Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket

By using analytic tools from stochastic analysis, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along fractals (which can be considered as models of simplified rough porous media). Unlike the regular space case, such parabolic type equations involving non-linear convection terms must take a different form, due to the fact that convection terms must be singular to the “linear part” which defines the heat semigroup. In order to study these parabolic type equations, a new kind of Sobolev inequalities for the Dirichlet form on the gasket will be established. These Sobolev inequalities, which are interesting on their own and in contrast to the case of Euclidean spaces, involve two \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}$$\end{document}Lp norms with respect to two mutually singular measures. By examining properties of singular convolutions of the associated heat semigroup, we derive the space-time regularity of solutions to these parabolic equations under a few technical conditions. The Burgers equations on the Sierpinski gasket are also studied, for which a maximum principle for solutions is derived using techniques from backward stochastic differential equations, and the existence, uniqueness, and regularity of its solutions are obtained.


Introduction
The analysis on fractals has attracted attentions of researchers in the last decades, not only for the reason that fractals are archetypal examples of spaces without suitable smooth structure, but also because fractals are examples of interesting models in statistical mechanics. Many objects in nature (e.g. percolation clusters in disordered media, complex biology systems, polymeric materials, and etc.) possess features of fractals (see e.g. [28] for details). Fractals appear as scaling limits of lattices. Lattice models (e.g. the Ising models and their variants) have been extensively studied in statistical mechanics, and properties for scaling limits have been derived using conformal field theory in dimension two.
Since a calculus on fractals is not available, the theory of Dirichlet forms on measure-metric spaces and stochastic calculus are the analytic tools employed for the study of analysis problems on fractals, and many interesting results have been established in the past decades.
Early works on analysis on fractals however have been focused mainly on diffusion processes and the corresponding Dirichlet forms (see e.g. [1][2][3]8,11,12,[21][22][23][24] and etc.). Brownian motion on the Sierpinski gasket was first constructed by Goldstein and Kusuoka as the limit of a sequence of (scaled) random walks on lattices (cf. [9,26]). Kigami [22] has obtained an analytic construction of the Dirichlet form via finite difference schemes. The construction of gradients of functions with finite energy has been given in Kusuoka in [25], where a significant difference between Euclidean spaces and fractals has also been revealed (see [25,Section 6]). On the Sierpinski gasket for example, volumes of sets and energies of functions are measured in terms of two mutually singular measures, the Hausdorff measure and Kusuoka's measure (see Sect. 2 below for definitions). By virtue of the results obtained in [25], gradients of functions on the Sierpinski gasket may be defined as square integrable functions with respect to Kusuoka's measure (cf. Sect. 2). Roughly speaking, the gradient of a function with finite energy is the square root of the density of its energy measure with respect to Kusuoka's measure. There have been interests in the understanding of gradients of functions and non-linear partial differential equations on fractals with nonlinearities involving first-order derivatives (see e.g. [16,[18][19][20]33] and references therein). A new class of semi-linear parabolic equations involving singular measures on the Sierpinski gasket was proposed and studied in [27], where, among other things, a Feynman-Kac representation was obtained assuming the existence of weak solutions.
In the present paper, we establish the existence and uniqueness of solutions to the semi-linear parabolic PDEs proposed in [27], and derive the regularity of solutions. A crucial ingredient in our argument is a new type of Sobolev inequalities on the Sierpinski gasket (and the infinite gasket) involving different measures (which can be mutually singular). To author's knowledge, this type of Sobolev inequalities on fractals has not been investigated before, and is of mathematical interests on its own. We formulate and study the Burgers equations on the gasket, which is an archetype of non-linear PDEs with non-Lipschitz coefficients, and also as a simplified model of flows in porous medium. The difficulty in our case is that there exists no suitable analogue of the Cole-Hopf transformation on the gasket. Instead we tackle the problem by using a Feynman-Kac representation and an iteration argument.
This paper is organized as follows. We introduce in Sect. 2 the notations and definitions which will be effective throughout the paper. Several preliminary results are also reviewed in the same section. In Sect. 3, we give the formulation and the proof of new Sobolev inequalities on the Sierpinski gasket (and the infinite gasket), which will be needed in latter sections. The optimal exponents and a sufficient and necessary condition for the validity of these inequalities are also given in this section. Section 4 is devoted to the semi-linear parabolic PDEs on the gasket, where we establish the existence and uniqueness and the regularity of solutions. In Sect. 5, we apply the results in previous sections to the study of the Burgers equations on the gasket, which are the analogues of the Burgers equations on R.
The results of this paper are presented only for the Sierpinski gasket in R 2 , we however believe that our results also hold for Sierpinski gaskets in higher dimensions. The main results and the arguments given in this paper can be adapted accordingly without difficulties.

Preliminaries
In this section, we set up several notations and definitions which will be in force throughout this paper.

Sierpinski gaskets
Let F i : R 2 → R 2 , i = 1, 2, 3 be the contractions defined by F 1 , m ∈ N. The (compact) Sierpinski gasket S and the infinite Sierpinski gasketŜ are defined to be the closures S = cl ( ∞ m=0 V m ) andŜ = cl ( ∞ m=0V m ) respectively.Ŝ can be written as a countable unionŜ = i∈Z τ i (S), where τ i : R 2 → R 2 , i ∈ Z are translations of R 2 such that τ i (S), i ∈ Z have non-overlapping interiors. To our purpose, the labelling of the translations τ i , i ∈ Z is immaterial. We should point out that there are many different infinite versions of S (see e.g. [32,Section 5]). TheŜ we use in the present paper is only one of them. Let As a convention, we define F [ω] 0 = Id. The Hausdorff measure on S is the unique Borel probability measure ν on S such that ν S [ω] m = 3 −m for all ω ∈ W * , m ∈ N , and the Hausdorff measure onŜ is the unique Borel measureν onŜ such that (ν • τ i )| S = ν for all i ∈ Z.

Standard Dirichlet forms
For each m ∈ N and any functions u, v on ∞ m=0 V m , let (u, u) exists (possibly infinite), and the limit will be denoted by E(u) for simplicity.
According to [24,Theorem 2.2.6], every function u ∈ F(S) uniquely extends to a continuous function on S, in other words, F(S) ⊆ C(S). (E, F(S)) is called the standard Dirichlet form on S, which is a regular local Dirichlet form on L 2 (S; ν). (E, F(S)) possesses the property of self-similarity in the sense that Let L be the self-adjoint non-positive operator on Dom(L) ⊆ L 2 (S; ν) associated with (E, F(S)). Let is also a regular local Dirichlet form on L 2 (S; ν) corresponding to Dirichlet boundary conditions. By replacing V m withV m in (2.1),Ê(u) can be defined similarly for any u ∈ C(Ŝ). Let F(Ŝ) be the completion of {u ∈ C(Ŝ) :Ê(u) < ∞} with respect to the norm E(·) 1/2 + · L 2 (ν) . It can be shown that F(Ŝ) ⊆ C 0 (Ŝ), where C 0 (Ŝ) is the space of continuous functions onŜ vanishing at infinity. Ê , F(Ŝ) is called the standard Dirichlet form onŜ, which is a regular local Dirichlet form on L 2 (Ŝ;ν). By definition Similar to E, the formÊ is self-similar in the sense that For any x, y ∈Ŝ, define R(x, y) by For every x, y ∈Ŝ, R(x, y) < ∞. Moreover, if x = y, then there exists a unique u ∈ F(Ŝ) such that u(x) = 1, u(y) = 0,Ê(u) = R(x, y) −1 (see [24,Theorem 2.3.4]). The function R(·, ·), called the resistance metric, is a metric onŜ satisfying for some universal constant C * ≥ 1, where d s = 2 log 3/ log 5, d w = log 5/ log 2, and d f = d w /(2d s ) are the spectral dimension, the walk dimension, and the fractal dimension ofŜ respectively (cf. [24,Lemma 3.3.5]). By the definition of R(·, ·), Let f be a function on V m . The m-harmonic function with boundary value f is defined to be the unique h ∈ F(S) such that h| Vm = f and that h • F [ω] m is a harmonic function for all ω ∈ W * . The energy of an m-harmonic function h can be calculated using E(h) = E (m) (h| Vm ).

Kusuoka measures and gradients
The Kusuoka measure μ on S, as defined in [25], is the unique Borel probability measure on S such that for all ω = ω 1 ω 2 ω 3 . . . ∈ W * , m ∈ N. The Kusuoka measureμ onŜ is the unique Borel measure onŜ such that (μ • τ i )| S = μ for all i ∈ Z. The Kusuoka measure μ (μ respectively) is singular to the Hausdorff measure ν (ν respectively) (cf. [25, p. 678]). If u ∈ F(S), then μ u denotes the energy measure of u, i.e. the Borel measure on S such that S φ dμ u = E(φu, u) − 2 −1 E(φ, u 2 ) for φ ∈ F(S). By [25,Theorem (5.4)], μ u μ for all u ∈ F(S) (see [13,15] for similar results on general fractals). Moreover, there exists a unique linear operator ∇ : F(S) → L 2 (μ), called the gradient operator on S, satisfying the following: (i) μ u = |∇u| 2 μ for all u ∈ F(S), and (ii) if h is the harmonic function with boundary value h(0, Remark 2. 1 We should point out that there exist several slight variants of gradients on fractals, which are introduced to address different problems (see, e. g. [4,6,14,23,27,30,33] and references therein). The definition of gradients on S adopted in the present paper was introduced in [27] via martingale representations, and can be regarded as the special case of the definition given in [14], where μ is the minimal energy-dominant measure (see [14, p. 3] for the definition).

Sobolev inequalities
The objective of this section is to establish some Sobolev inequalities involving different (probably mutually singular) measures on S andŜ (Theorems 3.6, 3.11 respectively), which is crucial to our study of some semi-linear parabolic equations on the gasket. A sufficient and necessary condition for the validity of these Sobolev inequalities (Theorems 3.8, 3.13) will be established as well.
To shed some light on the motivation of these inequalities, consider the following simple parabolic PDE on S ∂ t u dν = Lu dν + ∇u dμ.
Here the singular measures ν and μ must be involved as Lu is ν-a.e. defined while ∇u is only μ-a.e. defined. A precise interpretation of this equation will be given in Sect. 4. Let us assume for the moment that if u is a solution then one may test the equation against the solution to obtain For PDEs on Euclidean spaces, the measures ν and μ are equal to the Lebesgue measures, and therefore, the above differential inequality together with Grönwall's inequality yields the energy estimates and the existence and uniqueness of solutions. However, on S, the measures ν and μ are mutually singular, and hence the L 2 -norms · L 2 (ν) and · L 2 (μ) are in general incomparable. Thus, Grönwall's inequality does not apply in this case. For PDEs involving gradients on S, an appropriate comparison of · L 2 (ν) and · L 2 (μ) is necessary to obtaining energy estimates. In fact, for functions u ∈ F(S), the L 2 -norms u L 2 (ν) and u L 2 (μ) must be compared with the involvement of (an arbitrarily small portion of) the energy E(u) (see Corollary 3.18 below). This type of comparison is possible due to the Sobolev inequalities to be established in this section.
For convenience, C * will always denote a generic universal constant which may be different on various occasions.
Clearly,Ŝ can be written as the non-overlapping unionŜ = i∈Z S i,m for each m ∈ Z. Therefore,Ê(u) = i∈ZÊ | S i,m (u) for any u ∈ F(Ŝ) in view of (2.2) and (2.3).
The constant δ s is defined so that 5/3 = 3 1/δ s . Therefore, for every i and m, by We are now in a position to formulate the main results of this section. Letσ be a Borel measure onŜ satisfying the following condition: there exist constants Cσ ≥ 1 and 0 < δ ≤δ ≤ ∞,δ ≥ 1 such that being valid for all Borel sets implies the absolute continuity ofσ with respect tô ν.
We would like to point out that the condition (M.1) is general enough to include many cases of interests, some of important examples are listed below.
depending only on d. Therefore, the analogue onŜ of |x| −θ dx on R d would be a Borel measureσ ν satisfying the condition (M.1) with δ,δ given by Here we have used d s as the Sobolev dimension ofŜ (cf. Remark 3.7 below).
Supposeσ is a Borel measure onŜ satisfying the condition (M.1). Then then the pair of exponents given by (3.3) is optimal in the following sense: if where and hereafter C > 0 denotes a generic constant depending only on the constant Cσ in (M.1). Therefore, Similarly, when m ≤ 0, we have that The proof of (3.2) is done by optimising over m. SupposeÊ(u) 1/2 ≥ u L p (ν) . Consider the following two cases: Case 1: p ≤ q ≤ p/δ. Note that p ≤ p/δ forcesδ = 1 and therefore 1/(qδ) = 1/ p, a 1 = 0. Setting m → −∞ in (3.6) gives that in (3.6), we obtain that Suppose thatÊ(u) 1/2 < u L p (ν) . We consider the two cases.
Case 1: p ≤ q ≤ p/δ. In this case, a 2 = 0. Setting m = 0 in (3.5) gives that This proves (3.2) for q < ∞. Setting q → ∞ proves the case when q = ∞ as the constant C is independent of q. Suppose in addition that the condition (M.2) is satisfied, we prove that (a 1 , a 2 ) is the optimal pair of exponents. We first show that, for any dyadic triangle S ⊆Ŝ, there exists an h S ∈ F(Ŝ) such that To see this, suppose first that S = 2 −1 S for some m ∈ Z. Let h be the 1-harmonic function in S with boundary value Then h ∈ F(Ŝ) and satisfies (3.7). For a general dyadic triangle In view of (M.2), it is easily seen that

It follows from the above and (3.4) that
Remark 3.7 (i) Some comments are desired on the interpretation of the exponents appearing in the inequality (3.2). Recall that, on Euclidean space R d , the celebrated Gagliardo-Nirenberg inequality takes the form where a ∈ [0, 1] is given by 1 The case corresponding to setting of Dirichlet forms is the one when j = 0, m = 1 and r = 2, for which the exponent a is given by Comparing this to the denominator of (3.8), we see that the Sobolev dimension d should be given by (ii) The inequality (3.4) includes the analogue onŜ of a specific case of the weighted Sobolev inequalities on R d in [5]. The weighted Sobolev inequalities established in [5] take the form where α, β, γ < 0 satisfy 1/r + α/d > 0, The case corresponding to setting of Dirichlet forms is the one when α = β = 0, r = 2 and 1/ p ≥ 1/q + γ /d s > 0, for which the weighted inequality reads As remarked in Example 3.5.(iii), the analogue onŜ of |x| γ q dx on R d is a Borel measureσ onŜ satisfying the condition (M.1) with δ,δ given by 1/δ = 1/δ = 1 + γ q/d s . Therefore, the analogue of (3.9) onŜ should be This coincides with the result of (3.16) since the exponents for the measureσ are given by An additive version of (3.2), which is a corollary of (3.2) and Young's inequality, is derived in [17] for the study of vector fields on resistance spaces.
According to Theorem 3.6, the condition (M.1) is sufficient for the derivation of Sobolev inequalities. The following theorem states that this condition is also necessary for the validity of Sobolev inequalities of the form (3.4) with q < ∞.

Theorem 3.8 Letσ be a Borel measure onŜ. Suppose that there exist some constants
Proof Suppose that (3.4) holds. For any dyadic triangle S ⊆Ŝ, as shown in the proof of Theorem 3.6, there exists a piecewise harmonic function h S ∈ F(Ŝ) such that where the notationS and the relations and are the same as those in the proof of Theorem 3.6. Applying (3.4) to h S gives that Since q < ∞, it follows from the above that Therefore, the first part of (M.1) is satisfied withδ = ∞. Furthermore, for any dyadic triangle S with diam(S) ≥ 1, by (3.10),σ (S) 1/q ν(S) 1/ p asν(S) ≥ 1. Setting δ = p/q completes the proof.
Applying Theorem 3.6 to the cases whenσ =ν and whenσ =μ, we obtain the following.
The only thing needs a proof is thatδ = δ s in (b). Clearly, We show that sup from which the conclusion follows immediately.

Remark 3.10
Setting p = 1, q = 2 in (3.11) gives the Nash inequality onŜ (see [8,Theorem 4 Conclusions similar to that of Theorem 3.6 hold when the roles ofσ andν are exchanged. More specifically, letσ be a Borel measure onŜ satisfying the following condition: there exist constants Cσ ≥ 1 and 0 < δ ≤δ < ∞ such that for any dyadic triangle S ⊆Ŝ. For measuresσ satisfying (M.1'), we have Theorems 3.11 and 3.13 below, of which the proofs will be omitted as they are are similar to those of Theorems 3.6 and 3.8.

Remark 3.15
The value of δ in Corollary 3.14 follows from the fact that which will be given in another work by the present authors.
We end this section with the corresponding Sobolev inequalities on the compact gasket S, whose proof shall be omitted. Let σ be a finite Borel measure on S. For the compact gasket, only the first part of the condition (M.1) is relevant, i.e.
where δ ∈ (0, ∞] is a constant depending only on the Borel measure σ . (a) Suppose that σ satisfies (3.17). Then for any u ∈ F(S), where c is any constant satisfying min S u ≤ c ≤ max S u, and a = 1/ p − 1/(qδ) 20) and C > 0 is a constant depending only on the constant C σ in (3.17). Therefore, for any u ∈ F(S), Moreover, the exponent a given by (3.20) is optimal in the sense that if (3.19) holds for some a ∈ [0, 1], then a ≥ 1/ p−1/(qδ)  .21) gives the Nash inequality on S (see [8,Theorem 4.4] or [24,Theorem 5

Semi-linear parabolic PDEs
In this section, we study a type of semi-linear parabolic equations on S, for which energy estimates and existence and uniqueness of solutions are established (Theorem 4.16). Moreover, the regularity of solutions to these PDEs is derived under additional conditions. We consider the following initial-boundary value problem for semi-linear parabolic PDEs (see Definition 4.13 below for a precise interpretation) where ψ ∈ L 2 (ν), and the coefficient f : [0, T ]×S×R 2 → R satisfies the following: (i) There exists a constant K > 0 such that

Remark 4.1
There exist different formulations of non-linear PDEs on fractals. For example, a type of non-linear equations on fractals was considered by in [19], where the non-linearity f (∇u) is a bounded mapping f : L 2 (μ) → L 2 (ν). The equations studied there are essentially defined via a single measure (the Hausdorff measure ν). Therefore, the PDEs studied in this paper are different in essence from those considered in [19] in the way the gradients interact with the equations.
From now on, we shall use the notation f , g λ = S f g dλ for any Borel measure λ on S and any λ-a.e. defined functions f , g on S, whenever the integral is well-defined. As in the previous section, we denote by C * a generic universal constant which may vary on different occasions.
Let {P t } t≥0 be the Markov semigroup associated with the killed Brownian motion on S, the diffusion processes associated with the Dirichlet form (E, F(S\V 0 )). {P t } t≥0 admits a jointly continuous heat kernel p (t, x, y), which is C ∞ in t (cf. [3, Theorem 1.5]). The following result on heat kernel and resolvent kernel estimate was first proved in [3, Theorems 1.5, 1.8].
for some constant C α > 0 depending only on α.
In view of the joint continuity of p(t, x, y), the definition below is legitimate.

Remark 4.4 (i)
Let λ be a Radon measure on S. By the symmetry of p(t, ·, ·), it is easy to see that P t (gλ), f ν = g, P t f λ for all f ∈ L 2 (ν), g ∈ L 1 (λ). (ii) For any Radon measure λ on S, we have P t λ ∈ Dom(L) for t > 0. In fact, since p(t, x, y) ∈ C((0, ∞) × S × S), we have P t/2 λ ∈ C(S), which implies that P t λ = P t/2 (P t/2 λ) ∈ Dom(L). Moreover, P t λ ∈ C 1 (0, ∞; L 2 (ν)) and d dt P t λ = LP t λ. (iii) Notice that, due to the singularity between ν and μ, the contractivity P t (gμ) L 2 (ν) ≤ g L 2 (μ) , t > 0 is no longer valid in general. In fact, for g ∈ L 2 (μ), g = 0, we have To see this, suppose contrarily that lim t→0 P t (gμ) L 2 (ν) = sup t>0 P t (gμ) L 2 (ν) < ∞. Then there exists a unique g 0 ∈ L 2 (ν) such that lim t→0 P t (gμ) = g 0 weakly in L 2 (ν). On the other hand, for any v ∈ F(S\V 0 ), we have where the last equality follows from the uniform convergence lim t→0 P t v = v as a consequence of the convergence in F(S\V 0 ). By the density of F(S\V 0 ) in C(S), it is seen that gμ = g 0 ν, which contradicts the fact that ν and μ are mutually singular.
To study the semi-linear parabolic PDEs (4.1), let us first investigate the formal integral t 0 P t−s (g(s)μ) ds, (4.2) which is the formal solution to the equation ∂ t u dν = Lu dν + g(t) dμ. Since P t is not bounded from L 2 (μ) to L 2 (ν) (cf. Remark 4.4.(iii) above), there is a singularity in the integrand of (4.2) at s = t. We shall show that (4.2) is a well-defined function in the space L ∞ (0, T ; L 2 (ν)) ∩ L 2 (0, T ; F(S\V 0 )), and is jointly Hölder continuous if g(t) is uniformly bounded in L 2 (μ). To formulate the results, it is convenient to introduce several definitions.

Definition 4.5 For any
The space F −1 (S) is defined to be the · F −1 -completion of L 2 (ν).
We derive properties of the convolution (4.2) in the following lemmas. Then u δ ∈ L ∞ (0, T ; L 2 (ν)) ∩ L 2 (0, T ; F(S\V 0 )) and for any > 0, where C > 0 is a constant depending only on . Moreover, For any > 0 and each t ∈ (0, T ), testing (4.4) against u δ and applying Corollary 3.18 gives that where C > 0 denotes a generic constant depending only on which may vary on different occasions. By Grönwall's inequality and the fact that u δ (t) = 0, t ∈ [0, δ], we deduce By (4.4) again, for any v ∈ F(S\V 0 ), The above inequality also holds for v ∈ F(S). This can be seen by considering the F-orthogonal projection of v on F(S\V 0 ). Therefore, which, together with (4.5), implies the desired estimate for ∂ t u δ L 2 (0,T ;F −1 ) .

Lemma 4.9
The limit exists with respect to the norm · L ∞ (0,T ;L 2 (ν)) + · L 2 (0,T ;F ) , and satisfies Moreover, u(t) has a weak derivative ∂ t u in L 2 (0, T ; F −1 ), and Proof As before, we set g(t) = 0 for t < 0. Let δ, δ ∈ (0, T ) and w = u δ − u δ , where u δ are the functions defined by (4.3). By (4.4), we have from which it follows that The first term on the right hand side of (4.7) can be estimated in the same way as in the proof of Lemma 4.8, which yields that For the second term on the right hand side of (4.7), we have By the spectral decomposition, which, together with the fact that E((P δ − P δ )w(t)) ≤ E(w(t)), implies that Therefore, we deduce from (4.7) that It follows from the above inequality and Grönwall's inequality that Therefore, {u δ } is a Cauchy sequence with respect to the norm · L ∞ (0,T ;L 2 (ν)) + · L 2 (0,T ;F ) , which proves the convergence of (4.6). Moreover, the desired estimates for u follows readily from the similar estimates for u δ . Definition 4.10 By virtue of Lemma 4.9, the convolution t 0 P t−s (g(s)μ) ds can be defined to be the limit in (4.6).

Remark 4.12
The authors believe that 1/2 is the correct Hölder exponent in x ∈ S for (4.6) in general, which is suggested by the fact that a generic u ∈ F(S) has only Proof Let g(t) = 0 for t < 0. We first show that where C T > 0 is a constant depending only on T . Denote p s,x (y) = p(s, x, y). By the definition of u(t), we have By the Sobolev inequality (3.22), which, together with Lemma 4.2, implies that y).

Definition 4.13 A function u is called a weak solution to the PDE (4.1) if:
(WS.1) u ∈ L 2 (0, T ; F(S\V 0 )) and u has a weak derivative ∂ t u in L 2 (0, T ; F −1 (S)); , v μ will be L 2 (ν)-bounded, which contradicts with the singularity between ν and μ. Therefore, solutions to non-linear parabolic PDEs on S can only have mild regularity in general. This is a remarkable feature of non-linear PDEs on S, which suggests a significant distinction between the PDE theory on Euclidean spaces and that on fractals.

Proposition 4.15
Suppose that g ∈ L 2 (0, T ; L 2 (μ)). Then the initial and boundary problem to the PDE admits a unique weak solution u given by Proof Clearly, we only need to prove for the case when ψ = 0. Let u δ be the truncated convolution defined by (4.3), and let u be the convolution given by (4.6). For any v ∈ F(S\V 0 ), by (4.4), Since lim δ→0 P δ v = v uniformly, by considering a subsequence if necessary and setting δ → 0, we deduce that Therefore, u is a weak solution to (4.17).
The estimate (4.18) follows readily from Lemma 4.9, and the uniqueness of solutions is an immediate consequence of (4.18).
Therefore, there exists a u ∈ L 2 (0, T ; F(S\V 0 )) such that lim n→∞ u n − u * = 0. It is clear that u is a weak solution to (4.1), and the estimate (4.19) holds as E 1/2 (·) and · F are equivalent on F(S\V 0 ). This proves the existence. Suppose thatũ is a weak solution to (4.1) with initial valueψ. By an argument similar to (4.24) and (4.28), it can be shown that and that The estimate (4.20) follows readily from the above two inequalities. The uniqueness of solutions is now an immediate consequence of (4.20).
For general h, let h (t) = 1 t+ t h(s) ds, > 0. Then h is differentiable and satisfies (4.31) with L replaced by 2L. The above case gives that h (t) − h (s) ≤ 6L(t − s). It remains to apply the Lebesgue differentiation theorem to complete the proof of the lemma. Now by (4.30) and Jensen's inequality, the function h(t) = log[E(u(t))] satisfies (4.31) with L = C K . It follows from the previous lemma that Using the above inequality and (4.29) again, we deduce that which implies (4.21). We now prove the joint Hölder continuity. Let g(t, x) = f (t, x, u(t, x), ∇u(t, x)). Then u is the solution to the PDE ∂ t u dν = Lu dν + g(t)dμ.
We now can apply Lemma 4.11 and Proposition 4.15 and to deduce the desired joint Hölder continuity.

The Burgers equations
As an application of Theorem 4.16 and the Feynman-Kac representation for (backward) parabolic PDEs on S in [27,Theorem 3.19], we study the initial-boundary value problem for the following analogue on S of the Burgers equations on R where ψ ∈ F(S\V 0 ). We shall prove the existence and uniqueness of solutions to the Eq. (5.1), and derive the regularity of the solutions.
Remark 5. 1 We would like to point out a difference between the Burgers equations on S and those on R. The Burgers equations on R can be exactly solved with an explicit formula for the solutions via the Cole-Hopf transformation, and properties of solutions can be derived using the explicit formula. However, this Cole-Hopf type of transformation is not available on S. The Cole-Hopf transformation reduces the Burgers equation on R for u to a heat equation for −∇(log u). In contrast, on S, the formal expression L[∇(log u)] is not well-defined, since the gradient ∇(log u) is only μ-a.e. defined and therefore ∇(log u) / ∈ F(S) due to the singularity between ν and μ. Hence, different approaches must be employed for the study of (5.1).
Let us start with the Feynman-Kac representation for solutions to parabolic PDEs on S. Let {X t } t≥0 and {W t } t≥0 be Brownian motion and the representing martingale on S respectively, i.e. {X t } t≥0 is the diffusion process associated with the form (E, F(S)), and {W t } t≥0 is the unique martingale additive functional having μ as its energy measure such that M [u] [25,Theorem 5.4] and [27,Section 2]). The following result was given in [27,Theorem 3.19], and is an analogue on S of the representation theorem for semi-linear PDEs on R d established by Peng in [29]. See [27,Section 3] for the definition of solutions to backward stochastic differential equations (BSDEs) on S.
To verify the definition of {u n }, we must show that u n L ∞ (0,T ;L ∞ ) ≤ ψ L ∞ . Without loss of generality, we only need to show that u n (T ) L ∞ ≤ ψ L ∞ . By Theorem 5.2, (Y t , Z t ) = (u n (T − t, X t ), ∇u n (T − t, X t )) is the unique solution to the BSDE where σ (T ) = T ∧ inf{t > 0 : X t ∈ V 0 }, and For each x ∈ S\V 0 , we define a measureP x by dP x dP x = exp σ (T ) 0 u n−1 (T − r , X r ) dW r − 1 2 The measureP x is a probability measure. In fact, by [27,Corollary 4.3], the quadratic process W is exponentially integrable, i.e. Notice that which implies that E x T 0 Z 2 r d W r < ∞ for ν-a.e. x ∈ S and therefore, for all x ∈ S in view of the quasi-continuity of the function x → E x T 0 Z 2 r d W r and the fact that the empty set is the only subset of S having zero capacity since F(S) ⊆ C(S). Hence, Z r dW r is a P x -martingale for all x ∈ S. Moreover, it follows from the Girsanov theorem that {Y t } t≥0 is aP x -martingale, and therefore, which, together with the fact that | | ≤ ψ L ∞ , implies that u n (T ) L ∞ ≤ ψ L ∞ . Hence, we conclude that u n L ∞ (0,T ;L ∞ ) ≤ ψ L ∞ , and that the sequence {u n } is well-defined. Now, by Theorem 4.16, u n L 2 (0,T ;F ) + ∂ t u n L 2 (0,T ;F −1 ) ≤ CT , n ∈ N, where C > 0 is a generic constant depending only on ψ L ∞ which may vary on different occasions. Therefore, there exists a subsequence {u n k } and a u ∈ L 2 (0, T ; F(S\V 0 )) such that ∂ t u ∈ L 2 (0, T ; F −1 (S)), and lim k→∞ u n k = u, weakly in L 2 (0, T ; F(S\V 0 )), (5.4) lim k→∞ ∂ t u n k = ∂ t u, weakly in L 2 (0, T ; F −1 (S)). (5.5) Since u n L ∞ (0,T ;L ∞ ) ≤ ψ L ∞ , the sequence {u n ∇u n } n∈N + is bounded in L 2 (0, T ; L 2 (μ)). By considering a subsequence of {u n k } if necessary, we may assume that {u n k ∇u n k } is weakly convergent in L 2 (0, T ; L 2 (μ)). By the uniqueness of weak limits, lim k→∞ u n k ∇u n k = u∇u, weakly in L 2 (0, T ; L 2 (μ)). (5.6) Thus, it follows readily from (5.4)-(5.6) that u is a weak solution to (5.1). Moreover, the estimate u L ∞ (0,T ;L ∞ ) ≤ ψ L ∞ follows as a corollary of the inequalities u n L ∞ (0,T ;L ∞ ) ≤ ψ L ∞ . Testing (5.1) against u(t) and using the Sobolev inequality (3.22) gives that for any ∈ (0, 1) and a.e. t ∈ [0, T ], d dt u(t) 2 L 2 (ν) ≤ −E(u(t)) + ψ L ∞ E(u(t)) 1/2 + C u(t) L 2 (ν) E(u(t)) 1/2 .