FRACTIONAL STOCHASTIC HEAT EQUATION WITH PIECEWISE CONSTANT COEFFICIENTS arXiv:1910.12655v1 [math.PR] 28 Oct 2019 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR Abstract. We introduce a fractional stochastic heat equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution. 1. Introduction In the last years, stochastic partial differential equations (SPDEs) driven by different types of fractional noises have attracted a great attention of the probability community. In several cases, the noise is expressed as a function of a formal derivative of a one-dimensional fractional Brownian motion. A one-dimensional fractional Brownian motion (fBm) (B H (t))t≥0 with Hurst index H ∈ (0, 1) is a centered Gaussian process on some probability space (Ω, F , P) with covariance function  1  2H (1.1) t + s2H − |t − s|2H . E(B H (t)B H (s)) = 2 So, an fBm is a natural extension of a standard Brownian motion because taking H = 12 in (1.1) we get E(B 1/2 (t)B 1/2 (s)) = s ∧ t. Firstly introduced by Kolmogorov in [9], since the appearance of the paper [13], the interest in this process has increased enormously as an important ingredient of fractal models because of such characteristics as self-similarity, Hölder continuity and long-range dependence. Theory of stochastic calculus with respect to fBm has been developed, leading to the consideration of several types of SPDEs driven by a noise depending, in one way or another, on a fBm (e. g. [1–4, 8, 17, 18, 20] and references therein). In this paper, we consider the following SPDE: (
du(t, x) = Lu(t, x) dt + h u(t, x) W H (dt, x), t ∈ (0, T ], x ∈ R, (1.2) u(0, x) = 0, x ∈ R, where h is an affine function, h(z) = h1 z + h2 , (1.3) h1 , h2 ∈ R and the operator L is defined by   1 d d L= ρ(x)A(x) . 2ρ(x) dx dx Here the coefficients A and ρ have the following form A(x) = a1 1{x≤0} + a2 1{x>0} and ρ(x) = ρ1 1{x≤0} + ρ2 1{x>0} , 2010 Mathematics Subject Classification. 60G22, 60H15, 35R60. Key words and phrases. Stochastic partial differential equation; discontinuity of coefficients; fundamental solution; mild solution; infinite-dimensional fractional Brownian motion. 1 2 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR df ai , ρi (i = 1, 2) are positive constants, and dx denotes the derivative of f in the distributional sense. The term W H refers to an L2 (R)-valued fractional Brownian motion with Hurst index H ∈ ( 12 , 1), defined by W H (·, t) := ∞ X λj ej (·) BjH (t), (1.4) j=1 where BjH = {BjH (t), t ≥ 0}, j ∈ N is a sequence of one-dimensional fractional Brownian motions with the Hurst index H ∈ (1/2, 1) starting at the origin, {λj , j ∈ N} is a sequence of positive real numbers and {ej , j ∈ N} is an orthonormal basis of L2 (R), such that: sup kej k∞ < ∞, and j ∞ X j=1 λj < ∞, (1.5) where k · k∞ denotes the norm in L∞ (R). The series (1.4) converges a. s. in L2 (R) because of the assumptions (1.5), see [15]. Equation (1.2) can be considered as a stochastic counterpart of the parabolic equation ∂u(t, x) = Lu(t, x), (1.6) ∂t which arises in mathematical modelling of diffusion phenomena in many areas, such as ecology [5], biology [14] etc. The non-smoothness of the coefficients reflects the heterogeneity of the medium in which the process under study propagates. An explicit expression of the fundamendal solution of equation (1.6) was given in [6, 22–24]. SPDEs with the operator L have been introduced and investigated in [25, 26], with an additive noise W defined as a centered Gaussian field W = {W (t, C); t ∈ [0, T ], C ∈ Bb (R)} with covariance E(W (t, C)W (s, D)) = (t ∧ s)λ(C ∩ D), where λ denotes the Lebesgue measure. Comparing to those two articles, the equation (1.2) contains more complicated noise, which is multiplicative and fractional. Its study requires more sophisticated stochastic and Hilbert analysis tools. Other forms of SPDE (1.2), with other different operators and similar or different Gaussian noises, have been recently studied by many authors (e. g. [17, 19–21] and references therein). We make here a first step in the study of SPDEs of the form (1.2). More precisely, we prove existence and uniqueness of a mild solution to equation (1.2). In addition to the introduction of a Besov-type Banach space and the use of the Riemann– Liouville fractional derivatives, a part of the theory of generalized Lebesgue–Stieljes integration with respect to fBm with Hurst index H > 1/2 is employed, and the proofs require many integration techniques, calculation and analysis tools; they are particularly based on a deep characterization of the explicit form of the fundamental solution of (1.6). The paper is organized as follows. In the second section we present some useful characterizations and upper bounds of the fundamental solution of equation (1.6), and the third one is devoted to the proof of the existence and uniqueness of a mild solution to equation (1.2). Some technical lemmata are proved in the appendix. 2. Some properties of the fundamental solution The fundamental solution of the deterministic partial differential equation (1.6) is given by FRACTIONAL STOCHASTIC HEAT EQUATION " 3    1{y>0} 1{y≤0} (f (x) − f (y))2 + √ exp − √ a1 a2 2(t − s)   # (| f (x) | + | f (y) |)2 + β sign(y) exp − 1{s<t} , 2(t − s) 1 G(t − s, x, y) = p 2π(t − s)  with y y f (y) = √ 1{y≤0} + √ 1{y>0} , a1 a2 α = 1− √ √ a1 + a2 (α − 1) and β = √ . √ a1 − a2 (α − 1) ρ1 a 1 , ρ2 a 2 For the proof see, e. g., [22, 23] and [6]. The fundamental solution G satisfies the following properties. Lemma 2.1. For every 0 ≤ s < t ≤ T and x, y ∈ R, we have   1 (f (x) − f (y))2 , exp − |G(t − s, x, y)| ≤ Ca1 ,a2 √ 2(t − s) t−s   √ √1 + √1 with Ca1 ,a2 := 1+|β| . a a 1 2 2π Proof. We have for every x, y ∈ R and 0 ≤ s < t ≤ T ,     1{y>0} 1{y≤0} (f (x) − f (y))2 1 exp − + √ |G(t − s, x, y)| = p √ a1 a2 2(t − s) 2π(t − s)   (|f (x)| + |f (y)|)2 + β sign(y) exp − 2(t − s)     1 (f (x) − f (y))2 1 1 ≤p exp − √ +√ a1 a2 2(t − s) 2π(t − s)   2 (|f (x)| + |f (y)|) . + |β| exp − 2(t − s) Since the function x 7→ exp(−x2 ) is decreasing on the interval [0, +∞), we have !   (|f (x)| + |f (y)|)2 (f (x) − f (y))2 exp − ≤ exp − , 2(t − s) 2(t − s) and consequently (1 + |β|) |G(t − s, x, y)| ≤ p 2π(t − s)  1 1 √ +√ a1 a2    (f (x) − f (y))2 . exp − 2(t − s)  Corollary 2.1. For every 0 ≤ s < t ≤ T x ∈ R and y ∈ R, we have Z Z  max |G(t − s, z, y)| dz, |G(t − s, x, z)| dz ≤ C1 (a1 , a2 ), R with C1 (a1 , a2 ) =  R √1 a1 + √1 a2  √ √ (1 + |β|) max( a1 , a2 ). Proof. : By the virtue of Lemma 2.1, we get:  Z   Z (1 + |β|) 1 (f (z) − f (y))2 1 |G(t − s, z, y)| dz ≤ p dz. exp − √ +√ a1 a2 2(t − s) 2π(t − s) R R Further, !   Z 0 Z ( √za1 − f (y))2 (f (z) − f (y))2 dz = exp − dz exp − 2(t − s) 2(t − s) −∞ R 4 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR + Z 0 √z a By the change of variables U = √i −f (y) 2(t−s) ∞ exp − ( √za2 − f (y))2 2(t − s) dz , 2(t−s) ai =⇒ dU = √ ! dz. with i = 1 in the first integral and i = 2 in the second one, we obtain:   Z Z p √ √ (f (z) − f (y))2 dz ≤ max( a1 , a2 ) 2(t − s) exp(−U 2 ) dU exp − 2(t − s) R R p √ √ = max( a1 , a2 ) 2(t − s) π. (2.1) Thus, Z R |G(t − s, z, y)| dz ≤  1 1 √ +√ a1 a2  √ √ (1 + |β|) max( a1 , a2 ). Concerning the second integral, also by the virtue of Lemma 2.1 and (2.1), we get:  Z   Z 1 (f (x) − f (z))2 1 (1 + |β|) dz + exp − |G(t − s, x, z)| dz ≤ p √ √ a1 a2 2(t − s) 2π(t − s) R R   √ √ 1 1 ≤ √ +√ (1 + |β|) max( a1 , a2 ). a1 a2  Lemma 2.2. (i) For every η > 0, there exists a strictly positive constant C such that, for every 0 < t ≤ T and x, y ∈ R, Z Z  1 3 η η max |∂t G(t, z, y)| dz, |∂t G(t, x, z)| dz ≤ Ct− 2 η+ 2 . R R (ii) For every η > 0, there exists a strictly positive constant C such that, for every 0 ≤ s < t ≤ T and y ∈ R, Z η 1 5 ∂2 G(t − s, z, y) dz ≤ C(t − s)− 2 η+ 2 . ∂t∂s R Proof. (i) For every x, y ∈ R, we have   "   1 1 (f (x) − f (y))2 1 1 −3/2 |∂t G(t, x, y)| ≤ √ exp − + t √ √ a1 a2 2 2t 2π    (|f (x)| + |f (y)|)2 + |β| exp − 2t   n 1 (f (x) − f (y))2 (f (x) − f (y))2 + t−1/2 exp − 2 t2 2t   # 2 (|f (x)| + |f (y)|)2 o (|f (x)| + |f (y)|) exp − + |β| t2 2t "    1 (1 + |β|) −3/2 1 1 (f (x) − f (y))2 √ ≤ t exp − √ +√ a1 a2 2 2t 2π   n (f (x) − f (y))2 (f (x) − f (y))2 1 exp − + t−3/2 2 t 2t  #  (|f (x)| + |f (y)|)2 o (|f (x)| + |f (y)|)2 exp − + |β| t 2t FRACTIONAL STOCHASTIC HEAT EQUATION 5 Then, by using the fact that for every a, x, y, z > 0, we have (x + y + z)a ≤ C(xa + y a + z a ), we get η "   1 (1 + |β|)η − 3 η (f (x) − f (y))2 1 2 exp −η t √ +√ a1 a2 2η 2t     η (f (x) − f (y))2 1 3 n (f (x) − f (y))2 exp −η + η t− 2 η 2 t 2t  #  2η (|f (x)| + |f (y)|)2 o η (|f (x)| + |f (y)|) + |β| . exp −η tη 2t C |∂t G(t, x, y)| ≤ √ ( 2π)η η  Thus,  η 3 1 t− 2 η 1 η |∂t G(t, z, y)| dz ≤ √ +√ √ a1 a2 (2 2π)η R "   Z (f (z) − f (y))2 dz × (1 + |β|)η exp −η 2t R η   Z  (f (z) − f (y))2 (f (z) − f (y))2 + dz exp −η t 2t R η  #  Z  (|f (z)| + |f (y)|)2 (|f (z)| + |f (y)|)2 η dz . + |β| exp −η t 2t R Z On the one hand, by (2.1) we have   (f (z)−f (y))2 exp −η dz R 2t R ≤ √1 η √ √ √ max( a1 , a2 ) 2πt. On the other hand, we have Z  R η   (f (z) − f (y))2 (f (z) − f (y))2 dz exp −η t 2t ! Z +∞ ( √z − f (y))2 !η ( √za2 − f (y))2 a2 = dz exp −η t 2t 0 !η ! Z 0 ( √za1 − f (y))2 ( √za1 − f (y))2 dz exp −η + t 2t −∞ !η ! Z ( √za2 − f (y))2 ( √za2 − f (y))2 exp −η dz ≤ t 2t R !η ! Z ( √za1 − f (y))2 ( √za1 − f (y))2 + dz exp −η t 2t R  η+ 21 Z √ 2 √ √ 2 ≤ max( a1 , a2 ) t z 2η e−z dz. η R It is clear that the last integral converges for every η > 0. Therefore, we get Z  R (f (z) − f (y))2 t η   √ (f (z) − f (y))2 dz ≤ C(η, a1 , a2 ) t. exp −η 2t 6 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR Concerning the last integral, we have η   Z  (|f (z)| + |f (y)|)2 (|f (z)| + |f (y)|)2 dz exp −η t 2t R  η   Z (f (z) − f (y))2 (f (z) − f (y))2 = dz exp −η t 2t {zy≤0}  η   Z (f (z) + f (y))2 (f (z) + f (y))2 + dz exp −η t 2t {zy≥0}  η  Z  (f (z) − f (y))2 (f (z) − f (y))2 ≤ exp −η dz t 2t R η   Z  (f (z) + f (y))2 (f (z) + f (y))2 dz. exp −η + t 2t R By the same technique as above, we get η   Z  (|f (z)| + |f (y)|)2 (|f (z)| + |f (y)|)2 dz exp −η t 2t R r  η+ 21 √ √ 2 πt ≤ max( a1 , a2 ) . η 2 All this implies that Z R η 3 1 |∂t G(t, z, y)| dz ≤ C(η, a1 , a2 , β)t− 2 η+ 2 , where C(η, a1 , a2 , β) denotes a strictly positive constant depending only on η, a1 , a2 , β. Following the same steps, we can prove that Z 3 1 |∂t G(t, x, z)|η dz ≤ C(T, η, a1 , a2 , β)t− 2 η+ 2 R with the same constant C(η, a1 , a2 , β) as above. (ii) The mixed partial derivative of G equals     2 1{y≤0} 1{y>0} 1 3 ∂ (f (z) − f (y))2 − 25 + √ G(t − s, z, y) = √ (t − s) exp − √ ∂t∂s a1 a2 4 2(t − s) 2π   2 2 7 (f (z) − f (y)) (f (z) − f (y)) − 3(t − s)− 2 exp − 2(t − s) 2   2 9 (f (z) − f (y)) (f (z) − f (y))4 + (t − s)− 2 exp − 2(t − s) 4   2 5 (|f (z)| + |f (y)|) 3 + β sign(y)(t − s)− 2 exp − 4 2(t − s)   (|f (z)| + |f (y)|)2 (|f (z)| + |f (y)|)2 7 − 3β sign(y)(t − s)− 2 exp − 2(t − s) 2    2 (|f (z)| + |f (y)|) 9 (|f (z)| + |f (y)|)4 + β sign(y)(t − s)− 2 exp − . 2(t − s) 4 Therefore    (f (z) − f (y))2 ∂2 − 25 exp − G(t − s, z, y) ≤ C(t − s) ∂t∂s 2(t − s)   2 2 (f (z) − f (y)) (f (z) − f (y)) + exp − 2(t − s) 2(t − s)   2 (f (z) − f (y))4 (f (z) − f (y)) + exp − 2(t − s) 4(t − s)2 FRACTIONAL STOCHASTIC HEAT EQUATION 7     (|f (z)| + |f (y)|)2 (|f (z)| + |f (y)|)2 (|f (z)| + |f (y)|)2 + exp − + exp − 2(t − s) 2(t − s) 2(t − s)    2 4 (|f (z)| + |f (y)|) (|f (z)| + |f (y)|) + exp − . 2(t − s) 4(t − s)2 The rest of proof can be done similarly to that of (i).  Corollary 2.2. For all δ ∈ ( 13 , 1) there exists a strictly positive constant C > 0, such that, for every x, y ∈ R, for every s, t ∈ (0, T ] with s < t, Z Z  max G(t, z, y) − G(s, z, y) dz, G(t, x, z) − G(s, x, z) dz ≤ C s−δ (t − s)δ . R R Proof. For every fixed z, y ∈ R, 0 < s < t ≤ T and δ ∈ [0, 1] we have, by the triangular inequality, G(t, z, y) − G(s, z, y) = G(t, z, y) − G(s, z, y) ≤ 1−δ G(t, z, y) + G(s, z, y) G(t, z, y) − G(s, z, y)
1−δ δ δ G(t, z, y) − G(s, z, y) . By mean-value theorem, there exists t∗ ∈ (s, t) such that G(t, z, y) − G(s, z, y) δ ≤ (t − s)δ |∂t G(t∗ , z, y)|δ . 2 Therefore, by Lemma 2.1 and since for every x ∈ R, e−x ≤ 1, we have 1−δ  1 1 (t − s)δ |∂t G(t∗ , z, y)|δ . G(t, z, y) − G(s, z, y) ≤ C t− 2 + s− 2 Since for every s < t, we have t−1/2 < s−1/2 , by Lemma 2.2, we get: Z Z δ − 21 (1−δ) (t − s) |∂t G(t∗ , z, y)|δ dz G(t, z, y) − G(s, z, y) dz ≤ Cs R R 1 3 1 ≤ Cs− 2 (1−δ) (t − s)δ (t∗ )− 2 δ+ 2 . Note that for δ > 1 3 3 1 3 1 (t∗ )− 2 δ+ 2 ≤ s− 2 δ+ 2 . Consequently, we obtain: Z 3 1 1 G(t, z, y) − G(s, z, y) dz ≤ Cs− 2 (1−δ) s− 2 δ+ 2 (t − s)δ ≤ Cs−δ (t − s)δ . R Following the same technique we also get Z G(t, x, z) − G(s, x, z) dz ≤ C s−δ (t − s)δ .  R Lemma 2.3. For every δ ∈ ( 15 , 1) there exists a strictly positive constant C such that, for all 0 < σ < τ < s < t < T, Z |G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)| dz R ≤ C (t − s)δ (s − τ )−2δ (τ − σ)δ . (2.2) Proof. On the one hand, by Lemma 2.1, |G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)|   1 1 1 1 1 ≤ C (t − τ )− 2 + (s − τ )− 2 + (t − σ)− 2 + (s − σ)− 2 ≤ C(s − τ )− 2 . 8 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR On the other hand, by the mean value theorem, there exist θ∗ ∈ (s, t) and ρ∗ ∈ (σ, τ ) such that |G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)| Z tZ τ ∂2 ∂2 = G(θ − ρ, z, y) dρ dθ = G(θ∗ − ρ∗ , z, y) (t − s)(τ − σ). ∂θ∂ρ s σ ∂θ∂ρ Hence, for every δ ∈ (0, 1), we have Z |G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)| dz R Z 1−δ = |G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)| R δ × |G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)| dz Z δ ∂2 1 G(θ∗ − ρ∗ , z, y) dz. ≤ C(s − τ )− 2 (1−δ) (t − s)δ (τ − σ)δ R ∂θ∂ρ Taking into account Lemma 2.2 (ii), we obtain Z |G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)| dz R 1 5 1 ≤ C(s − τ )− 2 (1−δ) (t − s)δ (τ − σ)δ (θ∗ − ρ∗ )− 2 δ+ 2 . 5 1 5 1 If δ > 51 , then (θ∗ − ρ∗ )− 2 δ+ 2 ≤ (s − τ )− 2 δ+ 2 and we arrive at (2.2).  3. Mild solution 3.1. Definitions and notation. 3.1.1. Norms and spaces. Throughout the paper, the symbol C will denote a generic constant, the precise value of which is not important and may vary between different equations and inequalities. Let k · k2 be the norm in L2 (R). Let 0 < σ < 1. For every measurable function u : [0, T ] × R → R and t ∈ [0, T ] denote 2 Z t Z s ku(s, ·) − u(v, ·)k2 2 dv ds, kukσ,1,t := (s − v)σ+1 0 0 2 2 2 kukσ,2,t := sup ku(s, ·)k2 + kukσ,1,t . (3.1) s∈[0,t] We define also the following seminorm for f : [0, T ] → R and t ∈ [0, T ]:   Z v |f (u) − f (v)| |f (u) − f (z)| kf kσ,0,t := sup + dz (v − u)1−σ (z − u)2−σ 0≤u<v≤t u
For σ ∈ (0, 1) we denote by B σ,2 [0, T ]; L2(R) the following Banach space: n
B σ,2 [0, T ]; L2(R) := u : [0, T ] × R → R Lebesgue measurable mapping o 2 such that kukσ,2,T < ∞ . 3.1.2. Integration with respect to B H for every H ∈ ( 12 , 1). Let a, b ∈ R, a < b. Let ϕ ∈ L1 ([a, b]) and σ ∈ (0, 1). The Riemann–Liouville left- and right-sided fractional integrals of ϕ of order σ are defined for almost all x ∈ [a, b] by Z x 1 σ Ia+ ϕ(x) = (x − y)σ−1 ϕ(y) dy Γ(σ) a Z b 1 σ (x − y)σ−1 ϕ(y) dy. Ib− ϕ(x) = Γ(σ) x FRACTIONAL STOCHASTIC HEAT EQUATION 9 Let Iaσ+ (Lp )(resp. Ibσ− (Lp )) denote the image of Lp ([a, b]) by the operator (resp. Ibσ− ). If ϕ ∈ Iaσ+ (Lp )(resp. ϕ ∈ Ibσ− (Lp )) then the Riemann–Liouville left- and right-sided fractional derivatives are defined by   Z x 1 ϕ(x) ϕ(x) − ϕ(y) σ Da+ ϕ(x) = +σ dy , Γ(1 − σ) (x − a)σ (x − y)1+σ a ! Z b ϕ(x) ϕ(x) − ϕ(y) 1 +σ dy . Dbσ− ϕ(x) = 1+σ Γ(1 − σ) (b − x)σ x (y − x) Iaσ+ For every two functions ϕ, ψ : [a, b] → R such that Daσ+ ϕ ∈ L1 ([a, b]) and ∈ L∞ ([a, b]) where ψb− (x) = ψ(b− ) − ψ(x), we define the generalized Lebesgue–Stieljes integral by Z b Z b Daσ+ ϕ(x) Db1−σ (3.2) ϕ(x) dψ(x) := − ψb− (x) dx. Db1−σ − ψb− a a Furthermore, this integral admits the following bound:  Z x Z b Z b |ϕ(x)| |ϕ(x) − ϕ(y)| + dy dx. (3.3) ϕ(x) dψ(x) ≤ Cσ kψkσ,0,b (x − a)σ (x − y)1+σ a a a For more information on generalized Lebesgue–Stieljes integration with respect to fractional Brownian motion with Hurst index H > 1/2, the reader can see, e. g., [16] and references therein.
Now we suppose that σ ∈ (1 − H, 1/2), and denote by L L2 (R) the space of
linear operators on L2 (R). Let F : [0, T ] × Ω → L L2 (R) be an operator function such that  Z t Z T k(F (s, ω) − F (v, ω))ej k2 kF (s, ω)ej k2 + dv ds < ∞ a. s. (3.4) sup sσ (s − v)σ+1 j∈N 0 0 Following [15], we define the integral with respect to the L2 (R)-valued fractional Brownian motion by Z b Z b ∞ X λj F (s, ω)ej dBjH (s), (3.5) F (s, ω) dB H (s) = a a j=1 where the integral with respect to BjH is the path-wise generalized Lebesgue– Stieltjes integral defined in (3.2) and the convergence of the series should be understood as P-a. s. convergence in L2 (R). Applying assumptions (1.5) and [15, Proposition 2.1], we deduce that the integral (3.5) is well defined. From (3.3), one can obtain the following inequality for 0 ≤ a < b ≤ T : Z a b F (s, ω) dB H (s) 2 ≤ Cξσ,H,T sup i∈N + Z b a Z s a kF (s, ω)ei k2 (s − a)σ  kF (s, ω)ei − F (v, ω)ei k2 dv ds, (3.6) (s − v)1+σ where C = C(σ) is a constant, and the random variable ξσ,H,T := ∞ X λj BjH σ,0,T j=1 is finite a. s., see [15]. For more details on the integration with respect to the L2 (R)-valued fractional Brownian motion, see e. g. [15] and references therein. 10 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR 3.1.3. Mild solution. Definition 3.1. An L2 (R)-valued random process {u(t, ·), t ∈ [0, T ]} is called a mild solution to the problem (1.2) if it satisfies the following assumptions:
(1) u ∈ B σ,2 [0, T ]; L2(R) a.s. for some σ ∈ (1 − H, 1/2). (2) For any ∀t ∈ [0, T ] it holds that Z tZ ∞ X λj G(t − s, x, y) h(u(s, y)) ej (y) dy dBjH (s) a.s. (3.7) u(t, x) = 0 j=1 R Here, the integrals with respect to BjH , j ∈ N are the generalized Lebesgue–Stieltjes integrals defined in (3.2).
Let us introduce some notations for u ∈ B σ,2 [0, T ]; L2(R) . Namely, let Z := ςj,t (u)(s, x) G(t − s, x, y) h(u(s, y)) ej (y) dy, (3.8) R and ∗ ςj,t,s (u)(v, x) := ςj,t (u)(v, x) − ςj,s (u)(v, x). (3.9) Also let us denote the right-hand side of (3.7) by Z tZ ∞ X λj G(t − s, x, y) h(u(s, y)) ej (y) dy dBjH (s) (Au)(t, x) := = 0 j=1 ∞ X λj Z 0 j=1 t R ςj,t (u)(s, x) dBjH (s). Note that Au is an integral with respect to an L2 (R)-valued fractional Brownian motion, defined by (3.5). The condition (3.4) for it has the form  Z T Z s kςj,t (u)(s, ·)k2 kςj,t (u)(s, ·) − ςj,t (u)(v, ·)k2 sup + dv ds < ∞ a. s. sσ (s − v)σ+1 j∈N 0 0
In fact, this condition holds for any u ∈ B σ,2 [0, T ]; L2(R) . It will be checked in the proof of Proposition 3.1 below, in which we also establish that Au actually
determines an L2 (R)-valued stochastic process from the class B σ,2 [0, T ]; L2(R) .

3.2. A priori estimates. Let us fix H ∈ 21 , 1 and σ ∈ 1 − H, 21 . Abbreviate ξ = ξσ,H,T .

Proposition 3.1. Let u ∈ B σ,2 [0, T ]; L2(R) . Then Au ∈ B σ,2 [0, T ]; L2(R) a. s. Moreover, for any t ∈ (0, T ],   2 2 kAukσ,2,t ≤ Cξ 2 kukσ,2,t + 1 . Proof. Let δ ∈ (max{2σ, 31 }, 1) be fixed throughout the proof. It follows from (3.7) and Lemma A.1 that for any t ∈ (0, T ], k(Au)(t, ·)k2 = ∞ X j=1 λj Z 0 t ςj,t (u)(s, ·)dBjH (s) Z t 2  Z s kςj,t (u)(s, ·)k2 kςj,t (u)(s, ·) − ςj,t (u)(v, ·)k2 ≤ C ξ sup + dv ds sσ (s − v)σ+1 j∈N 0 0 Z tZ s Z t ku(s, ·) − u(v, ·)k2 ku(s, ·)k2 + 1 ds + dv ds ≤Cξ σ s (s − v)σ+1 0 0 0 FRACTIONAL STOCHASTIC HEAT EQUATION Z + t 0 − 2δ (t − s) s Z (s − v) 0 δ 2 −σ−1 11 ! (ku(v, ·)k2 + 1) dv ds . By the Cauchy–Schwarz inequality, we have Z 0 t Z ku(s, ·)k2 + 1 ds ≤ sσ Since σ < 1/2, we have t Z 0 Rt 1 0 s2σ t 0 1 ds s2σ 1/2 Z t 2 (ku(s, ·)k2 + 1) ds 0 ku(s, ·)k2 + 1 ds ≤ C sσ Z t  0 2 ku(s, ·)k2 +1  ds 1/2 . s−v t−v (3.10) in the last v 0 = . ds < ∞. Hence, By Fubini’s Theorem and applying the change of variables x = integral, we obtain Z t Z s δ − δ2 (t − s) (s − v) 2 −σ−1 (ku(v, ·)k2 + 1) dv ds 0 0  Z t Z t δ − 2δ −σ−1 2 (t − s) (s − v) = ds (ku(v, ·)k2 + 1) dv Z 1/2 t 0 =B ≤C ≤ Ct (t − v)−σ δ 2 1 Z 0 − σ, 1 −
δ 2  δ δ (1 − x)− 2 x 2 −σ−1 dx (ku(v, ·)k2 + 1) dv Z 0 t (t − v)−σ (ku(v, ·)k2 + 1) dv !Z t sup ku(v, ·)k2 + 1 v∈[0,t] 1−σ 0 (3.11) (t − v)−σ dv ! sup ku(v, ·)k2 + 1 , v∈[0,t] where B denotes the beta function defined, for every p, q > 0, by B(p, q) := R 1 p−1 t (1 − t)q−1 dt, and in the last inequality we used the fact that σ < 1/2. 0 Therefore, using (3.10) and (3.11), we obtain " ! Z t  1/2 2 2 2 1−σ k(Au)(t, ·)k2 ≤ C ξ t sup ku(v, ·)k2 + 1 + ku(s, ·)k2 + 1 ds v∈[0,t] + Z tZ 0 ≤Cξ + s 0 2 t sup v∈[0,t] 0 ≤ C ξ2 +t ku(s, ·) − u(v, ·)k2 dv ds (s − v)σ+1 2−2σ Z t Z 0 s 0 2 ku(v, ·)k2 sup v∈[0,t] 0 0 ! ku(s, ·) − u(v, ·)k2 dv ds (s − v)σ+1 (t2−2σ + t) Z t Z +1 #2 s 2 ku(v, ·)k2 ku(s, ·) − u(v, ·)k2 dv (s − v)σ+1 Z t  2 ku(s, ·)k2 + 1 ds + 0 2 ! +1 2 ! ! ds . 12 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR Since t2−2σ = t1−2σ t ≤ T 1−2σ t = Ct for σ < 1/2, we arrive at 2 ! Z t Z s ku(s, ·) − u(v, ·)k 2 2 2 k(Au)(t, ·)k2 ≤ C ξ 2 t sup ku(v, ·)k2 + 1 + dv ds . (s − v)σ+1 v∈[0,t] 0 0 (3.12) By the definition of the norm k.kσ,2,t , we get   2 2 k(Au)(t, ·)k2 ≤ C ξ 2 t kukσ,2,t + 1 . Consequently,   2 2 sup k(Au)(s, ·)k2 ≤ C ξ 2 kukσ,2,t + 1 , (3.13) s∈[0,t] because kuk2σ,2,s ≤ kuk2σ,2,t for s ≤ t. Futhermore, k(Au)(s, ·) − (Au)(v, ·)k2 Z s Z ∞ X = λj ςj,s (u)(z, ·)dBjH (z) + v j=1 ∞ X ≤ λj ∞ X 0 j=1 2 2 v Z λj [ςj,s (u)(z, ·) − 0  ςj,v (u)(z, ·)] dBjH (z) ςj,s (u)(z, ·)dBjH (z) v j=1 + s Z v [ςj,s (u)(z, ·) − ςj,v (u)(z, ·)] dBjH (z) . (3.14) 2 For the first term in the right-hand side of (3.14) we have that ∞ X s Z λj v j=1 ςj,s (u)(z, ·)dBjH (z) 2 ≤ C ξ sup j∈N Z s v  kςj,s (u)(z, ·)k2 + (z − v)σ Z z v  kςj,s (u)(z, ·) − ςj,s (u)(r, ·)k2 dr dz. (z − r)σ+1 Then, using Lemma A.1 and the same technique as used to prove (3.12), we get ∞ X Z λj s v j=1 2 ςj,s (u)(z, ·) dBjH (z) 2 2 ≤ C ξ (s − v) sup r∈[v,s] 2 ku(r, ·)k2 +1+ Z v s Z v z ku(z, ·) − u(r, ·)k2 dr (z − r)σ+1 2 ! dz . (3.15) For the second term in the right-hand side of (3.14) we have that Z v ∞ X λj [ςj,s (u)(z, ·) − ςj,v (u)(z, ·)] dBjH (z) j=1 0 2 ≤ C ξ sup j∈N + sup j∈N Z 0 Z vZ 0 v 0 z kςj,s (u)(z, ·) − ςj,v (u)(z, ·)k2 dz zσ [ςj,s (u)(z, ·) − ςj,v (u)(z, ·)] − [ςj,s (u)(r, ·) − ςj,v (u)(r, ·)] (z − r)σ+1 2 drdz ! FRACTIONAL STOCHASTIC HEAT EQUATION 13   =: C ξ L1 + L2 . Using Lemma A.2, we obtain δ δ kςj,s (u)(z, ·) − ςj,v (u)(z, ·)k2 ≤ C (v − z)− 2 (s − v) 2 (ku(z, ·)k2 + 1) . Thus, L1 ≤ C(s − v) δ 2 δ ≤ C(s − v) 2 δ = C(s − v) 2 ≤ C(s − v) δ v (v − z)− 2 (ku(z, ·)k2 + 1) dz zσ 0 !Z δ v (v − z)− 2 sup ku(z, ·)k2 + 1 dz zσ z∈[0,v] 0 ! Z sup ku(z, ·)k2 + 1 v −δ/2+1−σ B 1 − 2δ , 1 − σ z∈[0,v] δ 2 sup ku(z, ·)k2 + 1 z∈[0,v] !
because σ < 1/2 and δ < 1. Now by Lemma A.2, we have kςj,s (u)(z, ·) − ςj,v (u)(z, ·) − ςj,s (u)(r, ·) + ςj,v (u)(r, ·)k2  δ δ ≤ C (s − v) 2 (v − z)− 2 ku(z, ·) − u(r, ·)k2  δ δ + (s − v) 2 (v − z)−δ (z − r) 2 (ku(r, ·)k2 + 1) . Thus, L2 = sup j∈N ≤C Z vZ 0 (z − r)σ+1 0 Z vZ 0 +C [ςj,s (u)(z, ·) − ςj,v (u)(z, ·)] − [ςj,s (u)(r, ·) − ςj,v (u)(r, ·)] z δ z 0 Z vZ 0 δ z 0 δ δ δ (s − v) 2 (v − z)−δ (z − r) 2 (ku(r, ·)k2 + 1) drdz (z − r)σ+1 By Fubini’s theorem, we have Z vZ z δ L2,2 = (z − r) 2 −σ−1 (v − z)−δ (ku(r, ·)k2 + 1) dr dz Z v  Z0 v 0 δ = (ku(r, ·)k2 + 1) (z − r) 2 −σ−1 (v − z)−δ dz dr. 0 r Since 2σ < δ < 1, we see that Z v
δ δ (z − r) 2 −σ−1 (v − z)−δ dz = B 1 − δ, δ2 − σ (v − r)− 2 −σ . r Therefore, ≤C Z v 0 drdz (s − v) 2 (v − z)− 2 ku(z, ·) − u(r, ·)k2 drdz (z − r)σ+1 =: C(s − v) 2 (L2,1 + L2,2 ) . L2,2 ≤ C 2 δ (ku(r, ·)k2 + 1) (v − r)− 2 −σ dr !Z v sup ku(r, ·)k2 + 1 r∈[0,v] 0 δ (v − r)− 2 −σ dr (3.16) 14 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR ≤ Cv 1− 2δ −σ sup ku(r, ·)k2 + 1 r∈[0,v] ! ≤C ! sup ku(r, ·)k2 + 1 . r∈[0,v] (3.17) It follows from (3.16) and (3.17) that ! Z vZ z δ (v − z)− 2 ku(z, ·) − u(r, ·)k2 δ L2 ≤ C(s−v) 2 drdz + sup ku(r, ·)k2 + 1 , (z − r)σ+1 r∈[0,v] 0 0 and consequently, ∞ X λj j=1 Z v 0 ≤ C ξ(s−v)   ≤ C ξ L1 + L2 [ςj,s (u)(z, ·) − ςj,v (u)(z, ·)] dBjH (z) δ 2 sup ku(r, ·)k2 +1+ r∈[0,v] Z v z Z 0 0 2 ! δ (v − z)− 2 ku(z, ·) − u(r, ·)k2 drdz . (z − r)σ+1 (3.18) Hence, combining (3.14)–(3.18), we obtain " k(Au)(s, ·) − (Au)(v, ·)k2 ≤ C ξ (s − v)1/2 Z 1/2 + (s − v) v δ v r∈[0,v] Z δ 2 v 0 1 2 z Z ku(z, ·) − u(r, ·)k2 2 dr dz (z − r)σ+1 ! !1/2 sup ku(r, ·)k2 + 1 + (s − v) 2 + (s − v) s sup ku(r, ·)k2 + 1 r∈[v,s] ! Since (s − v) = (s − v) 1−δ 2 − 2δ (v − z) z Z 0 δ 2 (s − v) ≤ T # ku(z, ·) − u(r, ·)k2 dr dz . (z − r)σ+1 1−δ 2 δ (s − v) 2 , we see that " δ k(Au)(s, ·) − (Au)(v, ·)k2 ≤ C ξ(s − v) 2 sup ku(r, ·)k2 + 1 r∈[0,s] !1/2 ku(z, ·) − u(r, ·)k2 2 + dr dz (z − r)σ+1 v v # Z v Z z ku(z, ·) − u(r, ·)k δ 2 (v − z)− 2 + dr dz . (z − r)σ+1 0 0 Z s Z z Therefore, 2 kAukσ,1,t = Z t Z 0 s 0 k(Au)(s, ·) − (Au)(v, ·)k2 dv (s − v)σ+1 2 ds ≤ C ξ 2 Z tX 3 0 i=1 Hi2 (s) ds, where H1 (s) = sup ku(r, ·)k2 + 1 r∈[0,s] !Z s 0 δ (s − v) 2 −σ−1 dv, !1/2 ku(z, ·) − u(r, ·)k2 2 dr dz dv, H2 (s) = (s − v) (z − r)σ+1 0 v v Z z Z v Z s ku(z, ·) − u(r, ·)k2 δ −σ−1 − δ2 2 (v − z) H3 (s) = (s − v) dr dz dv. (z − r)σ+1 0 0 0 Z s δ 2 −σ−1 Z s Z z (3.19) FRACTIONAL STOCHASTIC HEAT EQUATION The first term can be bounded by ! H1 (s) = sup ku(r, ·)k2 + 1 s δ 2 −σ r∈[0,s] ≤C 15 ! sup ku(r, ·)k2 + 1 , r∈[0,s] (3.20) because 2δ > σ. Concerning H2 , we have H2 (s) ≤ kukσ,1,s Z s 0 δ (s − v) 2 −σ−1 dv ≤ C kukσ,1,s . (3.21) For the last one, applying Fubini’s theorem, we get Z z Z s Z s 2 ku(z, ·) − u(r, ·)k2 δ δ H32 (s) = dr dz (s − v) 2 −σ−1 (v − z)− 2 dv (z − r)σ+1 0 0 z 2 Z s Z z
2 ku(z, ·) − u(r, ·)k2 dr dz . ≤ B 1 − 2δ , δ2 − σ (s − z)−σ (z − r)σ+1 0 0 Since σ < 1/2, we can write, applying the Cauchy–Schwartz inequality, 2 Z s Z z Z s ku(z, ·) − u(r, ·)k2 2 (s − z)−2σ dz H32 (s) ≤ C dr dz ≤ C kukσ,1,s . σ+1 (z − r) 0 0 0 (3.22) Finally, combining (3.19)–(3.22), we get Z t    2 2 2 2 (3.23) kAukσ,1,t ≤ C ξ kukσ,2,s ds + 1 ≤ C ξ 2 kukσ,2,t + 1 , 0 kuk2σ,2,s kuk2σ,2,t because ≤ for s ≤ t. Hence, from (3.1), (3.13) and (3.23) we get the result.
Proposition 3.2. Let u, ũ ∈ B σ,2 [0, T ]; L2(R) . Then for all t ∈ [0, T ], Z t 2 2 2 kAu − Aũkσ,2,t ≤ Cξ ku − ũkσ,2,s ds.  0

Proof. Recall that σ ∈ 1 − H, . Fix δ ∈ max{σ, 31 }, 1 . By the definition of A, we can write Z t ∞ X λj [ςj,t (u)(s, ·) − ςj,t (ũ)(s, ·)] dBjH (s) k(Au)(t, ·) − (Aũ)(t, ·)k2 = 1 2 j=1 ≤ C ξ sup j∈N + Z 0 s Z t 0 2 kςj,t (u)(s, ·) − ςj,t (ũ)(s, ·)k2 sσ 0 ! kςj,t (u)(s, ·) − ςj,t (ũ)(s, ·) − ςj,t (u)(r, ·) + ςj,t (ũ)(r, ·)k2 dr ds. (s − r)σ+1 It follows from Lemma A.3 that k(Au)(t, ·) − (Aũ)(t, ·)k2 ≤ C ξ + Z s −δ (t − s) 0 (Z 0 t ku(s, ·) − ũ(s, ·)k2 sσ ! ku(r, ·) − ũ(r, ·)k2 dr ds (s − r)σ+1−δ ) ku(s, ·) − ũ(s, ·) − u(r, ·) + ũ(r, ·)k2 dr ds , + (s − r)σ+1 0 0
where the right-hand side is finite, because u, ũ ∈ B σ,2 [0, T ]; L2(R) . Z tZ s 16 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR First, the Cauchy–Schwarz inequality implies that for every σ < 12 Z t 1/2 Z t 1/2 Z t ku(s, ·) − ũ(s, ·)k2 2 −2σ ds ≤ s ds ku(s, ·) − ũ(s, ·)k ds 2 sσ 0 0 0 ! 1/2 Z t 2 ≤C sup ku(r, ·) − ũ(r, ·)k2 ds 0 r∈(0,s) Second, it follows from Fubini’s theorem and the Cauchy–Schwarz inequality that for every σ < δ < 1 Z tZ s (t − s)−δ ku(r, ·) − ũ(r, ·)k2 dr ds (s − r)σ+1−δ 0 0 Z t Z t = ku(r, ·) − ũ(r, ·)k2 (t − s)−δ (s − r)−σ−1+δ ds dr 0 r Z t = B(1 − δ, δ − σ) (t − r)−σ ku(r, ·) − ũ(r, ·)k2 dr 0 t ≤C Z ≤C Z 0 (t − r)−2σ dr  21 Z 0 t 0 t 2 sup ku(r, ·) − ũ(r, ·)k2 ds r∈(0,s) 2 ũ(r, ·)k2 sup ku(r, ·) − r∈(0,s) ds ! 21 ! 12 . Hence, k(Au)(t, ·) − (Aũ)(t, ·)k2 ≤ C ξ +t Z t Z 0 s 0 ( Z 0 t sup ku(r, ·) − r∈(0,s) 2 ũ(r, ·)k2 ds !1/2 ku(s, ·) − ũ(s, ·) − u(r, ·) + ũ(r, ·)k2 dr (s − r)σ+1 2 ) ds . Therefore, we obtain 2 sup k(Au)(s, ·) − (Aũ)(s, ·)k2 ≤ C ξ 2 s∈(0,t) Z 0 t 2 ku − ũkσ,2,s ds, On the other hand, we have that for r < s k(Au)(s, ·) − (Aũ)(s, ·) − (Au)(r, ·) + (Aũ)(r, ·)k2 Z s ∞ X λj [ςj,s (u)(v, ·) − ςj,s (ũ)(v, ·)] dBjH (v) ≤ r j=1 + ∞ X λj Z r [ςj,s (u)(v, ·) − ςj,s (ũ)(v, ·) − ςj,r (u)(v, ·) + ςj,r (ũ)(v, ·)] dBjH (v) 0 j=1 2 2 =: J1 + J2 . The first term can be bounded as follows Z s ∞ X λj [ςj,s (u)(v, ·) − ςj,s (ũ)(v, ·)] dBjH (v) J1 = r j=1 ≤ C ξ sup j∈N Z r s kςj,s (u)(v, ·) − ςj,s (ũ)(v, ·)k2 (v − r)σ 2 FRACTIONAL STOCHASTIC HEAT EQUATION + ! kςj,s (u)(v, ·) − ςj,s (ũ)(v, ·) − ςj,s (u)(z, ·) + ςj,s (ũ)(z, ·)k2 dz dv. (v − z)σ+1 v Z 17 r By Lemma A.3, Z J1 ≤ Cξ s Z v ku(v, ·) − ũ(v, ·)k2 ku(z, ·) − ũ(z, ·)k2 + (s − v)−δ dz σ (v − r) (v − z)σ+1−δ r ! Z v ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k2 + dz dv. (v − z)σ+1 r r The term J2 can be written and bounded as follows J2 = = ∞ X λj λj ≤ C ξ sup j∈N + Z r Z r [ςj,s (u)(v, ·) − ςj,s (ũ)(v, ·) − ςj,r (u)(v, ·) + ςj,r (ũ)(v, ·)] dBjH (v) 2 0 j=1 Z r 0 j=1 ∞ X Z   ∗ ∗ ςj,s,r (u)(v, ·) − ςj,s,r (ũ)(v, ·) dBjH (v) ∗ ∗ ςj,s,r (u)(v, ·) − ςj,s,r (ũ)(v, ·) 2 2 vσ 0 ∗ ∗ ∗ ∗ ςj,s,r (u)(v, ·) − ςj,s,r (ũ)(v, ·) − ςj,s,r (u)(z, ·) + ςj,s,r (ũ)(z, ·) v (v − z)σ+1 0 2 ! dz dv. By Lemma A.4, we get Z r δ ku(v, ·) − ũ(v, ·)k δ 2 J2 ≤ C ξ (s − r) 2 (r − v)− 2 vσ 0 Z v δ ku(z, ·) − ũ(z, ·)k δ 2 + (s − r) 2 (r − v)− 2 dz δ (v − z)σ+1− 2 0  Z v δ ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k δ 2 + (s − r) 2 (r − v)− 2 dz dv. (v − z)σ+1 0 Therefore Z t Z s 0 0 k(Au)(s, ·) − (Aũ)(s, ·) − (Au)(r, ·) + (Aũ)(r, ·)k2 dr (s − r)σ+1 =Cξ 2 2 ds Z tX 6 0 i=1 where s s G2i (s) ds, ku(v, ·) − ũ(v, ·)k2 dv dr, (v − r)σ 0 r Z s Z sZ v ku(z, ·) − ũ(z, ·)k2 (s − r)−σ−1 G2 (s) := (s − v)−δ dz dv dr, (v − z)σ+1−δ Z0 s Zr sZr v ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k2 G3 (s) := (s − r)−σ−1 dz dv dr, (v − z)σ+1 0 r r Z s Z r δ δ ku(v, ·) − ũ(v, ·)k 2 G4 (s) := (s − r) 2 −σ−1 (r − v)− 2 dv dr, vσ 0 0 Z rZ v Z s δ ku(z, ·) − ũ(z, ·)k δ 2 dz dv dr, (r − v)− 2 G5 (s) := (s − r) 2 −σ−1 δ σ+1− 2 (v − z) 0 0 0 G1 (s) := Z −σ−1 (s − r) Z 18 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR Z s δ (s − r) 2 −σ−1 0 Z rZ v δ ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k 2 dz dv dr. × (r − v)− 2 (v − z)σ+1 0 0 G6 (s) := Let us bound each of terms Gi , i = 1, . . . , 6. For every σ ∈ (0, 1/2) Z s  Z s −σ−1 −σ G1 (s) ≤ sup ku(z, ·) − ũ(z, ·)k2 (s − r) (v − r) dv dr z∈[0,s] 0 r = C sup ku(z, ·) − ũ(z, ·)k2 . z∈[0,s] and by the same technique we get for every δ > σ G2 (s) ≤ C sup ku(z, ·) − ũ(z, ·)k2 . z∈[0,s] By the Cauchy–Schwartz inequality, we obtain G3 (s) ≤ Z s Z s Z −σ− 21 (s−r) r 0 r v ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k2 dz (v − z)σ+1 2 dv ! 12 dr. Similarly to (3.21), we get the bound "Z Z 2 #1/2 v s ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k2 dz G3 (s) ≤ C dv . (v − z)σ+1 0 0 It is not hard to see that G4 (s) ≤ sup ku(z, ·) − ũ(z, ·)k2 z∈[0,s] Z s 0 δ (s − r) 2 −σ−1 ≤ C sup ku(z, ·) − ũ(z, ·)k2 . Z r 0 δ (r − v)− 2 v −σ dv dr z∈[0,s] In the same way, G5 (s) ≤ C sup ku(z, ·) − ũ(z, ·)k2 . z∈[0,s] Finally, the term G6 can be bounded similarly to (3.22). We obtain "Z Z 2 #1/2 v s ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k2 dz G6 (s) ≤ C dv . (v − z)σ+1 0 0 Then we have k(Au)(s, ·) − 2 (Aũ)(s, ·)kσ,2,t ≤ Cξ 2 Z 0 t 2 ku − ũkσ,2,s ds.  3.3. Existence and uniqueness of mild solution. Theorem 3.1. For every H ∈ ( 21 , 1), there exists a unique mild solution to the problem (1.2). Proof. Existence. We define the following sequence of random processes up : [0, T ] → L2 (R): u0 ≡ 0, up+1 = Aup , p ≥ 1. Reasoning by
induction and using Proposition 3.1, we easily get that up ∈ B σ,2 [0, T ]; L2(R) a. s. for every p ≥ 0. By Proposition 3.2, for any p ≥ 1, Z T kup+1 − up k2σ,2,T = kAup − Aup−1 k2σ,2,T ≤ Cξ 2 kup − up−1 k2σ,2,s ds. 0 FRACTIONAL STOCHASTIC HEAT EQUATION 19 By induction, we get 2 kup+1 − up kσ,2,t ≤ C p ξ 2p T Z Z s1 ··· 0 0 Z sp−1 2 ku1 − u0 kσ,2,sp dsp . . . ds2 ds1 . 0 Since u0 ≡ 0, we see that by Proposition 3.1,   2 2 2 2 ku1 − u0 kσ,2,sp = ku1 kσ,2,sp = kAu0 kσ,2,sp ≤ Cξ 2 ku0 kσ,2,sp + 1 = Cξ 2 . Hence, 2 kup+1 − up kσ,2,T ≤ Cξ 2
p+1 Then for m > n ≥ 0 we get kum − un kσ,2,t = ≤ Z T 0 Z s1 0 ··· Z m−1 X p=n (up+1 − up ) Cξ
2 p+1 p! dsp . . . ds2 ds1 = Cξ 2 0 m−1 X p=n sp−1 T σ,2,t p ≤ !1/2 m−1 X p=n
p+1 T p . p! kup+1 − up kσ,2,t →0 a. s. as m, n → ∞.
σ,2 2 Therefore, {up , p ≥ 0} is a Cauchy sequence in B [0, T ]; L (R) a. s. Then
σ,2 2 [0, T ]; L (R) such that there exists a process u∞ ∈ B kup − u∞ kσ,2,T → 0 a. s., as p → ∞. (3.24) Now we prove that u∞ is a mild solution. For any p ≥ 0, ku∞ − Au∞ k2σ,2,T ≤ 2 ku∞ − up+1 k2σ,2,T + 2 kup+1 − Au∞ k2σ,2,T 2 2 = 2 ku∞ − up+1 kσ,2,T + 2 kAup − Au∞ kσ,2,T Z T 2 2 ≤ 2 ku∞ − up+1 kσ,2,T + Cξ kup − u∞ kσ,2,s ds 0 2 ≤ 2 ku∞ − up+1 kσ,2,T + Cξ 2 T kup − u∞ kσ,2,T , a. s., by Proposition 3.2. Letting p → ∞ and taking into account (3.24), we get that u∞ = Au∞ a. s. Hence, u∞ is a mild solution. Uniqueness. Let u and ũ be two mild solutions. Then u = Au and ũ = Aũ a. s., and for all t ∈ [0, T ], Z t 2 2 2 ku − ũkσ,2,t = kAu − Aũkσ,2,t ≤ Cξ 2 ku − ũkσ,2,s ds a. s., 0 2 by Proposition 3.2. Then, by Gronwall’s lemma, ku − ũkσ,2,T = 0 a. s., which means the uniqueness of the mild solution.  Appendix A. In this appendix we establish upper bounds for L2 -norms of the functions ςj,t (u) ∗ (u) and their differences (see (3.8) and (3.9) for the definitions of that and ςj,t,s functions). The results of the appendix are used in the proofs of Propositions 3.1 and 3.2. Lemma A.1. Let {u(t, ·), t ∈ [0, T ]} be an L2 (R)-valued random process. Then (i) for all 0 < s < t < T , sup kςj,t (u)(s, ·)k2 ≤ C (ku(s, ·)k2 + 1) ; j∈N 20 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR (ii) for all 0 < r < s < t < T , and for any δ ∈ ( 13 , 1), sup kςj,t (u)(s, ·) − ςj,t (u)(r, ·)k2 j∈N o n δ δ ≤ C ku(s, ·) − u(r, ·)k2 + (t − s)− 2 (s − r) 2 (ku(r, ·)k2 + 1) . Proof. Let us start with the assertion (i) and produce the following transformations: Z Z Z 2 2 2 |ςj,t (u)(s, x)| dx = kςj,t (u)(s, ·)k2 = G(t − s, x, y) h(u(s, y)) ej (y) dy dx. R R R It follows from the Hölder inequality that  Z Z Z 2 2 |G(t − s, x, y)| dy |G(t − s, x, y)| |h(u(s, y))| |ej (y)|2 dy dx. kςj,t (s, ·)k2 ≤ R R R Corollary 2.1 implies that  Z Z 2 2 |G(t − s, x, y)| |h(u(s, y))| |ej (y)|2 dy dx. kςj,t (u)(s, ·)k2 ≤ C R By (1.3), for all z ∈ R, R
2 |h(z)| ≤ C z 2 + 1 . Therefore, applying Fubini’s Theorem, we get Z   2 2 |G(t − s, x, y)| |u(s, y)| + 1 |ej (y)|2 dx dy kςj,t (u)(s, ·)k2 ≤ C 2 Z ZR   2 2 |u(s, y)| + 1 |ej (y)| |G(t − s, x, y)| dx dy. =C R R Using again Corollary 2.1, we get kςj,t (u)(s, ·)k22 ≤ C Z  R  |u(s, y)|2 + 1 |ej (y)|2 dy. However,   Z   |u(s, y)|2 + 1 |ej (y)|2 dy ≤ C sup kej k2∞ ku(s, ·)k22 + kej k22 j∈N R 2  2 ≤ C ku(s, ·)k2 + 1 ,  (A.1) since supj∈N kej k∞ is bounded by (1.5), and kej k2 = 1 due to orthonormality. Consequently, the assertion (i) follows. Now, let us prove assertion (ii). On the one hand, obviously, Z 1/2 2 kςj,t (u)(s, ·) − ςj,t (u)(r, ·)k2 := |ςj,t (u)(s, x) − ςj,t (u)(r, x)| dx . R On the other hand, |ςj,t (u)(s, x) − ςj,t (u)(r, x)| Z   = G(t − s, x, y) h(u(s, y)) ej (y) − G(t − r, x, y) h(u(r, y)) ej (y) dy R Z h i = G(t − s, x, y) h(u(s, y)) − h(u(r, y)) ej (y) dy R Z h i G(t − s, x, y) − G(t − r, x, y) h(u(r, y)) ej (y) dy + R FRACTIONAL STOCHASTIC HEAT EQUATION 21 Z ≤ sup kej k∞ |G(t − s, x, y)| |h(u(s, y)) − h(u(r, y))| dy j∈N R Z + |G(t − s, x, y) − G(t − r, x, y)| |h(u(r, y))| |ej (y)| dy. (A.2) R By Holder’s inequality, we obtain from (A.2) that 2 |ςj,t (u)(s, x) − ςj,t (u)(r, x)| ≤ C Z R |G(t − s, x, y)| |h(u(s, y)) − h(u(r, y))| dy 2 Z 2 +C |G(t − s, x, y) − G(t − r, x, y)| |h(u(r, y))| |ej (y)| dy Z Z R |G(t − s, x, y)| dy |G(t − s, x, y)| |h(u(s, y)) − h(u(r, y))|2 dy ≤ C R R Z |G(t − s, x, y) − G(t − r, x, y)| dy +C ZR 2 2 (A.3) × |G(t − s, x, y) − G(t − r, x, y)| |h(u(r, y))| |ej (y)| dy. R Applying Corollaries 2.1 and 2.2, we get from (A.3) the following bounds Z 2 2 |G(t − s, x, y)| |h(u(s, y)) − h(u(r, y))| dy |ςj,t (u)(s, x) − ςj,t (u)(r, x)| ≤ C R Z 2 2 −δ δ + C(t − s) (s − r) |G(t − s, x, y) − G(t − r, x, y)| |h(u(r, y))| |ej (y)| dy R Z 2 ≤C |G(t − s, x, y)| |u(s, y) − u(r, y)| dy + C(t − s)−δ (s − r)δ ZR   2 2 × |G(t − s, x, y) − G(t − r, x, y)| |u(r, y)| + 1 |ej (y)| dy. R Applying Fubini’s theorem, we get Z |ςj,t (u)(s, x) − ςj,t (u)(r, x)|2 dx R Z Z 2 |u(s, y) − u(r, y)| ≤C |G(t − s, x, y)| dx dy R Z R  2 2 |u(r, y)| + 1 |ej (y)| + C(t − s)−δ (s − r)δ R Z  × |G(t − s, x, y) − G(t − r, x, y)| dx dy. R 2 ≤ C ku(s, ·) − u(r, ·)k2 + C(t − s)−δ (s − r)δ Z  Z Z   |u(r, y)|2 + 1 |ej (y)|2 |G(t − s, x, y)| dx + |G(t − r, x, y)| dx dy. × R R R Using again Corollary 2.1 and the bound (A.1), we get Z 2 2 |ςj,t (u)(s, x) − ςj,t (u)(r, x)| dx ≤ C ku(s, ·) − u(r, ·)k2 R Z   |u(r, y)|2 + 1 |ej (y)|2 dy + C(t − s)−δ (s − r)δ R  i h 2 2 ≤ C ku(s, ·) − u(r, ·)k2 + (t − s)−δ (s − r)δ ku(r, ·)k2 + 1 . 22 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR Consequently, kςj,t (u)(s, ·) − ςj,t (u)(r, ·)k2 h i δ δ ≤ C ku(s, ·) − u(r, ·)k2 + (t − s)− 2 (s − r) 2 (ku(r, ·)k2 + 1) . Hence, the statement (ii) is obtained.  Lemma A.2. Let {u(t, ·), t ∈ [0, T ]} be an L2 (R)-valued random process. Then (i) for all 0 < v < s < t < T and δ ∈ ( 13 , 1), δ δ sup kςj,t (u)(v, ·) − ςj,s (u)(v, ·)k2 ≤ C(s − v)− 2 (t − s) 2 (ku(v, ·)k2 + 1) , j∈N (ii) for all 0 < r < v < s < t < T and δ ∈ ( 31 , 1), δ ′ ∈ ( 51 , 1), sup kςj,t (u)(v, ·) − ςj,s (u)(v, ·) − ςj,t (u)(r, ·) + ςj,s (u)(r, ·)k2 j∈N  δ δ ≤ C (t − s) 2 (s − v)− 2 ku(v, ·) − u(r, ·)k2  ′ δ′ δ′ + (t − s) 2 (s − v)−δ (v − r) 2 (ku(r, ·)k2 + 1) . Proof. (i) First, we can produce the relations |ςj,t (u)(v, x) − ςj,s (u)(v, x)| Z   = G(t − v, x, y) h(u(v, y)) ej (y) − G(s − v, x, y) h(u(v, y)) ej (y) dy R Z h i G(t − v, x, y) − G(s − v, x, y) h(u(v, y)) ej (y) dy . = R By using Holder’s inequality, we obtain Z 2 |ςj,t (u)(v, x) − ςj,s (u)(v, x)| ≤ C |G(t − v, x, y) − G(s − v, x, y)| dy R Z × |G(t − v, x, y) − G(s − v, x, y)| |h(u(v, y))|2 |ej (y)|2 dy R Then, we can deduce from Corollary 2.1 2 |ςj,t (u)(v, x) − ςj,s (u)(v, x)| Z 2 2 |G(t − v, x, y) − G(s − v, x, y)| |h(u(v, y))| |ej (y)| dy ≤ C R Z   2 2 |G(t − v, x, y) − G(s − v, x, y)| |u(v, y)| + 1 |ej (y)| dy. ≤ C R Hence, it follows from Fubini theorem that Z 2 |ςj,t (u)(v, x) − ςj,s (u)(v, x)| dx R Z  Z   |u(v, y)|2 + 1 |ej (y)|2 |G(t − v, x, y) − G(s − v, x, y)| dx dy. ≤ C R R By Corollary 2.2, we obtain Z Z   2 2 2 −δ δ |ςj,t (u)(v, x) − ςj,s (u)(v, x)| dx ≤ C(s−v) (t−s) |u(v, y)| + 1 |ej (y)| dy. R R Consequently, by (A.1), δ δ kςj,t (u)(v, ·) − ςj,s (u)(v, ·)k2 ≤ C (s − v)− 2 (t − s) 2 (ku(v, ·)k2 + 1) . FRACTIONAL STOCHASTIC HEAT EQUATION 23 (ii) For the second statement, we have 2 kςj,t (u)(v, ·) − ςj,s (u)(v, ·) − ςj,t (u)(r, ·) + ςj,s (u)(r, ·)k2 Z 2 |ςj,t (u)(v, x) − ςj,s (u)(v, x) − ςj,t (u)(r, x) + ςj,s (u)(r, x)| dx = R Z Z h i = G(t − v, x, y) − G(s − v, x, y) h(u(v, y)) ej (y) dy R Z hR i 2 − G(t − r, x, y) − G(s − r, x, y) h(u(r, y)) ej (y) dy dx R Z Z h = R R i G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y) × h(u(r, y)) ej (y) dy + Z h R i G(t − v, x, y) − G(s − v, x, y) 2 h i × h(u(v, y)) − h(u(r, y)) ej (y) dy dx ≤C Z Z h i G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y) R R 2 × h(u(r, y)) ej (y) dy dx + C Z Z h i G(t − v, x, y) − G(s − v, x, y) R R 2 h i × h(u(v, y)) − h(u(r, y)) ej (y) dy dx = K1 + K2 . In order to get the upper bound for K1 , we apply Holder’s inequality and produce that Z Z h i K1 = G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y) R R 2 × h(u(r, y)) ej (y) dy dx  Z Z G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y) dy ≤ R Z R × |G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)| R  2 2 × |h(u(r, y))| |ej (y)| dy dx  Z Z ≤ C |G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)| dy R R Z × |G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)| R  2 2 × (|u(r, y)| + 1) |ej (y)| dy dx By Corollary 2.1, Z K1 ≤ C |G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)| R2 2 2 × (|u(r, y)| + 1) |ej (y)| dy dx 24 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR Lemma 2.3 and Fubini’s theorem allow us to get Z   |u(r, y)|2 + 1 |ej (y)|2 K1 ≤ C ZR × [G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)] dx dy R Z   2 2 δ′ −2δ ′ δ′ |u(r, y)| + 1 |ej (y)| dy. ≤ C (t − s) (s − v) (v − r) R Therefore, by (A.1), ′ ′ K1 ≤ C (t − s)δ (s − v)−2δ (v − r)δ ′   ku(r, ·)k22 + 1 . In order to get the upper bound for K2 , we apply again the Hölder’s inequality and get that Z Z h ih i 2 G(t − v, x, y) − G(s − v, x, y) h(u(v, y)) − h(u(r, y)) ej (y) dy dx K2 =  ZR ZR G(t − v, x, y) − G(s − v, x, y) dy ≤ R R Z  h i2 2 × G(t − v, x, y) − G(s − v, x, y) h(u(v, y)) − h(u(r, y)) |ej (y)| dy dx R  Z Z 2 ≤ C sup kej k∞ G(t − v, x, y) − G(s − v, x, y) dy j∈N × Z R R R 2  G(t − v, x, y) − G(s − v, x, y) u(v, y) − u(r, y) dy dx. By Corollary 2.1, Z K2 ≤ C 2 R2 G(t − v, x, y) − G(s − v, x, y) u(v, y) − u(r, y) dy dx Therefore, applying Corollary 2.2 and Fubini’s theorem we get that Z 2 δ −δ K2 ≤ C (t − s) (s − v) u(v, y) − u(r, y) dy R 2 = C (t − s)δ (s − v)−δ ku(v, ·) − u(r, ·)k2 ≤ C (t − s)δ (s − v)−δ ku(v, ·) − u(r, ·)k22 Consequently, 2 kςj,t (u)(v, ·) − ςj,s (u)(v, ·) − ςj,t (u)(r, ·) + ςj,s (u)(r, ·)k2  ≤ C (t − s)δ (s − v)−δ ku(v, ·) − u(r, ·)k22   ′ ′ ′ 2 + (t − s)δ (s − v)−2δ (v − r)δ ku(r, ·)k2 + 1 .  Lemma A.3. Let {u(t, ·), t ∈ [0, T ]} and {ũ(t, ·), t ∈ [0, T ]} be two L2 (R)-valued random processes. Then (i) for all 0 < s < t < T , sup kςj,t (u)(s, ·) − ςj,t (ũ)(s, ·)k2 ≤ C ku(s, ·) − ũ(s, ·)k2 , j∈N FRACTIONAL STOCHASTIC HEAT EQUATION 25 (ii) for all 0 < r < s < t < T and for every δ ∈ ( 13 , 1), sup kςj,t (u)(s, ·) − ςj,t (ũ)(s, ·) − ςj,t (u)(r, ·) + ςj,t (ũ)(r, ·)k2 j∈N ≤ C(t − s)−δ (s − r)δ ku(r, ·) − ũ(r, ·)k2 + C ku(s, ·) − ũ(s, ·) − u(r, ·) + ũ(r, ·)k2 . Proof. In order to prove (i), we can apply the Cauchy–Schwartz inequality and Corollary 2.1, and get the following inequalities: Z 2 2 |ςj,t (u)(s, x) − ςj,t (ũ)(s, x)| dx kςj,t (u)(s, ·) − ςj,t (ũ)(s, ·)k2 := R = Z 2 Z G(t − s, x, y) [h(u(s, y)) − h(ũ(s, y))] ej (y) dy dx R R  Z  Z Z 2 ≤C |G(t − s, x, y)|dy |G(t − s, x, y)| [h(u(s, y)) − h(ũ(s, y))] dy dx R R R Z  Z 2 [h(u(s, y)) − h(ũ(s, y))] ≤C |G(t − s, x, y)|dx dy R R ≤ C ku(s, ·) − ũ(s, ·)k22 , whence (i) follows. To proceed with (ii), we apply the Minkowski inequality in order to get the following bounds: 2 kςj,t (u)(s, ·) − ςj,t (ũ)(s, ·) − ςj,t (u)(r, ·) + ςj,t (ũ)(r, ·)k2 Z Z = [G(t − s, x, y) − G(t − r, x, y)] [h(u(r, y)) − h(ũ(r, y))] ej (y) R R 2 + G(t − s, x, y) [h(u(s, y)) − h(ũ(s, y)) − h(u(r, y)) + h(ũ(r, y))] ej (y) dy dx Z Z 2 ≤C |G(t − s, x, y) − G(t − r, x, y)| |h(u(r, y)) − h(ũ(r, y))| dy dx R +C R Z Z R R 2 |G(t − s, x, y)| |h(u(s, y)) − h(ũ(s, y)) − h(u(r, y)) + h(ũ(r, y))| dy dx =: H1 + H2 . It follows from the Cauchy–Schwartz inequality that  Z Z |G(t − s, x, y) − G(t − r, x, y)| dy H1 ≤ C R Z R  2 × |G(t − s, x, y) − G(t − r, x, y)| [h(u(r, y)) − h(ũ(r, y))] dy dx. R By Fubini’s theorem, Z H1 ≤ C (t − s)−δ (s − r)δ R Z  × |G(t − s, x, y) − G(t − r, x, y)| [h(u(r, y)) − h(ũ(r, y))]2 dy dx R ≤ C(t − s)−δ (s − r)δ  Z  Z 2 × [h(u(r, y)) − h(ũ(r, y))] |G(t − s, x, y) − G(t − r, x, y)| dx dy. R R Then, using Corollary 2.2 we obtain 2 H1 ≤ C (t − s)−2δ (s − r)2δ ku(r, ·) − ũ(r, ·)k2 . 26 YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR For the other term, we get by using Cauchy-Schwartz inequality, Lemma 2.1, and Fubini’s theorem,  Z Z |G(t − s, x, y)| dy H2 ≤ C R R Z  2 × |G(t − s, x, y)| [h(u(s, y)) − h(ũ(s, y)) − h(u(r, y)) + h(ũ(r, y))] dy dx R Z  [h(u(s, y)) − h(ũ(s, y)) − h(u(r, y)) + h(ũ(r, y))]2 ≤C R  Z × |G(t − s, x, y)|dx dy, R and Lemma 2.1 allow us to get 2 H2 ≤ C kh(u(s, ·)) − h(ũ(s, ·)) − h(u(r, ·)) + h(ũ(r, ·))k2 . It follows from (1.3) that 2 H2 ≤ C ku(s, ·) − ũ(s, ·) − u(r, ·) + ũ(r, ·)k2 . Hence, we get the result.  Lemma A.4. Let {u(t, ·), t ∈ [0, T ]} and {ũ(t, ·), t ∈ [0, T ]} be two L2 (R)-valued random processes. Then (i) for all 0 < v < s < t < T and for every δ ∈ ( 31 , 1), ∗ ∗ sup ςj,t,s (u)(v, ·) − ςj,t,s (ũ)(v, ·) j∈N δ δ ≤ C (t − s) 2 (s − v)− 2 ku(v, ·) − ũ(v, ·)k2 , 2 (ii) for all 0 < r < v < s < t < T and for any δ ∈ ( 15 , 1), δ ′ ∈ ( 31 , 1), ∗ ∗ ∗ ∗ sup ςj,t,s (u)(v, ·) − ςj,t,s (ũ)(v, ·) − ςj,t,s (u)(r, ·) + ςj,t,s (ũ)(r, ·) j∈N δ 2 δ ≤ C(t − s) 2 (s − v)−δ (v − r) 2 ku(r, ·) − ũ(r, ·)k2 δ′ δ′ + C(t − s) 2 (s − v)− 2 ku(s, ·) − ũ(s, ·) − u(r, ·) + ũ(r, ·)k2 . Proof. In order to prove (i), we can apply the Cauchy–Schwartz inequality, Corollaries 2.1 and 2.2, and get the following inequalities: Z 2 2 ∗ ∗ ∗ ∗ ςj,t,s (u)(v, ·) − ςj,t,s (ũ)(v, ·) 2 = ςj,t,s (u)(v, x) − ςj,t,s (ũ)(v, x) dx R = Z 2 Z [G(t − v, x, y) − G(s − v, x, y)] [h(u(v, y)) − h(ũ(v, y))] ej (y) dy dx R R  Z Z |G(t − v, x, y) − G(s − v, x, y)| dy ≤C R R Z  2 × |G(t − v, x, y) − G(s − v, x, y)| [h(u(v, y)) − h(ũ(v, y))] dy dx R Z  Z 2 [h(u(s, y)) − h(ũ(s, y))] ≤C |G(t − v, x, y) − G(s − v, x, y)|dx dy R R ≤ C(t − s)δ (s − v)−δ ku(s, ·) − ũ(s, ·)k22 . In the other hand, we have by using the fact that for every a, b ∈ R, (a + b)2 ≤ 2(a2 + b2 ) 2 ∗ ∗ ∗ ∗ ςj,t,s (u)(v, ·) − ςj,t,s (ũ)(v, ·) − ςj,t,s (u)(r, ·) + ςj,t,s (ũ)(r, ·) 2 Z ∗ ∗ ∗ ∗ = ςj,t,s (u)(v, x) − ςj,t,s (ũ)(v, x) − ςj,t,s (u)(r, x) + ςj,t,s (ũ)(r, x) R 2 dx FRACTIONAL STOCHASTIC HEAT EQUATION = Z Z R R 27 [G(t − v, x, y) − G(s − v, x, y)] × [h(u(v, y)) − h(ũ(v, y)) − h(u(r, y)) + h(ũ(r, y))]ej (y) dy Z + [h(u(r, y)) − h(ũ(r, y))] R 2 × [G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)]ej (y) dy dx Z Z ≤C [G(t − v, x, y) − G(s − v, x, y)] R R 2 × [h(u(v, y)) − h(ũ(v, y)) − h(u(r, y)) + h(ũ(r, y))]ej (y) dy dx Z Z +C [h(u(r, y)) − h(ũ(r, y))] R R 2 × [G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)]ej (y) dy dx =: D1 + D2 . Applying the Cauchy–Schwarz inequality and Fubini’s theorem, combined with Corollaries 2.1 and 2.2, we get Z Z [G(t − v, x, y) − G(s − v, x, y)] D1 = R R 2 × [h(u(v, y)) − h(ũ(v, y)) − h(u(r, y)) + h(ũ(r, y))]ej (y) dy dx  Z Z Z |G(t − v, x, y) − G(s − v, x, y)|dy ≤ |G(t − v, x, y) − G(s − v, x, y)| R R R  × |h(u(v, y)) − h(ũ(v, y)) − h(u(r, y)) + h(ũ(r, y))|2 dy dx ′ 2 ′ ≤ C (t − s)δ (s − v)−δ ku(s, ·) − ũ(s, ·) − u(r, ·) + ũ(r, ·)k2 . By the same technique as before and from Lemma 2.3, we obtain Z Z D2 = [G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)] R R 2 × [h(u(r, y)) − h(ũ(r, y))]ej (y) dy dx  Z Z |G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)|dy ≤ R Z R × |G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)| R  × |h(u(r, y)) − h(ũ(r, y))|2 dy dx ≤ C (t − s)δ (s − v)−2δ (v − r)δ ku(r, ·) − ũ(r, ·)k22 . Hence, we get the result.  Acknowledgments Y. Mishura and K. 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Zili, Construction d’une solution fondamentale d’une équation aux dérivées partielles à coefficients constants par morceaux, Bull. Sci. Math. 123 (1999) 115–155. [24] M. Zili, Fundamental solution of a parabolic partial differential equation with piecewise constant coefficients and admitting a generalized drift, Int. J. Appl. Math. 2 (2000) 1073–1110. [25] M. Zili and E. Zougar, One-dimensional stochastic heat equation with discontinuous conductance, Appl. Anal. 98 (2019), 2178–2191. [26] M. Zili and E. Zougar, Exact variations for stochastic heat equations with piecewise constant coefficients and application to parameter estimation, Theory Probab. Math. Statist. 100 (2019), 75–101. FRACTIONAL STOCHASTIC HEAT EQUATION 29 (Y. Mishura) Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska St., Kyiv, 01601, Ukraine E-mail address: myus@univ.kiev.ua (K. Ralchenko) Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska St., Kyiv, 01601, Ukraine E-mail address: k.ralchenko@gmail.com (M. Zili) University of Monastir, Department of Mathematics, Faculty of sciences of Monastir, Avenue de l’environnement, 5019 Monastir, Tunisia E-mail address: mounir.zili@fsm.rnu.tn (E. Zougar) University of Monastir, Department of Mathematics, Faculty of sciences of Monastir, Avenue de l’environnement, 5019 Monastir, Tunisia E-mail address: zougareya@gmail.com