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Moment bounds in spde’s with application to the stochastic wave equation Le Chen and Robert C. Dalang ∗ ∗ Institut de mathématiques 4 Ećole Polytechnique Fédérale de Lausanne 1 Station 8 0 2 CH-1015 Lausanne n Switzerland a J e-mails: le.chen@epfl.ch, robert.dalang@epfl.ch 5 2 ] R Abstract: We exhibit a class of properties of an spde that guarantees existence, P uniqueness and bounds on moments of the solution. These moment bounds are ex- . h pressed in terms of quantities related to the associated deterministic homogeneous t a p.d.e. With these, we can, for instance, obtain solutions to the stochastic heat m equation on the real line for initial data that falls in a certain class of Schwartz [ distributions, but our main focus is the stochastic wave equation on the real line with irregular initial data. We give bounds on higher moments, and for the hyper- 1 bolic Anderson model, explicit formulas for second moments. We establish weak v intermittency and obtain sharp bounds on exponential growth indices for certain 6 0 classes of initial conditions with unbounded support. Finally, we relate Hölder- 5 continuitypropertiesofthestochasticintegralpartofthesolutiontothestochastic 6 wave equation to integrability properties of the initial data, obtaining the optimal . 1 Hölder exponent. 0 4 MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60. 1 : Keywords: nonlinear stochastic wave equation, hyperbolic Anderson model, in- v termittency, growth indices, Hölder continuity. i X r a 1 Introduction Consider a partial differential operator in the time and space variables (t,x) and a space- time white noise W˙ (t,x), where t R∗ L= R 0 and x Rd, along with a function θ(t,x). ∈ + +\{ } ∈ Research partially supported by the Swiss National Foundation for Scientific Research. ∗ 1 We are interested in determining when the stochastic partial differential equation (spde) u(t,x) = ρ(u(t,x))θ(t,x)W˙ (t,x) , x Rd, t R∗ , (1.1) L ∈ ∈ + with appropriate initial conditions, admits as solution a random field (u(t,x), (t,x) R + Rd). In this case, we would like estimates and asymptotic properties of moments of u(∈t,x), a×s well as Ho¨lder-continuity properties. In this paper, we will develop such estimates for a wide class of operators , functions θ and initial conditions, with an emphasis on the stochastic L wave and heat equations. One basic example, which also was the starting point of this study, is the parabolic Andersonmodel. Inthiscase, d = 1, = ∂ κ2 ∂2 , ρ(x) = λxandθ 1. Theintermittency L ∂t− ∂x2 ≡ property of this equation, as defined in [7], is studied via the moment Lyapounov exponents, in which estimates of the moments play a key role. Indeed, recall that the upper and lower moment Lyapunov exponents for constant initial data are defined as follows: logE[ u(t,x) p] logE[ u(t,x) p] m (x) :=limsup | | , m (x) := liminf | | . (1.2) p t→+∞ t p t→+∞ t If the initial conditions are constants, then m (x) =: m and m (x) =: m do not depend on p p p p x. Intermittency is the property that m = m =: m and m < m /2 < < m /p < . p p p 1 2 p ··· ··· It is implied by the property m = 0 and m > 0 (see [7, Definition III.1.1, on p. 55]), which 1 2 is called full intermittency, while weak intermittency, defined in [29] and [17, Theorem 2.3] is the property m > 0 and m < + , for all p 2. 2 p ∞ ≥ Another property of the parabolic Anderson model is described by the behavior of exponential growth indices, initiated by Conus and Khoshnevisan in [17]. They defined 1 λ(p) :=sup α > 0 : limsup sup logE( u(t,x) p) > 0 , (1.3) t | | (cid:40) t→∞ |x|≥αt (cid:41) 1 λ(p) :=inf α > 0 : limsup sup logE( u(t,x) p) < 0 , (1.4) t | | (cid:40) t→∞ |x|≥αt (cid:41) This is again a property of moments of the solution u(t,x). In the recent paper [11], in the case θ 1, the authors have given minimal conditions on ≡ the initial data for existence, uniqueness and moments estimates in the parabolic Anderson model, building on the previous results of [2, 16]. The initial condition can be a signed measure, but not a Schwartz distribution that is not a measure, such as the derivative δ(cid:48) of 0 the Dirac delta function. Exact formulas for the second moments were determined for the parabolic Anderson model, along with sharp bounds for other moments and choices of the function ρ. Our program is to extend these kinds of results to many other classes of spde’s. Recall that an spde such as (1.1) is often rigorously formulated as an integral equation of the form u(t,x) = J (t,x)+ G(t s,x y)ρ(u(s,y))θ(s,y)W(ds,dy), (1.5) 0 − − (cid:90)(cid:90)R+×Rd 2 where J : R Rd represents the solution of the (deterministic) homogeneous p.d.e. with 0 + × the appropriate initial conditions, and G(t,x) is the fundamental solution of the p.d.e. The stochastic integral in (1.5) is defined in the sense of Walsh [46]. In a first stage, we shall focus on the equation (1.5), for given functions J and G satisfying suitable assumptions, 0 even if they are not specifically related to a partial differential operator . For this, the first L step is to develop a unified set of assumptions which are sufficient to guarantee the existence, uniqueness and moment estimates of the solution to (1.1). All of these assumptions should be satisfied for the J and G associated with the stochastic heat equation, so as to contain 0 the results of [11]. It will turn out that in fact, they can be verified for quite different equations, such as the stochastic wave equation, which we discuss in this paper, and the stochastic heat equation with fractional spatial derivatives as well as other equations, which will be discussed in forthcoming papers. The assumptions are given in Section 2.1. In particular, G must be a function with certain continuity and integrability properties, and must satisfy certain bounds, including tail control, and an L2-continuity property. Another assumption relates properties of the function J with those of G. Finally, a last set of assumptions concerns the function 0 K obtained by summing n-fold space-time convolutions of the square of G with itself. Our first theorem (Theorem 2.13) states that under these assumptions, we obtain existence, uniqueness and moment bounds of the solution to (1.5). When particularized to the stochastic heat equation, all the assumptions are satisfied and the bounds are the same as those obtained in [11]. Recall that θ(t,x) 1 in [11]. Here, as an application of our first theorem, we will show ≡ in Theorem 2.22 that by choosing θ so that θ(t,x) 0 as t 0 (which means that we taper → ↓ off the noise near t = 0), we can extend the class of admissible initial conditions in the stochastic heat equation beyond signed measures. And the more the noise near the origin is killed, the more irregular the initial condition may be. The balance between the admissible initial data and certain properties of the function θ is stated in Theorem 2.22. For instance, if θ(t,x) 1, then the initial data cannot go beyond measures; if θ(t,x) = tr 1 for some ≡ ∧ r > 0, then the initial data can be δ(k) for all integers k [0,2r + 1/2[ , where δ(k) is the 0 0 ∈ k-th distributional derivative of the Dirac delta function δ ; if θ(t,x) = exp( 1/t), then any 0 − Schwartz (or tempered) distribution can serve as the initial data (see Examples 2.24 and 2.25). Thesecondandmainapplicationinthispaperofourfirsttheoremconcernsthestochastic wave equation: ∂2 κ2 ∂2 u(t,x) = ρ(u(t,x))W˙ (t,x), x R, t R∗, ∂t2 − ∂x2 ∈ ∈ + (1.6) (cid:40)(cid:16) u(0, ) = g((cid:17)), ∂u(0, ) = µ( ), · · ∂t · · where R∗ = ]0, [ , W˙ is space-time white noise, ρ(u) is globally Lipschitz, κ > 0 is the + ∞ speed of wave propagation, g and µ are the (deterministic) initial position and velocity, respectively. The linear case, ρ(u) = λu, λ = 0, is called the hyperbolic Anderson model [23]. (cid:54) 3 This equation has been intensively studied during last two decades by many authors: see e.g., [6, 8, 9, 41, 46] for some early work, [20, 46] for an introduction, [23, 24] for the intermittency problems, [15, 21, 22, 25, 35, 42, 43] for the stochastic wave equation in the spatial domain Rd, d > 1, [26, 45] for regularity of the solution, [4, 5] for the stochastic wave equation with values in Riemannian manifolds, [13, 39, 40] for wave equations with polynomial nonlinearities, and [36, 37, 44] for smoothness of the law. Concerningintermittencyproperties, DalangandMuellershowedin[23]thatforthewave equation in spatial domain R3 with spatially homogeneous colored noise, with ρ(u) = u and constant initial position and velocity, the Lyapunov exponents m and m are both p p bounded, from above and below respectively, by some constant times p4/3. For the stochastic wave equation in spatial dimension 1, Conus et al [17] show that if the initial position and velocity are bounded and measurable functions, then the moment Lyapunov exponents satisfy m Cp3/2 for p 2, and m c(κ/2)1/2 for positive initial data. The difference in p 2 ≤ ≥ ≥ the exponents—3/2 versus 4/3 in the three dimensional wave equation—reflects the distinct nature of the driving noises. Recently Conus and Balan [1] studied the problem when the noise is Gaussian, spatially homogeneous and behaves in time like a fractional Brownian motion with Hurst index H > 1/2. Regarding exponential growth indices, Conus and Khoshnevisan [18, Theorem 5.1] show that for initial data with exponential decay at , 0 < λ(p) λ(p) < + , for all p 2. ±∞ ≤ ∞ ≥ They also show that if the initial data consists of functions with compact support, then λ(p) = λ(p) = κ, for all p 2. ≥ One objective of our study is to understand how irregular (and possibly unbounded) initial data affects the random field solutions to (1.6); another is to continue the study of moment Lyapounov exponents and exponential growth indices of [17, 18]. We will only assume that the initial position g belongs to L2 (R), the set of locally square integrable loc Borel functions, and the initial velocity µ belongs to (R), the set of locally finite Borel M measures. These assumptions are natural since the weak solution to the homogeneous wave equation is 1 J (t,x) := (g(x+κt)+g(x κt))+(µ G (t, ))(x) , (1.7) 0 κ 2 − ∗ ◦ where 1 G (t,x) = H(t)1 (x) κ [−κt,κt] 2 is the wave kernel function. Here, H(t) is the Heaviside function (i.e., H(t) = 1 if t 0 and ≥ 0 otherwise), and denotes convolution in the space variable. ∗ Regarding the spde (1.6), we interpret it in the integral (mild) form (1.5): u(t,x) = J (t,x)+I(t,x), (1.8) 0 where I(t,x) := G (t s,x y)ρ(u(s,y))W (ds,dy). κ − − (cid:90)(cid:90)[0,t]×R 4 We show that all the assumptions of Section 2.1 are verified for this equation. More importantly, the abstract bounds take an explicit form since the function can be evaluated K explicitly (see Theorem 3.1). This was also the case for the stochastic heat equation [11], but the formula for here is quite different than in this reference. We also obtain explicit K formulas for the second moment of the solution in the hyperbolic Anderson model, as well as sharp bounds for higher moments. These bounds also apply to other choices of ρ. For some particular choices of initial data (such as constant initial position and velocity, or vanishing initial position and Dirac initial velocity), the second moment of the solution takes a particularly simple form (see Corollaries 3.2 and 3.3 below). As an immediate consequence of Theorem 3.1, we obtain the result m Cp3/2 for p 2 p ≤ ≥ of [17] (see Theorem 3.11). We extend their lower bound on the upper Lyapunov exponent m to the lower Lyapounov exponent, by showing that m c(κ/2)1/2. In the case of the 2 2 ≥ Anderson model ρ(u) = λu, we show that m = m = λ (κ/2)1/2. 2 2 | | Concerning exponential growth indices, we use Theorem 3.1 to give specific upper and lower bounds on these indices. For instance, we show in Theorem 3.14 that if the initial position and velocity are bounded below by ce−β|x| and above by Ce−β˜|x|, with β β˜, then ≥ 1 1 l2 2 L2 2 κ 1+ λ(p) λ(p) κ 1+ , 8κβ2 ≤ ≤ ≤ 8κβ˜2 (cid:18) (cid:19) (cid:18) (cid:19) for certain explicit constants l and L. In the case of the Anderson model ρ(u) = λu and for ˜ p = 2 and β = β, we obtain λ2 1/2 λ(2) = λ(2) = κ 1+ . 8κβ2 (cid:18) (cid:19) Since the exponential growth indices of order 2 depend on the asymptotic behavior of E(u(t,x)2) as t , this equality highlights, in a somewhat surprising way, how the → ∞ initial data significantly affects the behavior of the solution for all time, despite the presence of the driving noise. A final question concerns the sample path regularity properties. Denote by C (D) the β1,β2 set of trajectories that are β -Ho¨lder continuous in time and β -Ho¨lder continuous in space 1 2 on the domain D R R, and let + ⊆ × C (D) := C (D) . β1−,β2− ∩α1∈]0,β1[ ∩α2∈]0,β2[ α1,α2 Carmona and Nualart [9, p.484–485] showed that if the initial position is constant and the initial velocity vanishes, then the solution is in C (R R) a.s. This property can 1/2−,1/2− + also be deduced from [45, Theorem 4.1]. The case where the sp×atial domain is R3 has been studied in [26, 20]. In [17], Conus et al establish Ho¨lder-continuity properties of x u(t,x) (t fixed). In (cid:55)→ particular, they show that if the initial position g is a 1/2-Ho¨lder-continuous function and the initial velocity is square-integrable, then x u(t,x) is (1 (cid:15))-Ho¨lder-continuous. The (cid:55)→ 2 − 5 assumption on the initial data is needed, since the Ho¨lder-continuity properties of the initial position are not smoothed out by the wave kernel but are transferred to J (t,x) via formula 0 (1.7). A related question concerns the stochastic term I(t,x) of (1.8), which represents the difference u(t,x) J (t,x) between the solution of (1.6) and the solution to the homogeneous 0 − wave equation. We are interested in understanding how properties of the initial data affect the regularity of (t,x) I(t,x). We show in Theorem 4.1 that the better the (local) (cid:55)→ integrability properties of the initial position g, the better the regularity of (t,x) I(t,x). In particular, if g L2γ(R), γ 1, and µ (R), then (t,x) I(t,x) b(cid:55)→elongs to C R∗ R ∈,whleorce 1+1 =≥1. Weshow∈inPMroposition4.2that(cid:55)→theHölder-exponents 21γ(cid:48)−,21γ(cid:48)− + × γ γ(cid:48) 1 are optimal. 2γ(cid:48) (cid:0) (cid:1) This paper is organized as follows. In Section 2, we study our abstract integral equation andpresentthemainresultinTheorem2.13. Theapplicationtothestochasticheatequation withdistribution-valuedinitialdataisgiveninSection2.3. Section3containstheapplication to the stochastic wave equation. The main results on existence, uniqueness and formulas and bounds on moments are stated in Section 3.1 and proved in Section 3.2. The weak intermittency property is established in Section 3.3. The bounds on exponential growth indices are given in Section 3.4, and proved in Section 3.5. Finally, Section 4 contains our results on Ho¨lder continuity of the solution of the stochastic wave equation. 2 Stochastic integral equation of space-time convolution type We begin by stating the main assumptions which will be needed in our theorem on existence, uniqueness and moment bounds. 2.1 Assumptions Let W (A) : A Rd , t 0 be a space-time white noise defined on a complete prob- t b ability space (Ω∈, B,P), wher≥e Rd is the collection of Borel sets with finite Lebesgue b (cid:8) F (cid:0) (cid:1) B(cid:9) measure. Let ( , t 0) be the standard filtration generated by this space-time white t noise, i.e., =Fσ W≥(A) : 0 s (cid:0) t,(cid:1)A Rd , where is the σ-field generated t s b F ≤ ≤ ∈ B ∨ N N by all P-null sets in . We use to denote the Lp(Ω)-norm. A random field Y(t,x), (cid:0) F ||·||p (cid:0) (cid:1)(cid:1) (t,x) R∗ Rd, is said to be adapted if for all (t,x) R∗ Rd, Y(t,x) is -measurable, and it∈is s+ai×d to be jointly measurable if it is measurab∈le w+i×th respect to (RF∗t Rd) . For p 2, if lim Y(t,x) Y (t(cid:48),x(cid:48)) = 0 for all (t,x) R∗ RBd, t+he×n Y is×saFid ≥ (t(cid:48),x(cid:48))→(t,x)|| − ||p ∈ + × to be Lp(Ω)-continuous. Let G, J : R Rd R be deterministic Borel functions. We use the convention that 0 + × (cid:55)→ G(t, ) 0 if t 0. In the following, we will use and to denote the time and space dummy · ≡ ≤ · ◦ variables respectively. 6 Definition 2.1. A random field (u(t,x), (t,x) R Rd), is called a solution to (1.5) if + ∈ × (1) u(t,x) is adapted and jointly measurable; (2) For all (t,x) R∗ Rd, G2( , )(cid:63) ρ(u( , )) 2θ2( , ) (t,x) < + , where (cid:63) denotes ∈ +× · ◦ || · ◦ ||2 · ◦ ∞ the simultaneous convolution in both space and time variables, and the function (t,x) I(t,x) from R Rd into(cid:0) L2(Ω) is (cid:2)continuous; (cid:3)(cid:1) (cid:55)→ + × (3) u(t,x) = J (t,x)+I(t,x), where for all (t,x) R Rd, 0 + ∈ × I(t,x) = G(t s,x y)ρ(u(s,y))θ(s,y)W (ds,dy), a.s. (2.1) − − (cid:90)(cid:90)R+×Rd We call I(t,x) the stochastic integral part of the random field solution. This stochastic integral is interpreted in the sense of Walsh [46]. Remark 2.2. Consider the stochastic wave equation (1.6) with g L2 (R) and µ = 0. ∈ loc In this case, J (t,x) = 1/2(g(κt+x)+g(κt x)). Since the initial position g may not 0 − be defined for every x, the function (t,x) J (t,x) may not be defined for certain (t,x). 0 (cid:55)→ Therefore, for these (t,x), u(t,x) may not be well-defined (see Example 3.4). Nevertheless, as we will show later, I(t,x) is always well defined for each (t,x) R R, and in most + ∈ × cases (when Assumption 2.14 below holds), it has a continuous version. Finally, we remark that for the stochastic heat equation with deterministic initial conditions, this problem does not arise because in that equation, (t,x) J (t,x) is continuous over R∗ R thanks to the (cid:55)→ 0 +× smoothing effect of the heat kernel. As in [21], a very first issue is whether the linear equation, where ρ(u) 1, admits a random field solution. For t R , and x,y Rd, this leads to examining the≡quantity + ∈ ∈ Θ(t,x,y) := dsdz G(t s,x z)G(t s,y z)θ2(s,z) . (2.2) − − − − (cid:90)(cid:90)[0,t]×Rd Clearly, 2Θ(t,x,y) Θ(t,x,x)+Θ(t,y,y). ≤ Assumption 2.3. G(t,x) is such that (i) Θ(t,x,x) < + for all (t,x) R Rd; + (ii) lim G∞(t(cid:48),x(cid:48)) = G(t,∈x), for×almost all (t,x) R Rd. (t(cid:48),x(cid:48))→(t,x) + ∈ × If θ(t,x) 1, d = 1 and if the underlying partial differential operator is ∂ , where ≡ ∂t −A A isthegeneratorofareal-valuedLévyprocesswiththeLévyexponentΨ(ξ), thenAssumption 2.3 (i) is equivalent to 1 dξ < + , for all β > 0, where Ψ(ξ) is the real part of 2π R β+2(cid:60)Ψ(ξ) ∞ (cid:60) Ψ(ξ): see [21, 29]. For the one-dimensional stochastic heat equation studied in [11], it is (cid:82) also clearly satisfied. For the stochastic wave equation (1.6), this assumption also holds: see (3.6). 7 Assumption 2.4. For all compact sets K R∗ Rd and all integers p 2, ⊆ + × ≥ sup 1+J2 θ2 (cid:63)G2 (t,x) < + . 0 ∞ (t,x)∈K (cid:0)(cid:0)(cid:2) (cid:3) (cid:1) (cid:1) We note that a related assumption appears in [9, Proposition 1.8]. The next three assumptions will be used to establish the Lp(Ω)-continuity in a Picard iteration. Assumption 2.5 is for kernel functions similar to the wave kernel and Assumptions 2.6–2.8 are for those similar to the heat kernel. We need some notation: for β ]0,1[ , τ > 0, α > 0 and (t,x) R∗ Rd, define ∈ ∈ + × B := (t(cid:48),x(cid:48)) R∗ Rd : βt t(cid:48) t+τ, x x(cid:48) α . (2.3) t,x,β,τ,α ∈ + × ≤ ≤ | − | ≤ Assumption 2.5 (Unifor(cid:8)mly bounded kernel functions). There exist thre(cid:9)e constants β ]0,1[, τ > 0 and α > 0 such that for all (t,x) R∗ Rd, for some constant C > 0, we hav∈e for all (t(cid:48),x(cid:48)) B and all (s,y) [0,t(cid:48)[ ∈Rd,+G×(t(cid:48) s,x(cid:48) y) C G(t+1 s,x y). t,x,β,τ,α ∈ ∈ × − − ≤ − − Assumption 2.6 (Tail control of kernel functions). There exists β ]0,1[ such that for all (t,x) R∗ Rd, for some constant a > 0, we have for all (t(cid:48),x(cid:48))∈ B and all s [0,t(cid:48)[∈and+y× Rd with y a, G(t(cid:48) s,x(cid:48) y) G(t+1 s,x y∈). t,x,β,1/2,1 ∈ ∈ | | ≥ − − ≤ − − Assumption 2.7. For all (t,x) R∗ Rd, ∈ + × lim dsdy θ(s,y)2(G(t(cid:48) s,x(cid:48) y) G(t s,x y))2 = 0. (t(cid:48),x(cid:48))→(t,x)(cid:90)(cid:90)R+×Rd − − − − − Note that this assumption can be more explicitly expressed in the following way: t∗ ds dy θ(s,y)2(G(t(cid:48) s,x(cid:48) y) G(t s,x y))2 − − − − − (cid:90)0 (cid:90)Rd tˆ + ds dy θ(s,y)2G2 tˆ s,xˆ y 0, (2.4) − − → (cid:90)t∗ (cid:90)Rd (cid:0) (cid:1) as (t(cid:48),x(cid:48)) (t,x), where → (t(cid:48),x(cid:48)) if t(cid:48) t, (t,x) if t(cid:48) t. (t ,x ) = ≤ and tˆ,xˆ = ≤ (2.5) ∗ ∗ (t,x) if t(cid:48) > t, (t(cid:48),x(cid:48)) if t(cid:48) > t. (cid:40) (cid:40) (cid:0) (cid:1) Assumption 2.8. For all compact sets K R∗ Rd, sup J (t,x) < . ⊆ +× (t,x)∈K | 0 | ∞ The remaining assumptions are mainly needed for control of the moments of the solution. We introduce some notation. For two functions f,g : R Rd R , define their θ-weighted + + × (cid:55)→ space-time convolution by (f (cid:66)g)(t,x) := θ2f (cid:63)g (t,x), for all (t,x) R Rd, + ∈ × (cid:0)(cid:0) (cid:1) (cid:1) 8 In the following, f(t,x) will play the role of J2(t,x), and g(t,x) of G2(t,x). In the Picard 0 iteration scheme, the expression (( ((f (cid:66)g )(cid:66)g )(cid:66) )(cid:66)g )(t,x) will appear, where 1 2 n g = g. Since (cid:66) is not associative in·g·e·neral (contrary to·t·h·e case θ 1), we need to handle i ≡ this formula with care. Definition 2.9. Let n 2 and let g : R Rd R , k = 1,...,n. Define the θ-weighted k + + multiple space-time conv≥olution, for (t,x), (×s,y) (cid:55)→R Rd with 0 s t, by + ∈ × ≤ ≤ (cid:66) (g ,g ,...,g )(t,x;s,y) n 1 2 n s := ds dy g (s s ,y y )θ2(t s+s ,x y +y ) n−1 n−1 n n−1 n−1 n−1 n−1 − − − − (cid:90)0 (cid:90)Rd sn−1 ds dy g (s s ,y y )θ2(t s+s ,x y +y ) n−2 n−2 n−1 n−1 n−2 n−1 n−2 n−2 n−2 × − − − − (cid:90)0 (cid:90)Rd s3 ds dy g (s s ,y y )θ2(t s+s ,x y +y ) 2 2 3 3 2 3 2 2 2 ×······× − − − − (cid:90)0 (cid:90)Rd s2 ds dy g (s s ,y y )θ2(t s+s ,x y +y )g (s ,y ). (2.6) 1 1 2 2 1 2 1 1 1 1 1 1 × − − − − (cid:90)0 (cid:90)Rd Notice that (cid:66) (g ,...,g )(t,x;t,x) = (( ((g (cid:66)g )(cid:66)g )(cid:66) )(cid:66)g )(t,x), n 1 n 1 2 3 n ··· ··· where the r.h.s. has n 1 convolutions. By the change of variables − τ = s s , τ = s s , , τ = s s , and 1 n−1 2 n−2 n−1 1 − − ··· − (2.7) z = y y , z = y y , , z = y y , 1 n−1 2 n−2 n−1 1 − − ··· − and Fubini’s theorem, the multiple convolution (cid:66) has an equivalent definition: n (cid:66) (g ,g ,...,g )(t,x;s,y) n 1 2 n s = dτ dz θ2(t τ ,x z )g (s τ ,y z ) n−1 n−1 n−1 n−1 1 n−1 n−1 − − − − (cid:90)0 (cid:90)Rd τn−1 dτ dz θ2(t τ ,x z )g (τ τ ,z z ) × n−2 n−2 − n−2 − n−2 2 n−1 − n−2 n−1 − n−2 (2.8) (cid:90)0 (cid:90)Rd τ3 dτ dz θ2(t τ ,x z )g (τ τ ,z z ) 2 2 2 2 n−2 3 2 3 2 ×······× − − − − (cid:90)0 (cid:90)Rd τ2 dτ dz θ2(t τ ,x z )g (τ τ ,z z )g (τ ,z ) . 1 1 1 1 n−1 2 1 2 1 n 1 1 × − − − − (cid:90)0 (cid:90)Rd Lemma 2.10. Let f,g : R Rd R , k = 1,...,n + 1, and n 2. Then for all k + + (t,x) R Rd, we have × (cid:55)→ ≥ + ∈ × (( ((f (cid:66)g )(cid:66)g )(cid:66) )(cid:66)g )(t,x) = (f (cid:66)(cid:66) (g ,...,g )(t,x; , ))(t,x), (2.9) 1 2 n n 1 n ··· ··· · ◦ 9 (f (cid:66)(cid:66) (g ,...,g )(t,x; , ))(t,x) = ((f (cid:66)g )(cid:66)(cid:66) (g ,...,g )(t,x; , ))(t,x), (2.10) n 1 n 1 n−1 2 n · ◦ · ◦ and t ds dy(f (cid:66)(cid:66) (g ,...,g )(s,y; , ))(s,y) θ2(s,y)g (t s,x y) n 1 n n+1 · ◦ − − (cid:90)0 (cid:90)Rd = (f (cid:66)(cid:66) (g ,...,g )(t,x; , ))(t,x). (2.11) n+1 1 n+1 · ◦ Note that (s,y) appears twice in the term f (cid:66)(cid:66) ( ) on the l.h.s. of (2.11). The proof n ··· of Lemma 2.10 is straightforward; see [10, Lemma 3.2.6] for details. When n = 2, for f and g : R Rd R , (cid:66) (f,g)(t,x;t,x) = (f (cid:66)g)(t,x) and + + 2 × (cid:55)→ s (cid:66) (f,g)(t,x;s,y) = ds dy g(s s ,y y )θ2(t s+s ,x y +y )f (s ,y) 2 0 0 0 0 0 0 0 0 − − − − (cid:90)0 (cid:90)Rd (2.12) s = dτ dz θ2(t τ ,x z )f (s τ ,y z )g(τ ,z ). (2.13) 0 0 0 0 0 0 0 0 − − − − (cid:90)0 (cid:90)Rd In particular, if θ(t,x) 1, then (cid:66) reduces to the standard space-time convolution (cid:63) (as is 2 the case for (cid:66)), in whic≡h case the first two variables (t,x) do not play a role. We call (2.12) and (2.6) the forward formulas, and (2.13) and (2.8) the backward formulas. For λ R, define (t,x;λ) := λ2G2(t,x), and for n N∗, 0 ∈ L ∈ (t,x;s,y;λ) := (cid:66) ( , ;λ),..., ( , ;λ) (t,x;s,y) n n+1 0 0 L L · ◦ L · ◦ for all (t,x),(s,y) R∗ Rd with s (cid:0)t. By convention, (t,x(cid:1);s,y;λ) = λ2G2(s,y). For n N, define ∈ + × ≤ L0 ∈ (t,x;λ) := (1(cid:66) (t,x; , ;λ))(t,x). n n H L · ◦ By definition, both and are non-negative. We use the following conventions: n n L H (t,x;s,y) := (t,x;s,y; λ), (t,x;s,y) := (t,x;s,y; L ), n n n ρ L L L K (2.14) (t,x;s,y) := (t,x;s,y; l ), (t,x;s,y) := (t,x;s,y; a z L ), p 2 , n n ρ n n p,ς p ρ L L L L ≥ where the constant a ( 2) is defined by p,ς (cid:98) ≤ 2(p−1)/p ς = 0, p > 2, (cid:54) a := √2 ς = 0, p > 2, (2.15) p,ς  1 p = 2, and z is the optimal universal constant in the Burkholder-Davis-Gundy inequality (see p [18, Theorem 1.4]) and so z = 1 and z 2√p for all p 2. Note that the kernel 2 p ≤ ≥ function (t,x;s,y) depends on the parameters p and ς, which is usually clear from the n L context. Similarly, define (t,x), (t,x) and (t,x). The same conventions will apply n n n H H H to (t,x;(cid:98)s,y), (t,x;s,y), (t,x;s,y) and (t,x;s,y) below. K K K K (cid:98) 1(cid:98)0