Well-posedness of renormalized solutions for a stochastic $p$-Laplace equation with $L^1$-initial data

We consider a $p$-Laplace evolution problem with stochastic forcing on a bounded domain $D\subset\mathbb{R}^d$ with homogeneous Dirichlet boundary conditions for $1<p<\infty$. The additive noise term is given by a stochastic integral in the sense of It\^{o}. The technical difficulties arise from the merely integrable random initial data $u_0$ under consideration. Due to the poor regularity of the initial data, estimates in $W^{1,p}_0(D)$ are available with respect to truncations of the solution only and therefore well-posedness results have to be formulated in the sense of generalized solutions. We extend the notion of renormalized solution for this type of SPDEs, show well-posedness in this setting and study the Markov properties of solutions.


Introduction.
1.1. Motivation of the study. We are interested in the study of well-posedness for a p-Laplace evolution problem with stochastic forcing on a bounded domain D ⊂ R d with homogeneous Dirichlet boundary conditions for 1 < p < ∞. For p = 2, we are in the case of the classical Laplace operator, for arbitrary 1 < p < ∞, u → − div (|∇u| p−2 ∇u) is a monotonone operator on the Sobolev space W 1,p 0 (D) that is singular for p < 2 and degenerate for p > 2. Evolution equations of p-Laplace type may appear as continuity equations in the study of gases flowing in pipes of uniform cross sectional areas and in models of filtration of an incompressible fluid through a porous medium (see [3,14]): In the case of a turbulent regime, a nonlinear version of the Darcy law of p-power law type for 1 < p < 2 is more appropriate (see [14]). Turbulence is often associated with the presence of randomness (see [10] and the references therein). Adding random influences to the model, we also take uncertainties and multiscale interactions into account. Randomness may be introduced as random external force by adding an Itô integral on the right-hand side of the equation and by considering random initial values. Consequently, the equation becomes a stochastic partial differential equation (SPDE) and the solution is then a stochastic process.
For square-integrable initial data u 0 , the stochastic p-Laplace evolution problem can be solved with classical methods for nonlinear, monotone SPDEs (see, e.g. [26], [24]). Existence and regularity results for stochastic p-Laplacian-type systems have been proposed in [9]. In our contribution, we focus on more general, merely integrable random initial data. The well-posedness of quasilinear, degenerate hyperbolic-parabolic SPDEs with L 1 random initial data has already been addressed [18] in the framework of kinetic solutions, but, to the best of our knowledge, these results do not apply in our situation. On the other hand, there has been an extensive study on nonlinear evolution PDEs with initial data and righthand side in L 1 (see, e.g., [8,6,7]). From these results it is well known that the deterministic p-Laplace evolution problem with L 1 -data is not well-posed in the variational setting for 1 < p < d, were d ∈ N is the space dimension. For this reason, the problem is formulated in the framework of renormalized solutions. The notion of renormalization summarizes different strategies to get rid of infinities (see [13]) that may appear in physical models. The notion of renormalized solutions has been introduced to partial differential equations by Di Perna and Lions in the study of Boltzmann equation (see [15]) and then extended to many elliptic and parabolic problems (see, e.g., [16], [4,7,5] and the references therein). The main idea is to make a nonlinear change of unknown v = S(u) in the equation, where S is chosen in order to remove infinite quantities of the solution u. For SPDEs, this concept has been applied for stochastic transport equations in [1,11] and for the Boltzmann equation with stochastic kinetic transport in [27]. For many physically relevant singular SPDEs, a slightly different notion of renormalization has recently been developed (see [19,20] and the references therein). For these cases, solutions may be obtained as limits of classical solutions to regularized problems with addition of diverging correction terms. These counterterms arise from a renormalization group which is defined in terms of an associated regularity structure.
In this contribution, it is our aim to extend the notion of renormalized solutions in the sense of [7] for the stochastic p-Laplace evolution problem with random initial data in L 1 and to show well-posedness in this framework.
Due to the poor regularity of the initial data u 0 , a-priori estimates on ∇u are not available and therefore the well-posedness result has to be formulated in the sense of a generalized solution, more precisely in the framework of renormalized solutions.
In Section 2, we show the existence of a unique strong solution to (1) in the case where the initial value u 0 is an element of L 2 (Ω × D). After that, we establish a contraction principle that shows that a sequence of strong solutions is a Cauchy sequence in L 1 (Ω; C([0, T ]; L 1 (D))) whenever the sequence of initial values is a Cauchy sequence in L 1 (Ω × D). In Section 4 we prove a version of the Itô formula which makes it possible to define renormalized solutions to equation (1). Section 5 contains the definition of renormalized solutions to (1), in Section 6 we show the existence of such a solution and Section 7 contains the uniqueness result, which is a consequence of the L 1 -contraction principle for renormalized solutions stated in Theorem 7.1. Another consequence of the contraction principle is the continuous dependence of the renormalized solution on the initial value in L 1 which is necessary for the proof of the Markov property. In the Hilbert space setting, these arguments are classical nowadays, see, e.g., [24]. In Section 8 we give a brief overview of the arguments in the L 1 setting and study regularity properties of the associated semigroup.
For the sake of clarity of the presentation, we restrict ourselves to the case of a real-valued Brownian motion. However, it is straightforward to extend our results for additive stochastic perturbations by a stochastic integral in the sense of Itô with respect to a cylindrical Wiener process in L 2 (D). In this case, the vector space of Hilbert-Schmidt Operators from L 2 (D) to R can be identified with L 2 (D). For multiplicative stochastic perturbations, a more general version of Proposition 9.1 is necessary and the study of this case will be subject to a forthcoming work.

Strong solutions.
Theorem 2.1. Let the conditions in the introduction be satisfied. Furthermore, let u 0 ∈ L 2 (Ω × D) be F 0 -measurable. Then there exists a unique strong solution to (1), i.e., an F t -adapted stochastic process u : for all t ∈ [0, T ] and a.s. in Ω.
Remark 2.2. Since we know from all terms except the term t 0 div (|∇u| p−2 ∇u) ds that these terms are elements of L 2 (D) for all t ∈ [0, T ] and a.s. in Ω it follows that t 0 div(|∇u| p−2 ∇u) ds ∈ L 2 (D) for all t ∈ [0, T ] and a.s. in Ω. Therefore this equation is an equation in L 2 (D).
Proof. The existence result is a consequence of [23], Chapter II, Theorem 2.1 and Corollary 2.1. Following the notations therein, we set E = R, H = L 2 (D), V = W 1,p 0 (D) ∩ L 2 (D) for 1 < p < 2 and V = W 1,p 0 (D) for p ≥ 2. In particular, V → H → V with continuous embedding in accordance with [23], p.1251 conditions a.)-d.). Moreover, we have A : . We remark that A does not depend on (t, ω) ∈ [0, T ] × Ω and that B does not depend on u ∈ V . Therefore it follows immediately that conditions (A1)-(A5) of [23] are satisfied for all 1 < p < ∞. The uniqueness is a consequence of [23], Chapter II, Theorem 3.1, which applies under the same assumptions.
3. Contraction principle. Theorem 3.1. Let u 0 , v 0 ∈ L 2 (Ω × D) and u and v strong solutions to the problem (1) with initial value u 0 and v 0 , respectively. Then Proof. We subtract the equations for u and v and we get for all t ∈ [0, T ] and a.s. in Ω. We apply the Itô formula pointwise a.s. with respect to x ∈ D with an approximation of the absolute value N δ in (2) which is defined as follows (see, e.g., Proposition 5 in [28]): For δ > 0, let ρ(r) = ce 1 r 2 −1 1 {|r|≤1} such that R ρ(s) ds = 1 and ρ δ (r) = 1 δ ρ( r δ ) be the classical symmetric mollifier sequence approximating the Dirac mass with support [−δ, δ]. Then, η δ (r) = 2 r 0 ρ δ (s) ds is a regular nondecreasing Lipschitz approximation of the sign function with Lipschitz constant 2c eδ . Now, N δ (r) := r 0 η δ (s) ds. Discarding the nonnegative term coming from the p-Laplace and passing to the limit with δ → 0 + yields for all t ∈ [0, T ] and a.s. in Ω. 4. Itô formula and renormalization. In order to find an appropriate notion of renormalized solutions to (1), we prove an Itô formula in the L 1 -framework. We remark that the combined Itô chain and product rule from [10], Appendix A4 does not apply to our situation for two reasons. Firstly, we take the bouded domain D ⊂ R d into account in our regularizing procedure by adding a cutoff function (see Appendix, Subsection 9.1). Secondly, the spacial regularities are different in our case.
For two Banach spaces X, Y let L(X; Y ) denote the Banach space of bounded, linear operators from X to Y and L(X) denote the space of bounded linear operators from X to X respectively.
For the sake of completeness, we recall the following regularization procedure which has been introduced similarly by Fellah and Pardoux in [17]: There exists a sequence of operators and all n ∈ N ii.) For any n ∈ N and any Banach space Proof. See Appendix, Subsection 9.1.
In the following, we refer to [2], p.33 for a definition of the progressively measurable functions on Ω × (0, T ). For 1 ≤ p < ∞, with a slight abuse of notation we will call a function f ∈ L p (Ω × Q T ) progressively measurable if it belongs to the set of dP ⊗ dt-equivalence classes of progressively measurable functions with finite L p -norm.
5. Renormalized solutions. Let us assume that there exists a strong solution u to (1) in the sense of Theorem 2.1. We observe that for initial data u 0 merely in L 1 , the Itô formula for the square of the norm (see, e.g., [26]) can not be applied and consequently the natural a priori estimate for ∇u in we find that there exists a constant C(k) ≥ 0 depending on the truncation level k > 0, such that As in the deterministic case, the notion of renormalized solutions takes this information into account : (17) holds true for all t ∈ [0, T ] and a.s. in Ω. (iii) The following energy dissipation condition holds true: From the chain rule for Sobolev functions it follows that a.s. in Ω × Q T and therefore from (i) it follows that all the terms in (17) are welldefined. In general, for the renormalized solution u, ∇u may not be in L p (Ω × Q T ) d and therefore (iii) is an additional condition which can not be derived from (ii). However, for u ∈ L 1 (Ω × Q T ) satisfying (i), we can define a generalized gradient (still denoted by ∇u) by setting or equivalently, in differential form, , a.s. in Ω and for any S ∈ C 2 (R) with supp(S ) compact, and, since the right-hand side of (20) is in L 2 (D), also in L 2 (D).
Remark 5.2. If u is a renormalized solution to (1), thanks to (20), the Itô formula from Proposition 4.2 still holds true for S(u) for any S ∈ C 2 (R) with supp(S ) compact such that S(u) ∈ W 1,p 0 (D) a.s. in Ω × (0, T ). Indeed, in this case (3) is satisfied for the progressively measurable functions dr as a test function in (66), we may pass to the limit with l → ∞ and we find that u is a strong solution to (1). 6. Existence of renormalized solutions. In this Section, we fix u 0 ∈ L 1 (Ω × D) F 0 -measurable. Let (u n 0 ) n ⊂ L 2 (Ω × D) be an F 0 -measurable sequence such that |u n 0 | ≤ |u 0 | for all n ∈ N and lim n→∞ u n 0 = u 0 in L 1 (Ω × D) and in L 1 (D) for a.e. ω ∈ Ω. A possible choice is u n 0 = T n (u 0 ), n ∈ N. Theorem 6.1. There exists a renormalized solution to (1) with initial datum u 0 . Theorem 6.1 will be proved succesively in the following Lemmas. Lemma 6.2. For n ∈ N, let u n be a strong solution to (1) with respect to the initial value u n 0 . Then there exists an F t -adapted stochastic process u : for all n, m ∈ N a.s. in Ω and in L 1 (Ω). Thus, (u n ) n is a Cauchy sequence both in C([0, T ]; L 1 (D)) and in L 1 (Ω; C([0, T ]; L 1 (D))). Consequently, there exists u ∈ L 1 (Ω; C([0, T ]; L 1 (D))) satisfying (21). In particular, we have u n (t) → u(t) in L 1 (D) a.s. in Ω and for all t ∈ [0, T ]. As a limit function of a sequence of F t -measurable functions we may conclude that u(t) is F t -measurable.
In the following, we will show that the process u from Lemma 6.2 is the renormalized solution to (1) with initial datum u 0 in the sense of Definition 5.1. Lemma 6.3. For n ∈ N, let u n be a strong solution to (1) with respect to the initial value u n 0 . Let u be defined as in Lemma 6.2. Then, u satisfies (i) and (ii) from Definition 5.1.
Proof. Now, let u n be a strong solution to (1) with initial value u n 0 , i.e., for all t ∈ [0, T ] and a.s. in Ω. We apply the Itô formula introduced in Proposition 4.2 to equality (22). Therefore we know that for all holds true for all t ∈ [0, T ] and a.s. in Ω. In the following, passing to a suitable, not relabeled subsequence if necessary, and taking the limit for n → ∞, we will show that (23) is also satisfied by u and u 0 respectively and therefore (ii) from Definition 5.1 holds. First, we plug S(r) = r 0 T k (r) dr and ψ = 1 into (23) and taking expectation we get for all k > 0, all t ∈ [0, T ] and a.s. in Ω. The first term on the left hand side of (24) is nonnegative. The right-hand side of (24) can be majorized by and from (24) and (25) it follows that Since u n → u in L 1 (Ω; C([0, T ]; L 1 (D))), for n → ∞, passing to a not relabeled subsequence if necessary, u n → u a.s. in Ω × Q T for n → ∞. Since |T k (u n )| ≤ k a.s.
in Ω × Q T for all n ∈ N and any k > 0, from Lebesgue's dominated convergence theorem it follows that T k (u n ) → T k (u) for all k > 0 in L p (Ω × Q T ) for n → ∞. Recalling (26), we may conclude that, for k > 0 fixed, (T k (u n )) n is bounded in L p (Ω; L p (0, T ; W 1,p 0 (D))). From the convergence of (T k (u n )) n and from this boundedness we may conclude, passing to a not relabeled subsequence if necessary, that T k (u n ) T k (u) in L p (Ω; L p (0, T ; W 1,p 0 (D))) for n → ∞, and any fixed k > 0 which claims (i) of Definition 5.1.
Furthermore, from (26) it follows that |∇T k (u n )| p−2 ∇T k (u n ) is bounded in L p (Ω × Q T ) d . Consequently, for any fixed k > 0 there exists a not relabeled subsequence of n such that |∇T k (u n )| p−2 ∇T k (u n ) σ k in L p (Ω × Q T ) d .

NIKLAS SAPOUNTZOGLOU AND ALEKSANDRA ZIMMERMANN
Obviously, the proof of property (ii) in Definition 5.1 is done as far as we can show that )) for all k > 0. This will be done in the following technical lemmas that are inspired by Theorem 2 and Lemma 2 in [6].
Remark 6.4. In Lemma 6.5 and in the following we use the notation lim Proof. Using the Itô product rule (see Proposition 9.1) yields for all t ∈ [0, T ] and a.s. in Ω. Using t = T and passing to the limit yields lim n,m→∞ The rest of the proof is the same as the proof of [6], Theorem 2.
Lemma 6.6. For n ∈ N, let u n be a strong solution to (1) with respect to the initial value u n 0 . Let u be defined as in Lemma 6.2. Then, Especially, we have )) for n → ∞ and for all k > 0.
Proof. Since u n and u m are strong solutions to (1), we consider the difference of the corresponding equations. Using T k (u n − u m ) as a test function it yields DT k (u n (T ) − u m (T )) dx SinceT k is nonnegative we may conclude that lim n,m→∞ for all k > 0. We set

It is
Let us define θ k k (r) := T k+k (r) − T k (r).

NIKLAS SAPOUNTZOGLOU AND ALEKSANDRA ZIMMERMANN
The first term on the left hand side is nonnegative and the integrand of the second term on the right hand side can be estimated as follows . Multiplying by δ and passing to the limit with n → ∞ yields We can estimate that δθ which finally shows the validity of equality (28). Since equality (28) holds true, it follows that Minty's trick yields σ k = |∇T k (u)| p−2 ∇T k (u). We may conclude by using equality (30) that which ends the proof of Lemma 6.6.
For the proof of Theorem 6.1 is left to show that the energy dissipation condition (iii) from Definition 5.1 holds true. Proof. For fixed l > 0, let h l : R → R be defined as in Remark (5.2). We plug S(r) = r 0 h l (r)(T k+1 (r) − T k (r)) dr and Ψ ≡ 1 in (23) and take expectation to obtain where h l (u n )|∇u n | p dx ds, for all t ∈ [0, T ]. We can pass to the limit with l → ∞ in (32) by Lebesgue's Dominated Convergence theorem. We obtain where |∇u n | p dx ds, Since u n → u in L 1 (Ω; C([0, T ]; L 1 (D))) and u n 0 → u 0 in L 1 (Ω × D), for n → ∞, it follows that Now, the term J 3 desires our attention. For any σ > 0 we have Thanks to the convergence of (u n ), there exists a constant C ≥ 0 not depending on the parameters k, n and σ such that Thus, and therefore, passing to the limit with σ → ∞, from (36) and the nonnegativity of J 3 it follows that lim sup Combining (33), (34) and (37), and using the nonnegativity of J 2 , we arrive at (31).
We have for n → ∞ in L r (Ω × Q T ) for any 1 ≤ r < ∞ and a.e. in Ω × Q T . From Lemma 6.6 we recall that for any k > 0, for n → ∞, thus, passing to a not relabeled subsequence if necessary, also a.s. in Ω × Q T . Since ∇T k (u) = 0 a.s. on {|u| = m} for any m ≥ 0, from Fatou's Lemma it follows that and the energy dissipation condition (iii) follows combining (31) with (38).
7. Uniqueness of renormalized solutions. In the following, we formulate a contraction principle that yields immediately both uniqueness and continuous dependence on the initial values for renormalized solutions.
Theorem 7.1. Let u, v be renormalized solutions to (1) with initial data u 0 ∈ L 1 (Ω × D) and v 0 ∈ L 1 (Ω × D), respectively. Then we get Proof. This proof is inspired by the uniqueness proof in [8]. We know that u satisfies the SPDE for all S ∈ C 2 (R) such that supp S compact. Moreover, v satisfies an analogous SPDE. Subtracting both equalities yields in Ω. Now we set S(r) := T σ s (r) for r ∈ R and s, σ > 0 and define T σ s as follows: Firstly, we define for all r ∈ R Then we set T σ s (r) := r 0 (T σ s ) (τ ) dτ . Furthermore we have the weak derivative (T σ s ) (r) = − 1 σ sign(r), if s < |r| < s + σ, 0, otherwise.

NIKLAS SAPOUNTZOGLOU AND ALEKSANDRA ZIMMERMANN
For ω ∈ Ω and t ∈ [0, T ] fixed, we pass to the limit with σ → 0 firstly, then we pass to the limit k → 0 and finally we let s → ∞. Before we do so, we have to give some remarks on T σ s . By definition of (T σ s ) we see immediately that (T σ s ) (r) → χ {|r|≤s} pointwise for all r ∈ R as σ → 0. Since |(T σ s ) | ≤ 1 on R we have, (T σ s ) (u) → χ {|u|≤s} in L 1 (Q T ) a.s. in Ω and a.e. in Ω × Q T as σ → 0. An analogous result holds true for v instead of u.
For 0 < σ < 1 and fixed s > 0, we have supp(T σ s ) ⊂ [−s − 1, s + 1]. Therefore T σ s is bounded in L ∞ (R) for fixed s and we may conclude T σ s (u) → T s (u) a.e. in Ω × Q T and in L 1 (Q T ) a.s. in Ω as σ → 0. Furthermore, we have ∇T σ for σ → 0 a.s. in Ω. Let us consider I 1 . By Lebesgue's Theorem it follows a.s. in Ω, for all t ∈ [0, T ]. We write For 0 < σ < 1 we have and we want to show that there exists a sequence (s j ) j∈N ⊂ N such that s j → ∞ as j → ∞ and lim j→∞ lim sup According to Lemma 6 in [8] it is sufficient to show that for any s ∈ N there exists a nonnegative function F ∈ L 1 (Ω × (0, t) × D) such that for a.e. ω ∈ Ω where s is exchanged by s j in I 1 3 . For symmetry reasons, we only have to show the existence of a nonnegative function F ∈ L 1 (Ω × (0, t) × D) such that for fixed ω ∈ Ω To this end, we plug S(u(t)) = 1 σ u(t) 0 h l (τ )(T s+σ (τ ) − T s (τ )) dτ and Ψ ≡ 1 in the renormalized formulation for u to obtain a.s. in Ω, where h l (τ )(T s+σ (τ ) − T s (τ )) dτ dx, It is straightforward to pass to the limit with l → ∞ for a.e. ω ∈ Ω in L 1 , L 3 , L 4 , L 5 and L 6 . We have a.s. in Ω. In order to pass to the limit with l → ∞ in L 2 , we recall that from the energy dissipation condition (iii) it follows that, passing to a not relabeled subsequence if necessary, a.s. in Ω, where (T s+σ (τ ) − T s (τ )) dτ dx, We set F =: Moreover, a.e. in {|τ | = s} and from Lebesgue's dominated convergence theorem it follows that Using the Itô isometry and Hölder inequality we obtain Now let us consider the integrand of I 2 3 pointwise in Q t for a fixed w ∈ Ω. We have (T σ s ) (u) → 0 a.e. in Q T as σ → 0. Hence the whole integrand of I 2 3 tends to 0 a.e. in Q t as σ → 0. W.l.o.g. assume that u ≥ v at some point in Q t . Then we have in L 2 (Ω) for every t ∈ [0, T ]. Indeed, by Itô isometry and Lebesgue's Theorem it follows with similar arguments as in the proof of (46) that In the following we show that there exists a subsequence (s j ) j∈N ⊂ N with lim j→∞ s j = +∞ that may vary with ω ∈ Ω such that Since Φ 2 (ω) is a nonnegative, integrable function for a.e. ω ∈ Ω, the assertion follows from Lemma 6 in [8]. From (44) -(54) it follows that 8. Markov property. Note that it is possible to replace the starting time 0 by a starting time r ∈ [0, T ]. In this case, we consider the filtration starting at time r, i.e., (F t ) t∈[r,T ] . Then,β t := β t − β r , t ∈ [r, T ], is a Brownian motion with respect to (F t ) t∈[r,T ] such that σ(β t , t ≥ r) is independent of F r (see, e.g., Remark 3.2. in [2]). Moreover, the augmentationF t of σ(β t , t ≥ r) is right-continuous and independent of F r . Furthermore, we have dβ t = dβ t and all results and arguments still hold true in the case of a starting time r ∈ [0, T ] and F r -measurable initial conditions u r ∈ L 1 (Ω × D). In this section, we denote by u(t, r, u r ) the unique renormalized solution of (1) starting in u r at time r for t, r ∈ [0, T ] with r ≤ t and u r ∈ L 1 (Ω×D) F r -measurable.
Proposition 8.1. For all r, s, t ∈ [0, T ] with r ≤ s ≤ t and all u r ∈ L 1 (Ω × D) F r -measurable we have u(t, s, u(s, r, u r )) = u(t, r, u r ) a.s. in Ω.
The unique renormalized solution u(t) = u(t, r, u r ), t ∈ [r, T ], of (1) starting in u r at time r satisfies the Markov property in the following sense: For every bounded and B(L 1 (D))-measurable function G : L 1 (D) → R and all s, t ∈ [r, T ] with s ≤ t we have E[G(u(t))|F s ](ω) = E[G(u(t, s, u(s, r, u r )(ω)))] for a.e. ω ∈ Ω.
It is only left to prove that ψ is B(L 1 (D)) ⊗F t -measurable. Since G is B(L 1 (D)) measurable it is left to show that φ : L 1 (D) × Ω → L 1 (D), φ(x, ω) = u(t, s, x)(ω) is B(L 1 (D))⊗F t −B(L 1 (D))-measurable. To this end we show that φ is Carathéodory, i.e., Since it is possible to choose the filtrationF t instead of the filtration (F t ) t∈[s,T ] , Theorem 6.1 yields that for fixed x ∈ L 1 (D) the function u(t, s, x) isF t -measurable. Moreover, Theorem 7.1 yields that the mapping in (ii) is a contraction for almost every ω ∈ Ω, especially it is continuous. Now, Lemma 4.1. in [2] is applicable and yields the assertion.
For s, t ∈ [0, T ], s ≤ t and x ∈ L 1 (D) we set P s,t : where B b (L 1 (D)) denotes the space of all bounded Borel functions from L 1 (D) to R. Moreover, we set P t := P 0,t . As a consequence of Theorem 8.2 we obtain the Chapman-Kolmogorov property: Proof. Let r, s, t ∈ [0, T ], r ≤ s ≤ t, x ∈ L 1 (D) and ϕ ∈ B b (L 1 (D)). From Theorem 8.2 it follows that = E[P s,t (ϕ)(u(s, r, x))] = P r,s (P s,t (ϕ))(x).
Setting t = τ + s yields the assertion.
Proposition 8.6. The family P s,t , s, t ∈ [0, T ], s ≤ t, has the e-property in the sense of [22], i.e.: For all ϕ ∈ Lip b (L 1 (D)), x ∈ L 1 (D) and > 0 there exists δ > 0 such that for all z ∈ B(x, δ) and all 0 ≤ s ≤ t ≤ T : where Lip b (L 1 (D)) denotes the space of all bounded Lipschitz continuous functions from L 1 (D) to R.
This yields the assertion. 9. Appendix. In particular, (D n ) n∈N is an increasing sequence of domains in D such that D n ⊂⊂ D n+1 ⊂ D for all n ∈ N with n∈N D n = D.