ Λ n , by the definition of the measure µ we have that Then summing over all v ∈ Λ n gives

The continuity theorems
Theorem 3.14 (First continuity theorem). Let (E, ∆) be a complete separable metric space. Let ξ = (ξ x ) x∈F be an E-valued process indexed by F and let C, β, γ > 0 such that for all x, y ∈ F . Then there exists a version of ξ which is almost surely essentially γ β -Hölder continuous with respect to R.
Proof. The set F is uncountable, but F * = ∞ n=0 F n Λ is countable and dense in F . We may therefore consider the countable set (ξ x ) x∈F * without issues of measurability. Let δ ∈ (0, γ β ) and define the measurable event We then define the random variablesξ x for x ∈ F bŷ for some arbitrary fixed x 0 ∈ E. Thenξ := (ξ x ) x∈F is measurable and essentially γ β -Hölder continuous. If P[Ω δ ] = 1 for all δ ∈ Q ∩ (0, γ β ) thenξ is also a version of ξ. This is becauseξ x is then the almost-sure limit of (ξ y ) y∈F * as y → x, so applying Fatou's lemma to the estimate in the statement of this theorem shows thatξ x = ξ x almost surely. It therefore suffices to show that P[Ω δ ] = 1 for all δ ∈ (0, γ β ). We define the random variable to be the Hölder norm of ξ restricted to F * , and we observe that Ω δ = {H δ < ∞} by the completeness of E. For n ≥ 0 we also define the random variables By Proposition 3.13, Then using the Markov inequality and Proposition 3.12, for some constant C > 0. Now δβ < γ so ∞ n=0 P K n > 2 −nδ < ∞, so by the Borel-Cantelli lemma we have that lim sup n→∞ (2 nδ K n ) ≤ 1 almost surely.
In particular there exists an almost surely finite postive random variable J such that K n ≤ 2 −nδ J for all n ≥ 0 almost surely. Now recall the constant c g from Lemma 3.9. Let x, y ∈ F * be distinct points, and let m 0 be the greatest integer such that y ∈ D 1 m0 (x) (which exists by Proposition 3.12). Then there exists w, v ∈ Λ m0 such that x ∈ F w , y ∈ F v and there exists some z ∈ F w ∩ F v . In fact by [15,Proposition 1.3.5] and the definition of a partition, we can choose z to be in ψ w Otherwise, there exists m > m 0 such that x ∈ F m Λ and we construct a finite sequence This can be done in the following way: assume that we already have We then have by Lemma 3.9 that We can make the same estimate for y and z. Therefore we conclude that Now m 0 was chosen such that y / ∈ D 1 m0+1 (x), so we use Proposition 3.12 to conclude that R(x, y) > c 5 2 −(m0+1) . Thus we find that for all s, t ∈ [0, 1] and all x, y ∈ F . Then there exists a version of ξ which is almost surely essentially γ∧γ β -Hölder continuous with respect to G = ([0, 1] × F, R ∞ ).
Proof. This proof proceeds in much the same way as in Theorem 3.14, so we only give an outline.
then G * is countable and dense in G. Then for each n ≥ 0 we define a relation * n on G n by (s, x) * n (t, y) if and only if either (|s − t| = 2 −n and x = y) or (s = t and x ∼ n y). Notice that this implies that if (s, x) * n (t, y) then R ∞ ((s, x), (t, y)) ≤ (c 6 ∨ 1)2 −n , by Proposition 3.12. Then as before we can define K n = sup p,q∈G n p * nq ∆(ξ p , ξ q ).
Since both [0, 1] and (F, R) are bounded, for δ ∈ (0, γ∧γ β ) this satisfies So as in Theorem 3.14, there exists an almost surely finite positive random variable J such that K n ≤ 2 −nδ J for all n ≥ 0 almost surely. The sets analogous to D 0 n (x) and D 1 n (x) in Theorem 3.14 are given bŷ where s − = max{k2 −n : k ∈ Z, k2 −n < s}, s − = min{k2 −n : k ∈ Z, k2 −n > s}. Using Proposition 3.12 it is simple to verify the analogous result that for all n ≥ 0 and all p ∈ G, where B ∞ denotes the closed R ∞ -balls of G. Now if (s, x), (t, y) ∈ G * are distinct points, let m 0 be the greatest integer such that (t, y) ∈ D 1 m0 (s, x). Then there exists w, v ∈ Λ m0 and τ 1 , and there exists some In fact just as in the proof of Theorem 3.14 we may pick (τ, z) such that We can then estimate ∆(ξ sx , ξ ty ) by constructing suitable finite sequences of points from (s, x) to (τ, z) and from (t, y) to (τ, z), similar to the proof of Theorem 3.14.
The rest of the details of the proof are left up to the reader; it suffices to adapt the proof of Theorem 3.14, using for example c g + 2 instead of c g .
We now extend the previous result to this section's main theorem, which includes spatial and temporal Hölder exponents.
(3). For every x ∈ F the map t →ξ tx is almost surely essentially δ 2 -Hölder continuous with respect to the Euclidean metric where Then by Theorem 3.14 there exists a version (ξ x ) x∈F of (ξ tx ) x∈F which is almost surely essentially δ 1 -Hölder continuous with respect to R. Now using (1), (ξ x ) x∈F and (ξ tx ) x∈F are both almost surely continuous on the separable space F (see for example F * ⊆ F ) so we must in fact have that (ξ x ) x∈F = (ξ tx ) x∈F almost surely. We conclude that (ξ tx ) x∈F is almost surely essentially δ 1 -Hölder continuous with respect to R. The proof of (3) is conceptually identical -we use the standard Kolmogorov continuity theorem for

Pointwise regularity
Before we talk about Hölder continuity of the solution u to (1.1) we show that the point evaluations u(t, x) for (t, x) ∈ [0, ∞) × F are indeed well-defined random variables.
Recall from Proposition 2.7 and the subsequent discussion that so it follows from Itō's isometry for H-valued stochastic integrals that For each k ≥ 1 define the real-valued stochastic process X b,k = (X b,k t ) t≥0 by so we have the series representation Evidently X b,k is a centred real continuous Gaussian process. We compute its covariance to be if λ b k > 0 and we identify X b,k to be a centred Ornstein-Uhlenbeck process with unit volatility and rate parameter λ b k . If λ b k = 0 then X b,k is simply a standard Wiener process. It is easy to check that the family (X b,k ) ∞ k=1 is independent. Remark 4.1. We give an alternative view on the series representation (4.2). Let u be the solution to (1.1) in the case u 0 = 0, so that u = W b α . We take an eigenfunction expansion of (1.1): for each k ∈ N, whereû(·, k) := ϕ b k , u(·) µ is a real-valued process. This is analogous to using Fourier methods to solve differential equations on R n . Now using standard theory we see that {ϕ b * k W } ∞ k=1 is a family of independent real-valued standard Wiener processes. It follows that (4.3) is just a family of decoupled one-dimensional SDEs, and the solution to the kth SDE can be found to be exactly (

Resolvent density
By a density argument it follows that for all f ∈ H, (3). ρ N λ is non-negative (easy to see from (2) and fact that , symmetric and bounded. We define (for now) c 7 (λ) > 0 such that Using symmetry this Hölder continuity result holds in the first argument as well.
By an identical argument to [2,Theorem 7.20], ρ D λ exists and satisfies the analogous results with (E, D 0 ) and S D . By the reproducing kernel property it follows that for every x ∈ F , ρ D λ (x, ·) must be the D λ -orthogonal projection of ρ N λ (x, ·) onto D 0 . We now choose c 7 (λ) large enough that it does not depend on the value of b ∈ {N, D} for (3) and (4).
The Hölder continuity property of the resolvent densities ρ b λ described above is the subject of this section. We seek to strengthen it into Lipschitz continuity. Definition 4.4. Let B ⊆ F be closed. Let g B : F × F → R be the Green function on F with boundary B, see [16,Chapter 4].
The properties of the Green function that we require are given in [16,Theorem 4.1]. Note in particular that every Green function is symmetric and uniformly Lipschitz in (F, R); this is the main tool of our proof.
Fix λ > 0. We prove the result for b = D first. Let g D = g F 0 , the Green function associated with Dirichlet boundary conditions. Then for x, y ∈ F , ≤ 2R(y, y ).
Doing the same estimate with y, y interchanged gives the required result. Now for the case b = N , fix x 0 ∈ F an arbitrary point. We see that , g {x0} (y, ·) µ , and the rest of the proof is identical to the b = D case.
As before, by the symmetry of ρ b λ the above Lipschitz continuity property in fact holds in both of its arguments.

Pointwise regularity of solution
We return to the SPDE (1.1). The next lemma is based on an argument in [6, Section 7.2].
Proof. Let g * ∈ H * be the bounded linear functional f → f, g µ . We see by Itō's isometry that E u(t), g 2 µ = E g * (u(t)) 2 where the last equality is a result of the self-adjointness of the operator We know from the functional calculus for self-adjoint operators that is a well-defined real-valued centred Gaussian random variable. There exists a constant c 8 > 0 such that for all x ∈ F , t ∈ [0, ∞) and n ≥ 0 we have that Proof. Note that ϕ b k ∈ D(L b ) for each k, so ϕ b k is continuous and so ϕ b k (x) is well-defined. By the definition of u(t, x) as a sum of real-valued centred Gaussian random variables we need only prove that it is square-integrable and that the approximation estimate holds. Let x ∈ F . The theorem is trivial for t = 0 so let t ∈ (0, ∞). By Lemma 4.6 we have that  Writing u in its series representation (4.2) and using the independence of the X b,k and the fact that It follows that the left-hand side tends to zero as m, n → ∞. By Theorem 2.11 we know for all x ∈ F , n ≥ 0 and t ∈ [0, ∞), therefore by the completeness of the sequence space 2 there must exist a unique sequence (y k ) ∞ . Thus by Fatou's lemma we can identify the sequence (y k ). We must have Equivalently by (4.2), E u(t, x) 2 < ∞ (so we have proven square-integrability) and lim n→∞ E ( u(t), f x n µ − u(t, x)) 2 = 0.
By virtue of the previous theorem it is possible to interpret solutions u to the SPDE (1.1) as random maps u : [0, ∞) × F → R, where as usual we have suppressed the dependence of u on the underlying probability space. It therefore makes sense to consider issues of continuity of u on [0, ∞) × F .

Hölder regularity
The aim of this section is to use our continuity theorems of Section 3 to prove Hölder regularity results for a version of the family defined in Theorem 4.8, and then show that this version can be identified with the original solution to (1.1). We wish to use Theorem 3.17 and Corollary 3.19, so we need estimates on the expected spatial and temporal increments of the solution.

Spatial estimate
Proposition 5.1. Let T > 0. Let u = (u(t, x)) (t,x)∈[0,∞)×F be the family defined in Theorem 4.8. Then there exists a constant C 2 > 0 such that for all t ∈ [0, T ] and all x, y ∈ F . Proof. Recall from Theorem 4.8 that and an analogous result holds for y. Thus by Lemma 4.6, where we have used Proposition 4.5, Proposition 3.12 and the definition of f x n (similarly to the proof of Theorem 4.8). Hence again by Proposition 4.5, ≤ 2e 2T R(x, y).

Temporal estimates
For the time estimates we can save ourselves some work by noticing that if X b,k is an Ornstein-Uhlenbeck process then for any s, t ∈ [0, ∞). Therefore regardless of whether X b,k is an Ornstein-Uhlenbeck or Wiener process we have that Now since (using the independence of the X b,k ) it follows that it suffices to find estimates of the above in the case s = 0.
We start with a method similar to the proof of [ for t ∈ (0, ∞), where a, b ∈ R are constants. Then the following hold: (1). If a, b ≥ 0 then σ ab (t) diverges.
(2). If a ∈ R, b < 0 then there exists C a,b > 0 such that σ ab (t) ≤ C a,b t for all t.
For (3) we must consider two cases depending on the value of b. First assume that for all k ∈ N, and we are back to the case a < 0, b = 1. It follows that for all s, t ∈ [0, T ] and all x ∈ F .
Proof. We assume that λ b 1 > 0 to streamline our calculations. The case λ b 1 = 0 is left as an exercise. By the discussion at the start of this section we may assume that s = 0. Fix x ∈ F and t ∈ [0, T ]. Recall the constant c 3 from Proposition 2.7. By independence of the and we are within the scope of Lemma 5.2 (as long as t > 0, though the case t = 0 is trivial). We find that if α > d s − 1 then the sum converges and there exists c > 0 such for all x ∈ F , t ∈ [0, T ].
Therefore using Lemma 5.2, we have for all α ≥ 0 that This is the method used in [21].
We now prove an alternative estimate that is weaker for large α but holds for all α ≥ 0.
By the functional calculus for self-adjoint operators, for all (t, x) ∈ [0, T ] × F and all integer n ≥ 0. We assume now that t > 0, and our aim is to choose n ≥ 0 to minimise the above expression. Fixing t ∈ (0, T ], define g : The function g has a unique stationary point which is a global minimum at Since t ≤ T we have by the definition of c 8 that y 0 ≥ 0. Since y 0 is not necessarily an integer we choose n = y 0 . Then g is increasing in [y 0 , ∞) so we have that E u(t, x) 2 ≤ 2g(n) ≤ 2g(y 0 + 1).
. This inequality obviously also holds in the case t = 0.

Hölder regularity of solution
We are now ready to prove the Hölder regularity result. It will turn out that the continuous version of u(t, x) can be interpreted as an H-valued process, and is a version of the original H-valued solution to (1.1) found in Theorem 2.11. Recall R ∞ the natural supremum metric on R × F given by R ∞ ((s, x), (t, y)) = max{|s − t|, R(x, y)}.
Proof. Take T > 0 and consider u T , the restriction of u to [0, T ] × F . It is an easily verifiable fact that for every p ∈ N there exists a constant C p > 0 such that if Z is any centred real Gaussian random variable then We also know that u T is a centred Gaussian process on [0, T ] × F by Theorem 4.8.
We will treat the case α ≤ ds 2 , which is precisely the region of values of α for which Proposition 5.5 will give us a better temporal Hölder exponent than Proposition 5.3. Propositions 5.1 and 5.5 then give us the estimates for all s, t ∈ [0, T ] and all x, y ∈ F . Taking p arbitrarily large and then using Corollary 3.19 we get a versionũ T of u T (that is,ũ T is a version of u on [0, T ] × F ) that satisfies the Hölder regularity conditions of the theorem for the given value of T . This works because any two almost surely continuous versions of u T must coincide almost surely since [0, T ] × F is separable. If now T > T and we construct a versionũ T of u on [0, T ] × F in the same way, thenũ T must agree withũ T on [0, T ] × F almost surely since both are almost surely continuous on [0, T ] × F which is separable. Therefore let T = n for n ∈ N and let Ω be the almost sure event and we are done. Now if α > ds 2 then we use the temporal estimate of Proposition 5.3 rather than the temporal estimate of Proposition 5.5 in (5.1).
Proof. From Theorem 5.6,ũ is almost surely continuous in [0, ∞) × F . Eachũ(t, x) is a well-defined random variable so by [1,Lemma 4.51],ũ is jointly measurable. Using continuity again, this implies thatũ(t, ·) ∈ H for all t ∈ [0, ∞) almost surely. We also have that t →ũ(t, ·) is a continuous function from [0, ∞) to H for each ω ∈ Ω and that eachũ(t, ·) is a Borel measurable map from Ω to H; the latter follows from the joint continuity ofũ and the fact that the Borel σ-algebra of H is generated by the bounded linear functionals on H.
For each n ≥ 1 and (t, Then each u (n) is obviously jointly measurable, and by (4.5), u (n) (t, x) →ũ(t, x) in L 2 (P) for each (t, x) ∈ [0, ∞) × F . In fact by joint measurability we also have that where in the last line we have used (4.5). Then by Tonelli's theorem, In particular, this implies that u (n) (t, ·), ϕ b k µ → ũ(t, ·), ϕ b k µ in L 2 (P) as n → ∞, for every t ∈ [0, ∞) and every k ≥ 1. Recall that if u 0 = 0 then the solution to (1.1) is simply given by the series It follows that for all t ∈ [0, ∞) and n ≥ k we have that almost surely. Thereforeũ(t, ·) = W b α (t) almost surely for all t ∈ [0, ∞) and we are done.
Remark 5.8. In [13] it is shown that under some mild conditions on the p.c.f.s.s. set , it can be embedded into Euclidean space in such a way that its resistance metric R is uniformly equivalent to some power of the Euclidean metric. Therefore in this case the conclusion of Theorem 5.6 holds with respect to the spatial Euclidean metric, albeit with a different Hölder exponent. An example given in [13, Section 3] is the n-dimensional Sierpinski gasket for n ≥ 2, see Example 1.3(2) of the present paper. This fractal has a natural embedding in R n , and it is shown that in this case we have a constant c > 0 such that  1), viewed as a function of α, is linearly strictly increasing in some neighbourhood of α = 0. Intuitively this phenomenon should occur for any F . We therefore conjecture that the Hö exponent in Theorem 5.6(1) is not sharp when α ∈ (0, d s ), and is in fact equal to the exponent obtained in Theorem 2.11 for the solution interpreted as an H-valued process.
We have shown regularity properties of the solution to (1.1) in the case u 0 = 0, and henceforth we assume that we are dealing with the continuous version of this solution.
If we now take an arbitrary initial condition u 0 = f ∈ H, then obviously the same results may not hold since f may be very rough. We can however prove continuity in almost the entire domain [0, ∞) × F .

Invariant measure
We conclude with a brief description of the long-time behaviour of the solutions to (1.1). In this section we allow the initial condition u 0 to be an H-valued random variable which is independent of W . Definition 6.1. An invariant measure for the SPDE (1.1) is a probability measure ν ∞ on L 2 (F, µ) = H such that if u is the solution to (1.1) with random initial condition u 0 ∼ ν ∞ (independent of W ) then u(t) ∼ ν ∞ for all t > 0.
In the following theorems, let (Z k ) ∞ k=1 be a sequence of independent and identically distributed one-dimensional standard Gaussian random variables.
If u is a solution to (1.1) in the case b = D then u(t) converges weakly to ν D ∞ as t → ∞ regardless of its initial distribution Law(u 0 ).
Proof. We first show that the definition of ν D ∞ makes sense. We have that < ∞, so ν D ∞ is indeed a well-defined probability measure on H. Now suppose u has initial distribution u 0 ∼ ν D ∞ . By Definition 2.1 and (2.1) we have that where (Z k ) ∞ k=1 and (X D,k ) ∞ k=1 are understood to be independent. Recall that λ D 1 > 0 by Remark 2.9, so for every k, X D,k is a centred Ornstein-Uhlenbeck process with unit volatility and rate parameter λ D k . Moreover, t → e −λ D k t (2λ D k ) − 1 2 Z k + X D,k t is an Ornstein-Uhlenbeck process with unit volatility, rate parameter λ D k and initial distribution given by the law of (2λ D k ) − 1 2 Z k , which turns out to be exactly its invariant measure (which we leave as an exercise for the reader). Thus the law of u(t) is equal to ν D ∞ for all t > 0, so ν D ∞ is an invariant measure. We also see that for all f ∈ H, In the case b = N we do not have nearly as neat a result, but there exists a decomposition of u into two independent processes, one of which has similar invariance properties to the b = D case and the other of which is simply a Brownian motion. Let denote convolution of measures. For a measure ν on H, let π * 1 ν denote the pushforward of ν with respect to π 1 , which is a measure on H 1 .
Let u be a solution to (1.1) with b = N . Then there exists a one-dimensional standard Wiener process B = (B(t)) t≥0 which is adapted to the filtration generated by W such that B and u − B are independent processes, and if u has initial distribution u 0 ∼ ν ν N ∞ for some probability measure ν on H 1 then u(t) − B(t) ∼ ν ν N ∞ for all t > 0. Moreover, if u has initial distribution u 0 ∼ ν 0 for some probability measure ν 0 on H then u(t) − B(t) converges weakly to (π * 1 ν 0 ) ν N ∞ as t → ∞.
Proof. Recall that λ N 1 = 0 and that λ N k > 0 for k ≥ 2. Just as in the b = D case we can prove that ν N ∞ is a well-defined probability measure on H ⊥ 1 , and indeed on H as well. We From our discussion after (4.1) we know that X N,1 is a standard one-dimensional Wiener process, and by Remark 2.9, ϕ N 1 ≡ 1. From the representation (4.2) of W b α as a sum of independent stochastic processes it is clear that u − B is independent of B. Now suppose u has initial distribution u 0 ∼ ν ν N ∞ for some probability measure ν on H 1 . The space H 1 is one-dimensional so let Z 0 be a real-valued random variable such that Z 0 ϕ N 1 has law ν. By Definition 2.1 and (2.1) we can write where again Z 0 , (Z k ) ∞ k=1 and (X N,k ) ∞ k=1 are understood to be independent. Now we observe that π ⊥ 1 = 1 (0,∞) (−L N ) using the functional calculus on self-adjoint operators, and in particular π ⊥ 1 commutes with functions of L N , including S N t . Then define π ⊥ 1 u(t) = u(t) − Z 0 + X N,1 t =: u 1 (t), where we have identified scalars c ∈ R with their associated constant functions cϕ N 1 ∈ H. It is then easily verifiable that π ⊥ 1 W is a cylindrical Wiener process on H ⊥ 1 and that u 1 is the (mild) solution to the SPDE on H ⊥ 1 given by Note that all operators in (6.1) commute with π ⊥ 1 and so can be identified with their restriction to H ⊥ 1 for the purposes of the above SPDE. By definition, Now just as in the b = D case, using [4, Theorem 11.20] (or [10, Proposition 5.23]) we find that ν N ∞ is the unique invariant measure for (6.1) and that the solution to (6.1) converges weakly to ν N ∞ for any initial distribution on H ⊥ 1 . So we have that for all t > 0, We observe that if u has deterministic initial value u 0 = f ∈ H then this is equivalent to u 0 being distributed according to the convolution of Dirac measures δ f1 δ f2 for f 1 = π 1 (f ) ∈ H 1 , f 2 = π ⊥ 1 (f ) ∈ H ⊥ 1 . By doing the usual eigenfunction expansion we have that u(t) = f 1 + X N,1 + u 1 (t) where u 1 is now the solution to (6.1) with initial value f 2 . Thus u(t) − B(t) = f 1 + u 1 (t) which converges weakly to δ f1 ν N ∞ . Now assume u has an arbitrary initial probability distribution u 0 ∼ ν 0 in H. By conditioning first on the value of π 1 (u 0 ) ∈ H 1 , then on the value of π ⊥ 1 (u 0 ) ∈ H ⊥ 1 and then using the dominated convergence theorem we find that E[g(u(t) − B(t))] converges to ((π * 1 ν 0 ) ν N ∞ )(g) for any continuous and bounded function g on H. So we have weak convergence of u(t) − B(t) to (π * 1 ν 0 ) ν N ∞ .