This is the accepted version of the journal article: Balan, Raluca M.; Quer i Sardanyons, Lluís; Song, Jian. «Hölder continuity for the Parabolic Anderson Model with space-time homogeneous Gaussian noise». Acta Mathematica Scientia, Vol. 39, Issue 3 (May 2019), p. 717-730. DOI 10.1007/s10473-019-0306-3 This version is available at https://ddd.uab.cat/record/274597 under the terms of the license Hölder continuity for the Parabolic Anderson Model with space-time homogeneous Gaussian noise Raluca M. Balan∗ Lluı́s Quer-Sardanyons† Jian Song‡ July 15, 2018 Abstract In this article, we consider the Parabolic Anderson Model with constant initial condition, driven by a space-time homogeneous Gaussian noise, with general covari- ance function in time and spatial spectral measure satisfying Dalang’s condition. First, we prove that the solution (in the Skorohod sense) exists and is continuous in Lp (Ω). Then, we show that the solution has a modification whose sample paths are Hölder continuous in space and time, with optimal exponents, and under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered in the case of the white noise in time). This improves similar results which were obtained in [5, 9] under more restrictive conditions, and with sub-optimal exponents for Hölder continuity. MSC 2010: Primary 60H15; 60H07 Keywords: Gaussian noise; stochastic partial differential equations; Malliavin calculus 1 Introduction In this article, we consider the Parabolic Anderson Model on Rd , d ≥ 1, with constant initial condition:  ∂u 1   (t, x) = ∆u(t, x) + u(t, x)Ẇ (t, x), t > 0, x ∈ Rd , ∂t 2 (1)  u(0, x) = 1, x ∈ Rd .  ∗ Corresponding author. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, K1N 6N5, Canada. E-mail address: rbalan@uottawa.ca. Research sup- ported by a grant from the Natural Sciences and Engineering Research Council of Canada. † Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Catalonia, Spain. E-mail address: quer@mat.uab.cat. Research supported by the grant MTM2015- 67802P. ‡ Department of Mathematics. University of Hong Kong. E-mail address: txjsong@hku.hk. 1 The noise W is given by a zero-mean Gaussian process {W (ϕ); ϕ ∈ D(Rd+1 )} defined on a complete probability space (Ω, F, P ), with covariance Z E[W (ϕ1 )W (ϕ2 )] = γ(t − s)f (x − y)ϕ1 (t, x)ϕ2 (s, y)dxdydtds =: J(ϕ1 , ϕ2 ), R2 ×R2d where γ : R → [0, ∞] and f : Rd → [0, ∞] are continuous, symmetric, locally integrable functions, such that γ(t) < ∞ if and only if t 6= 0; f (x) < ∞ if and only if x 6= 0. Here D(R ) is the space of C ∞ -functions on Rd+1 with compact support. We denote by d+1 H the completion of D(Rd+1 ) with respect to h·, ·iH defined by hϕ1 , ϕ2 iH = J(ϕ1 , ϕ2 ). We assume that f is non-negative-definite (in the sense of distributions), i.e. Z (ϕ ∗ ϕ)(x)f e (x)dx ≥ 0, for all ϕ ∈ S(Rd ), Rd where ϕ(x) e = ϕ(−x) and S(Rd ) is the space of rapidly decreasing C ∞ -functions on Rd . By the Bochner-Schwartz theorem, there exists a tempered measure µ on Rd such that f = Fµ, where Fµ denotes the Fourier transform of µ in the space SC0 (Rd ) of C-valued tempered distributions on Rd , i.e. Z Z f (x)ϕ(x)dx = Fϕ(ξ)µ(dξ) for all ϕ ∈ SC (Rd ), Rd Rd where SC (R d is the space of C-valued rapidly decreasing C ∞ -functions on Rd , and R ) −iξ·x Fϕ(x) = Rd e ϕ(x)dx is the Fourier transform of ϕ. Here ξ · x denotes the scalar d product in R . Similarly, we assume that γ is non-negative-definite (in the sense of dis- tributions), and so there exists a tempered measure ν on R such that γ = Fν in SC0 (R). We denote by G the fundamental solution of the heat equation on Rd : |x|2   1 G(t, x) = exp − , t > 0, x ∈ Rd , (2πt)d/2 2t where | · | denotes the Euclidean norm. Note that G(t, ·) ∈ S(Rd ) and t|ξ|2   FG(t, ·)(ξ) = exp − for all ξ ∈ Rd . 2 Similarly to the wave equation in spatial dimension d ≤ 2, we have the following definition of the solution (see Definition 5.1 of [2]). Definition 1.1. A square-integrable process u = {u(t, x); t ≥ 0, x ∈ Rd } with u(0, x) = 1 for all x ∈ Rd is a (mild Skorohod) solution of equation (1) if u has a jointly measurable modification (denoted also by u) such that sup(t,x)∈[0,T ]×Rd E|u(t, x)|2 < ∞ for all T > 0, and for any t > 0 and x ∈ Rd , the following equality holds in L2 (Ω): Z tZ u(t, x) = 1 + G(t − s, x − y)u(s, y)W (δs, δy), (2) 0 Rd where the stochastic integral is understood in the Skorohod sense and the process v (t,x) = {v (t,x) (s, y) = 1[0,t] (s)G(t − s, x − y)u(s, y); s ≥ 0, y ∈ Rd } is Skorohod integrable. 2 We refer the reader to [5, 2, 1] for more details regarding the concept of Skorohod solution, and to [7] for the background on Malliavin calculus. From [5], we know that equation (1) has a unique Skorohod solution u, provided that Dalang’s condition holds: Z 1 2 µ(dξ) < ∞. (3) Rd 1 + |ξ| We should point our that this condition was considered for the first time in [4] for equations driven by Gaussian noise which was white in time and had the same spatial covariance structure as above. A Feynman-Kac representation for the solution u was obtained in [5], under additional conditions on γ and µ. Theorem 4.9 of [5] shows that if the function γ satisfies: 0 ≤ γ(t) ≤ Cβ |t|−β for all t ∈ R, (4) for some β ∈ (0, 1), and the measure µ satisfies: Z  1−β−α 1 µ(dξ) < ∞ for some α ∈ (0, 1 − β), Rd 1 + |ξ|2 then for any p ≥ 2 and T > 0, there exists a constant C > 0 depending on p, T, β and α such that for any t, t0 ∈ [0, T ] and x, x0 ∈ Rd , α ku(t, x) − u(t0 , x0 )kp ≤ C |t − t0 | 2 + |x − x0 |α ,  and therefore, on any compact set [0, T ] × K, u has a modification which is θ1 -Hölder continuous in time for any θ1 ∈ (0, α2 ), and θ2 -Hölder continuous in space for any θ2 ∈ (0, α). This result is based on the Feynman-Kac representation of the solution, and does not yield optimal exponents for the order of Hölder continuity (see Remark 4.10 of [5]). The goal of the present article is to obtain the optimal exponents for the order of Hölder continuity of the solution u, under the following minimal condition on µ: Z  η 1 µ(dξ) < ∞ for some η ∈ (0, 1). (5) Rd 1 + |ξ|2 We note that this condition is the same as for the white noise in time: see Theorem 2.1 of [8] for the case when the initial condition is given by a bounded Hölder continuous function, and Theorem 1.8 of [6] for the case when the initial condition is given by a measure. We emphasize that do not impose any constraints on the temporal covariance function γ. More precisely, in Theorem 3.2 below, we show that if the measure µ satisfies (5), then u has a modification which which is θ1 -Hölder continuous in time for any θ1 ∈ (0, 1−η 2 ), and θ2 -Hölder continuous in x for any θ2 ∈ (0, 1 − η). The proof of this theorem relies on a careful analysis of the p-th moments of the increments of the solution u, which is related to the problem of continuity of u in Lp (Ω). Therefore, in Theorem 2.2 below, we show that under condition (3), u is Lp (Ω)-continuous on [0, ∞) × Rd for any p ≥ 2. A special effort is dedicated to the left continuity in time, a delicate issue for the heat equation, which is usually left out in the literature. 3 Theorem 5.9 of [9] gives the Hölder continuity of the solution to equation (1) under condition (5), with the same exponent θ2 < 1 − η in space as in our Theorem 3.2, but with exponent θ1 < [(1 − η) ∧ (1 − β)]/2 in time, where β is given by (4). Our proofs use simplified versions of the arguments contained in [1] for the heat equa- tion with initial condition given by a measure. But the results that we present here do not follow directly from Theorems 1.1.(a) and 1.4 of [1] whose conclusions are valid for time intervals which do not include 0. The novelty of our results stems from the fact that they are valid for time intervals of the form [0, T ]. This article is organized as follows. In Section 2, we prove the existence and uniqueness of the Skorohod solution u, and the fact that u is continuous in Lp (Ω). In Section 3, we show that u has a modification whose sample paths are Hölder continuous with optimal exponents, as mentioned above. 2 Existence of solution and continuity in Lp(Ω) In this section, we show the existence and uniqueness of the solution to equation (1) and its continuity in Lp (Ω) on [0, ∞) × Rd , using the method of [1] for the heat equation with initial condition given by a measure. We note that the existence of solution to equation (1) was established also in Theorem 3.2 of [5], using a different method. Intuitively, if it exists, the solution to equation (1) has the Wiener chaos expansion: X u(t, x) = 1 + In (fn (·, t, x)), (6) n≥1 where In is the multiple Wiener integral of order n with respect to W , and fn (t1 , x1 , . . . , tn , xn , t, x) = G(t − tn , x − xn ) . . . G(t2 − t1 , x2 − x1 )1{0 0, we let Z k(t) = |FG(t, ·)(ξ)|2 µ(dξ), Rd h0 (t) = 1 and for n ≥ 1, Z hn (t) = k(t2 − t1 ) . . . k(tn − tn−1 )k(t − tn )dt1 . . . dtn . 0 0 and t > 0, we let X Xp H(t; γ) = γ n hn (t) and H(t; e γ) = γ n hn (t). n≥0 n≥0 By Lemma 3.8 of [1], we know that under Dalang’s condition, H(t; γ) < ∞ and H(t; e γ) < ∞ for any t > 0 and γ > 0. The following result is a particular case of Lemma 3.4 of [1]. Lemma 2.1. For any t > 0, Z sup |FG(t, ·)(ξ + η)|2 µ(dξ) = k(t). η∈Rd Rd We denote by k · kp the norm in Lp (Ω). We will use frequently the fact that the k · kp -norms are equivalent on a fixed Wiener chaos space Hn . More precisely, kF kp ≤ (p − 1)n/2 kF k2 for all F ∈ Hn (7) (see the last line of page 62 of [7]). Theorem 2.2. Under condition (3), the series on the right-hand side of (6) converges in L2 (Ω) and the process u given by (6) is the unique solution to equation (1). Moreover, for any p ≥ 2, u is Lp (Ω)-continuous on [0, ∞) × Rd , and sup ku(t, x)kp < ∞, f or all T > 0. (t,x)∈[0,T ]×Rd Proof. Step 1. (Summability of the series) We first show that the series on the right-hand side of (6) converges in L2 (Ω), which is equivalent to: X 1 αn (t) < ∞. n≥0 n! By relation (4.15) in [2], Z αn (t) ≤ Γnt ψn (t, t)dt, [0,t]n Rt where t = (t1 , . . . , tn ), Γt = 2 0 γ(s)ds, Z ψn (t, t) = |FG(u1 , ·)(ξ1 )|2 . . . |FG(un−1 , ·)(ξ1 + . . . + ξn−1 )|2 Rnd × |FG(un , ·)(ξ1 + . . . + ξn )|2 µ(dξ1 ) . . . µ(dξn ), (8) uj = tρ(j+1) − tρ(j) for j = 1, . . . , n, tρ(n+1) = t, and ρ is the permutation of {1, . . . , n} such that tρ(1) < . . . < tρ(n) . Note that n Z ! n Y Y 2 ψn (t, t) ≤ sup |FG(uj , ·)(ξj + η)| µ(dξj ) = k(uj ), j=1 η∈Rd Rd j=1 5 where the last equality follows by Lemma 2.1. Hence, XZ Yn n αn (t) ≤ Γt k(tρ(j+1) − tρ(j) )dt ρ∈Sn 0 0 kJn (t + h, x) − Jn (t, x)kp ≤ (p − 1)n/2 kJn (t + h, x) − Jn (t, x)k2  1/2   n/2 2 ≤ (p − 1) An (t, h) + Bn (t, h) , (9) n! where An (t, h) = (n!)2 kfen (·, t + h, x)1[0,t]n − fen (·, t, x)k2H⊗n Bn (t, h) = (n!)2 kfen (·, t + h, x)1[0,t+h]n \[0,t]n k2H⊗n . Note that An (t, h) and Bn (t, h) do not depend on x. We first treat An (t, h). Using relations (7.6) and (7.7) of [2], we see that Z n (n) An (t, h) ≤ Γt ψt,h (t, t)dt, (10) [0,t]n 6 where t = (t1 , . . . , tn ), Z (n) ψt,h (t, t) = |FG(u1 , ·)(ξ1 )|2 . . . |FG(un−1 , ·)(ξ1 + . . . + ξn−1 )|2 Rnd × |F[G(un + h, ·) − G(un , ·)](ξ1 + . . . + ξn )|2 µ(dξ1 ) . . . µ(dξn ), (11) and u1 , . . . , un are the same as in Step 1 above. We use the fact that  2 2 2 |F[G(un + h, ·) − G(un , ·)](ξ1 + . . . + ξn )|2 = e−(un +h)|ξ1 +...+ξn | /2 − e−un |ξ1 +...+ξn | /2  2 2 −h|ξ1 +...+ξn |2 /2 = |FG(un , ·)(ξ1 + . . . + ξn )| 1 − e converges to 0 when h → 0, and is bounded by |FG(un , ·)(ξ1 + . . . + ξn )|2 (since h > 0) . Note that Z n Y 2 2 |FG(u1 , ·)(ξ1 )| . . . |FG(un , ·)(ξ1 + . . . + ξn )| µ(dξ1 ) . . . µ(dξn ) ≤ k(uj ). Rnd j=1 (n) (n) Qn theorem, ψt,h (t, t) → 0 as h → n0. Moreover, ψt,h (t, t) is By the dominated convergence bounded by the function j=1 k(uj ), which is integrable on [0, t] , its integral being equal to n!hn (t). By the dominated convergence theorem, An (t, h) → 0 as h → 0. Next, we treat Bn (t, h). Let Dt,h = [0, t + h]n \[0, t]n . Using relation (7.10) of [2], we see that Z n (n) Bn (t, h) ≤ Γt+h γt,h (t, t)1Dt,h (t)dt, (12) [0,t+h]n where Z (n) γt,h (t, t) = |FG(u1 , ·)(ξ1 )|2 . . . |FG(un−1 , ·)(ξ1 + . . . + ξn−1 )|2 Rnd |FG(un + h)(ξ1 + . . . + ξn )|2 µ(dξ1 ) . . . µ(dξn ). (13) We use the fact that 2 |FG(un + h, ·)(ξ1 + . . . + ξn )|2 = |FG(un , ·)(ξ1 + . . . + ξn )|2 e−h|ξ1 +...+ξn | converges to |FG(un , ·)(ξ1 + . . . + ξn )|2 as h → 0, and is bounded by |FG(un , ·)(ξ1 + . . . + ξn )|2 (since h > 0). By the dominated convergence theorem, Z (n) lim γt,h (t, t) = |FG(u1 , ·)(ξ1 )|2 . . . |FG(un−1 , ·)(ξ1 + . . . + ξn−1 )|2 h→0 Rnd × |FG(un , ·)(ξ1 + . . . + ξn )|2 µ(dξ1 ) . . . µ(dξn ). (n) But 1Dt,h → 0 as h → 0 and γt,h (t, t)1Dt,h is bounded by the function nj=1 k(uj ), which Q is integrable on [0, t]n . By the dominated convergence theorem, Bn (t, h) → 0 as h → 0. We consider now the left-continuity in time of Jn . We will prove that for any t > 0, lim kJn (t − h, x) − Jn (t, x)kp = 0 uniformly in x ∈ Rd . h↓0 7 We argue as in Step 2 of the proof of Lemma B.3 of [1]. By (7), for any t > 0, x ∈ Rd and h ∈ (0, t) kJn (t − h, x) − Jn (t, x)kp ≤ (p − 1)n/2 kJn (t − h, x) − Jn (t, x)k2  1/2   n/2 2 0 0 ≤ (p − 1) A (t, h) + Bn (t, h) , (14) n! n where A0n (t, h) = (n!)2 kfen (·, t − h, x) − fen (·, t, x)1[0,t−h]n k2H⊗n Bn0 (t, h) = (n!)2 kfen (·, t, x)1[0,t]n \[0,t−h]n k2H⊗n . Note that A0n (t, h) and Bn0 (t, h) do not depend on x. We first treat A0n (t, h). Note that Z n Z (n)0 (n)0 Y A0n (t, h) = γ(tj − sj )ψt,h (t, s)dtds ≤ Γnt−h ψt,h (t, t)dt, [0,t−h]2n j=1 [0,t−h]n where Z (n)0 (n) (n) ψt,h (t, s) = F[gt (·, t, x) − gt (·, t − h, x)](ξ1 , . . . , ξn ) Rnd (n) (n) × F[gs (·, t, x) − gs (·, t − h, x)](ξ1 , . . . , ξn ) µ(dξ1 ) . . . µ(dξn ) (n) and gt (·, t, x) = n!fen (t1 , ·, . . . , tn , ·, t, x). Let (t1 , . . . , tn ) ∈ [0, t − h]n and ρ be the permutation of 1, . . . , n such that tρ(1) < . . . < tρ(n) . Let uj = tρ(j+1) − tρ(j) for any j = 1, . . . , n − 1. Note that (n) Fgt (·, t − h, x)(ξ1 , . . . , ξn ) = e−i(ξ1 +...+ξn )·x FG(u1 , ·)(ξρ(1) ) . . . FG(un−1 , ·)(ξρ(1) + . . . + ξρ(n−1) ) × FG(t − h − tρ(n) , ·)(ξρ(1) + . . . + ξρ(n) ) and (n) Fgt (·, t, x)(ξ1 , . . . , ξn ) = e−i(ξ1 +...+ξn )·x FG(u1 , ·)(ξρ(1) ) . . . FG(un−1 , ·)(ξρ(1) + . . . + ξρ(n−1) ) × FG(t − tρ(n) , ·)(ξρ(1) + . . . + ξρ(n) ). 8 Let un = t − tρ(n) . Note that un − h > 0 since tρ(n) < t − h. It follows that Z (n)0 ψt,h (t, t) = |FG(u1 , ·)(ξ1 )|2 . . . |FG(un−1 , ·)(ξ1 + . . . + ξn−1 )|2 Rnd × |F[G(un − h, ·) − G(un , ·)](ξ1 + . . . + ξn )|2 µ(dξ1 ) . . . µ(dξn ) n−1 Z ! Y ≤ sup |FG(uj , ·)(ξj + η)|2 µ(dξj ) j=1 η∈Rd Rd Z × sup |FG(un − h, ·)(ξn + η) − FG(un , ·)(ξn + η)|2 µ(dξn ) η∈Rd Rd n−1 Y Z = k(uj ) · sup |FG(un − h, ·)(ξn + η) − FG(un , ·)(ξn + η)|2 µ(dξn ). j=1 η∈Rd Rd Using the inequality (1 − e−x )2 ≤ 1 − e−x ≤ min(x, 1), we obtain: 2 2 2 |FG(un − h, ·)(ξ) − FG(un , ·)(ξ)|2 = e−(un −h)|ξ| 1 − e−h|ξ| /2 h|ξ|2   −(un −h)|ξ|2 ≤e min ,1 2 2   −(un −h)|ξ|2 h|ξ| −(un −h)|ξ|2 = min e ,e . 2 We bound separately the two terms appearing in the minimum above. For the first term, we use inequality (B.27) of [1] which we recall below: for any A > 0 and x ≥ 0,   2  2 2 −1 A 2 exp −Ax x ≤ A exp − x . e 2 We obtain: h −(un −h)|ξ|2 2 1 h 2 e |ξ| ≤ · e−(un −h)|ξ| /2 . 2 e un − h 2 2 For the second term, we use the fact that e−(un −h)|ξ| ≤ e−(un −h)|ξ| /2 . Hence,   1 h 2 2 |FG(un − h, ·)(ξ) − FG(un , ·)(ξ)| ≤ min · , 1 e−(un −h)|ξ| /2 e un − h and Z sup |FG(un − h, ·)(ξ + η) − FG(un , ·)(ξ + η)|2 η∈Rd Rd   Z 1 h 2 ≤ min · , 1 sup e−(un −h)|ξ+η| /2 µ(dξ) e un − h η∈Rd Rd     1 h un − h = min · ,1 k , e un − h 2 9 using Lemma 2.1 for the last equality. It follows that n−1     t − tn − h Z 0 n Y 1 h An (t, h) ≤ Γt−h n! k(tj+1 − tj ) min · ,1 k dt 0 0, h > 0 and z ∈ Rd , Z t sup |FG(t + h, ·)(ξ + η) − FG(t, ·)(ξ + η)|2 µ(dξ) ≤ Chθ t−θ k , (18) η∈Rd Rd 2 Z t sup |e−i(ξ+η)·z − 1|2 |FG(t, ·)(ξ + η)|2 µ(dξ) ≤ C|z|2θ t−θ k , (19) η∈Rd Rd 2 where R θ = 1 − η, C > 0 is a constant depending only on θ, and we recall that k(t) = Rd |FG(t, ·)(ξ)|2 µ(dξ). Proof. We use the same estimates as in the proof of Theorem 1.4 of [1]. More precisely, we will use inequality (5.2) of [1] which we recall below: for any A > 0,   2 2θ −θ A 2 exp(−Ax )x ≤ Cθ A exp − x for all x ≥ 0, (20) 2 where Cθ > 0 is a constant depending on θ. Using the fact that (1−e−x )2 ≤ (1−e−x )θ ≤ xθ  2 2 for any x ≥ 0, it follows that 1 − e−h|ξ| /2 ≤ 2−θ hθ |ξ|2θ . Hence, for any ξ ∈ Rd , 2  2 2 2 |FG(t + h, ·)(ξ) − FG(t, ·)(ξ)|2 = e−t|ξ| 1 − e−h|ξ| /2 ≤ 2−θ hθ e−t|ξ| |ξ|2θ 2 /2 ≤ 2−θ Cθ hθ t−θ e−t|ξ| = 2−θ Cθ hθ t−θ |FG(t/2, ·)(ξ)|2 It follows that Z sup |FG(t + h, ·)(ξ + η) − FG(t, ·)(ξ + η)|2 µ(dξ) η∈Rd Rd Z −θ θ −θ ≤ 2 Cθ h t sup |FG(t/2, ·)(ξ + η)|2 µ(dξ) η∈Rd Rd = 2−θ Cθ hθ t−θ k(t/2), where the last equality is due to Lemma 2.1. This proves (18). To prove (19), we use the inequality: for any x ∈ R, |1 − eix |2 = 2(1 − cos x)2 ≤ Kθ |x|2θ , where Kθ > 0 is a constant depending on θ. Combining this with Cauchy-Schwarz inequality |ξ · z| ≤ |ξ||z| and inequality (20), we obtain that for any ξ ∈ Rd , 2 2 /2 |e−iξ·z − 1|2 |FG(t, ·)(ξ)|2 ≤ Kθ |z|2θ |ξ|2θ e−t|ξ| ≤ Kθ Cθ |z|2θ t−θ e−t|ξ| . 11 Hence Z Z −i(ξ+η)·z 2θ −θ 2 /2 sup |e 2 2 − 1| |FG(t, ·)(ξ + η)| µ(dξ) ≤ Kθ Cθ |z| t sup e−t|ξ+η| µ(dξ) η∈Rd Rd η∈Rd Rd and relation (19) follows by Lemma 2.1, exactly as above. From Lemma 5.1 of [1], we know that condition (5) is equivalent to Z 1 k(s) 1−η ds < ∞. (21) 0 s Theorem 3.2. Under condition (5), for any p ≥ 2 and T > 0, there exists a constant C > 0 depending on p, T and η such that for any t, t0 ∈ [0, T ] and x, x0 ∈ Rd , 1−η ku(t, x) − u(t0 , x0 )kp ≤ C |t − t0 | + |x − x0 |1−η .  2 Consequently, for any T > 0 and for any compact set K ⊂ Rd , the solution {u(t, x); t ∈ [0, T ], x ∈ K} to equation (1) has a modification which is θ1 -Hölder continuous in time for any θ1 ∈ (0, 1−η 2 ), and θ2 -Hölder continuous in space for any θ2 ∈ (0, 1 − η). Proof. We argue as in the proof of Theorem 8.3 of [2]. Let θ = 1 − η. Step 1. (increments in time) Assume that t0 > t and let h = t0 − t. By Minkowski’s inequality and (14), X ku(t + h, x) − u(t, x)kp ≤ kJn (t + h, x) − Jn (t, x)kp n≥1  1/2 X n/2 2  ≤ (p − 1) An (t, h) + Bn (t, h) . (22) n≥1 n! We first treat An (t, h). Recall estimate (10) for An (t, h) that we obtained above. Using (n) definition (11) of ψt,h (t, t), followed by Lemma 2.1 and (18), we see that n−1 Z ! (n) Y 2 ψt,h (t, t) ≤ sup |FG(uj , ·)(ξj + η)| µ(dξj ) j=1 η∈Rd Rd Z × sup |FG(un + h, ·)(ξn + η) − FG(un , ·)(ξn + η)|2 µ(dξn ) η∈Rd Rd n−1 Y ≤ Ch θ k(uj )u−θ n k(un /2). j=1 12 Rt We let ρt = 0 s−θ k(s)ds. Then n−1 t − tρ(n) XZ Y   θ −θ An (t, h) ≤ Ch Γnt k(tρ(j+1) − tρ(j) )(t − tρ(n) ) k dt ρ∈Sn 0 0, there exists a constant CT > 0 depending on T such that  1/2 X n/2 1 (1) (p − 1) An (t, h) ≤ CT hθ/2 . (23) n≥1 n! Next, we treat Bn (t, h). Recall estimate (12) for Bn (t, h) that we obtained above. (n) Using definition (13) of γt,h (t, t), followed by Lemma 2.1, we see that n−1 Z ! (n) Y 2 γt,h (t, t) ≤ sup |FG(uj , ·)(ξj + η)| µ(dξj ) j=1 η∈Rd Rd Z n−1 Y 2 × sup |FG(un + h, ·)(ξn + η)| µ(dξn ) = k(uj )k(un + h), η∈Rd Rd j=1 Note that Γt+h ≤ ΓT since t + h = t0 ≤ T . We observe that if (t1 , . . . , tn ) ∈ Dt,h then there exists at least one index i with ti > t. So, [ Dt,h = {(t1 , . . . , tn ); 0 ≤ tρ(1) ≤ . . . ≤ tρ(n) , t < tρ(n) ≤ t + h}. ρ∈Sn 13 It follows that XZ t+h Z n−1 Y Bn (t, h) ≤ ΓnT k(tρ(j+1) − tρ(j) )k(t − tρ(n) + h)dt ρ∈Sn t 0 0 is a constant depending on T . Step 2. (increments in space) Let z = x0 − x. By Minkowski’s inequality and (15), X ku(t, x + z) − u(t, x)kp ≤ kJn (t, x + z) − Jn (t, x)kp n≥1  1/2 X n/2 1 ≤ (p − 1) Cn (t, z) . (25) n≥1 n! (n) Recall estimate (16) for Cn (t, h) that we obtained above. Using definition (17) of ψt,z (t, t), followed by Lemma 2.1 and Proposition 3.1, we see that n−1 Z ! 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