Considering stochastic partial differential equations of parabolic type with
random coefficients in vector-valued H\"older spaces, we obtain a sharp
Schauder estimate. As an application, the existence and uniqueness of solution
to the Cauchy problem is also proved.Comment: This is an abridged version. A full version is submitted separatel

Du, Kai

Liu, Jiakun

English

arXiv

Kai Du

Jiakun Liu

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 354 (2016) 371–375Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comPartial differential equations/Probability theoryA Schauder estimate for stochastic PDEsUne estimée de Schauder pour des EDPs stochastiquesKai Du, Jiakun Liu 1Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australiaa r t i c l e i n f o a b s t r a c tArticle history:Received 18 September 2015Accepted 25 January 2016Available online 11 February 2016Presented by the Editorial BoardConsidering stochastic partial differential equations of parabolic type with random coefficients in vector-valued Hölder spaces, we establish a sharp Schauder theory. The existence and uniqueness of solutions to the Cauchy problem is obtained.© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous considérons des équations aux dérivées partielles stochastiques, du type parabolique et à coefficients aléatoires dans des espaces de Hölder à valeurs vectorielles. Nous obtenons une estimée de Schauder optimale, puis nous utilisons cette estimée pour prouver l’existence et l’unicité de la solution du problème de Cauchy.© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionWe consider the second-order stochastic partial differential equations (SPDEs) of the Itô typedu = (aijuxi x j + biuxi + cu + f )dt + (σ ikuxi + νku + gk)dwkt , (1.1)in Rn × (0, ∞), where wk are countable independent standard Wiener processes defined on a filtered complete probability space (, F , (Ft)t∈R, P) for k = 1, 2, · · · . The matrix a = (aij) is symmetric, and the uniform parabolic condition is assumed throughout the paper, namely there is a constant λ > 0 such that2aij − σ ikσ jk ≥ λδi j on Rn × (0,∞) × , (1.2)where δi j is the Kronecker delta. The random fields u, aij, bi, f are all real-valued, while σ i, ν and g take values in 2. One of the most important examples of (1.1) is the Zakai equation arising in the nonlinear filtering problem [15].E-mail addresses: kaid@uow.edu.au (K. Du), jiakunl@uow.edu.au (J. Liu).1 Research of Liu was supported by the Australian Research Council DE140101366.http://dx.doi.org/10.1016/j.crma.2016.01.0101631-073X/© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.372 K. Du, J. Liu / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 371–375The regularity of solutions to (1.1) in Sobolev spaces has already been investigated by many researchers. Various aspects of L2-theory were studied since 1970s, see [11,9,13,1] and references therein. Later on, a complete Lp -theory was established by Krylov in 1990s, see [7,8]. By using Sobolev’s embedding, one then has the regularity in Hölder spaces, which is however not sharp. As an open problem mentioned in [8], one desires a sharp C2+α-theory in the sense that not only that for f , gbelonging to a proper space F , the solution belongs to some kind of stochastic C2+α-spaces, but also that every element of this stochastic space can be obtained as a solution for certain f , g belonging to the same F .The purpose of this paper is to establish a Schauder theory of Equation (1.1), which is sharp in the above sense. In order to state our main results, we first introduce a notion of quasi-classical solutions.Definition 1.1. A random field u is called a quasi-classical solution to (1.1) if(1) for each t ∈ (0, ∞), u(·, t) is a twice strongly differentiable function from Rn to Lγω := Lγ (; R) for some γ ≥ 2; and(2) for each x ∈ Rn , the process u(x, ·) satisfies (1.1) in the Itô integral form with respect to the time variable.If furthermore, u(·, t, ω) ∈ C2(Rn) for any (t, ω) ∈ (0, ∞) × , then u is a classical solution to (1.1).Analogously to classical Hölder spaces, we can define the Lγω-valued Hölder spaces Cm+αx (QT ; Lγω) and Cm+α,α/2x,t (QT ; Lγω), where T > 0, QT = Rn × (0, T ), and Lγω := Lγ (; R) is a Banach space equipped with the norm ‖ξ‖Lγω := (E|ξ |γ )1/γ . More specifically, we define Cm+αx (QT ; Lγω) to be the set of all Lγω-valued strongly continuous functions u such that|u|m+α;QT := sup(x,t)∈QT|β|≤m‖Dβu(x, t)‖Lγω + supt, x=y|β|=m‖Dβu(x, t) − Dβu(y, t)‖Lγω|x − y|α < ∞. (1.3)Using the parabolic module |X |p := |x| + √|t| for X = (x, t) ∈ Rn × R, we define Cm+α,α/2x,t (QT ; Lγω) to be the set of all u ∈ Cm+αx (QT ; Lγω) such that|u|(m+α,α/2);QT := |u|m;Q T + sup|β|=m, X =Y‖Dβu(X) − Dβu(Y )‖Lγω|X − Y |αp < ∞. (1.4)Similarly, we can define the norms (1.3) and (1.4) over a domain Q =O × I , for any domains O ⊂ Rn and I ⊂ R.Our main result is the following.Theorem 1.1. Assume that the classical Cαx -norms of aij, bi, c, σ i, σ ix, ν, νx are all dominated by a constant K uniformly in (t, ω) ∈(0, T ) × , and the condition (1.2) is satisfied. If f ∈ Cαx (QT ; Lγω), g ∈ C1+αx (QT ; Lγω) for some γ ≥ 2, then Equation (1.1) with a zero initial condition admits a unique quasi-classical solution u in C2+α,α/2x,t (QT ; Lγω).We remark that the problem with nonzero initial value can be easily reduced to our case by a simple transform. We also remark that by an anisotropic Kolmogorov continuity theorem (see [2]), if αγ > n + 2, the above obtained quasi-classical solution u has a C2+δ,δ/2 modification for 0 < δ < α − (n + 2)/γ as a classical solution to (1.1).In order to prove the solvability in Theorem 1.1, by means of the standard method of continuity, it suffices to establish the following a priori estimate.Theorem 1.2. Under the hypotheses of Theorem 1.1, letting u ∈ C2,0loc (QT ; Lγω) be a quasi-classical solution to (1.1) and u(·, 0) = 0, there is a positive constant C depending only on n, λ, γ , α, and K such that|u|(2+α,α/2);QT ≤ CeC T (| f |α;QT + |g|1+α;QT ). (1.5)The Hölder regularity in spaces Cm+αx (QT ; Lγω) for Equation (1.1) was previously investigated by Rozovsky [12], and later was improved by Mikulevicius [10]. However, both works addressed only the equations with nonrandom coefficients and with no derivatives of the unknown function in the stochastic term, namely aij is deterministic and σ ik ≡ 0. Moreover, both previous works did not obtain the time-continuity of second-order derivatives of u, comparing to our estimate (1.5) and Theorem 1.1.The Schauder estimate we obtained in Theorem 1.2 is sharp in the sense that mentioned in [8], and is for the general form (1.1) with natural assumptions, where all coefficients are random. The approach to C2+α -theory in [10] was based on several delicate estimates for the heat kernel. Our method is completely different and more straightforward by combining certain integral estimates and a perturbation argument of Wang [14]. A sketch of proof of Theorem 1.2 is given in Section 2. Full details in addition to applications and further remarks are contained in our separate paper [3].K. Du, J. Liu / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 371–375 3732. Schauder estimatesIn this section, we give an outline of the proof of our main estimate (1.5). For simplicity we will first deal with a simplified model equation, and then extend to the general ones.Consider the model equationdu = (aijui j + f )dt + (σ ikui + gk)dwkt , (2.1)where aij, σ ik are predictable processes, independent of x, satisfying the condition (1.2). We shall consider the model equa-tion in the entire space Rn × R. Suppose that f (t, ·) and gx(t, ·) are Dini continuous with respect to x uniformly in t , namely1ˆ0(r)rdr < ∞,where(r) = supt∈R, |x−y|≤r(‖ f (t, x) − f (t, y)‖Lγω + ‖gx(t, x) − gx(t, y)‖Lγω ).For any r > 0, we denoteBr(x) = {y ∈ Rn : |y − x| < r}, Q r(x, t) = Br(x) × (t − r2, t), (2.2)and further define Br = Br(0) and Q r = Q r(0, 0).Lemma 2.1. Let u ∈ C2,0x,t (Q 1; Lγω) be a quasi-classical solution to (2.1). Then there is a positive constant C , depending only on n, λ and γ such that for any X, Y ∈ Q 1/4 ,‖uxx(X) − uxx(Y )‖Lγω ≤ C⎡⎣δM1 +δˆ0(r)rdr + δ1ˆδ(r)r2dr⎤⎦ , (2.3)where δ = |X − Y |p and M1 = |u|0;Q 1 + | f |0;Q 1 + |g|1;Q 1 .An important consequence of Lemma 2.1 is the fundamental Schauder estimate that the solution u ∈ C2+α,α/2x,t (Q 1/4; Lγω)when f ∈ Cαx (Q 1; Lγω) and g ∈ C1+αx (Q 1; Lγω) for some α ∈ (0, 1).Outline of proof. Without loss of generality, we may assume X = 0. Let ρ = 1/2, and denoteQ κ = Q ρκ = Q ρκ (0,0), κ = 0,1,2, · · · .Construct a sequence of Cauchy problemsduκ = [aijuκi j + f (0, t)]dt + [σ ikuκi + gk(0, t) + gkx(0, t) · x]dwkt in Q κ,uκ = u on ∂ Q κ.Claim 1. For each κ, there is a unique generalised solution uκ such that uκ(·, t) ∈ Lγ (; Cm(Bε)) for any m ≥ 0 and ε ∈ (0, ρκ). Moreover, for any r < ρκ there is a constant C = C(n, γ ) such that‖u‖Lγ (;L2(Q r)) ≤ C(r2‖ f ‖Lγ (;L2(Q r)) + r‖g‖Lγ (;L2(Q r))). (2.4)Proof. In fact, for γ = 2, the unique solvability and interior smoothness of uκ follows from [5, Theorem 2.1]. For γ ≥ 2, higher order Lγω-integrability (2.4) can be achieved by a truncation technique. Claim 2. There is a constant C = C(n, λ, γ ) such that|Dm(uκ − uκ+1)|0;Q κ+2 ≤ Cρ(2−m)κ(ρκ), m = 1,2, . . . . (2.5)374 K. Du, J. Liu / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 371–375Proof. Note that (uκ − uκ+1) satisfies a homogeneous equation. By a delicate computation, we have|Dm(uκ − uκ+1)|0;Q κ+2 ≤ Cρ−mκ∥∥∥∥ Q κ+1(uκ − uκ+1)2 dX∥∥∥∥1/2Lγ /2ω=: Iκ,m.On the other hand, (uκ − u) satisfies a zero initial condition. By Claim 1,Jκ :=∥∥∥∥ Q κ(uκ − u)2 dX∥∥∥∥1/2Lγ /2ω≤ Cρ2κ(ρκ).Thus, Claim 2 is proved, sinceIκ,m ≤ Cρ−mκ( Jκ + Jκ+1) ≤ Cρ(2−m)κ(ρκ).It is worth remarking that instead of using the maximum principle to estimate the term |Dm(uκ − uκ+1)|0;Q κ+2 as in [14], we obtain the inequality (2.5) by subtle integral estimates. Claim 3. {uκxx(0)} converges in Lγω (here 0 ∈ Rn+1), and the limit is uxx(0).Proof. By Claim 2 and the assumption of Dini continuity,∑κ≥1|(uκ − uκ+1)xx|0;Q κ+2 ≤ C∑κ≥1(ρκ) ≤ C1ˆ0(r)rdr < ∞,which implies that uκxx(0) converges in Lγω . Since γ ≥ 2, it suffices to show thatlimκ→∞‖uκxx(0) − uxx(0)‖L2ω = 0, (2.6)which can also be achieved straightforward by our integral estimates. Now for any Y = (y, s) ∈ Q 1/4 we can select an κ such that |Y |p ∈ [ρκ+2, ρκ+1). By decomposition, one has‖uxx(Y ) − uxx(0)‖Lγω≤ ‖uκxx(Y ) − uκxx(0)‖Lγω + ‖uκxx(0) − uxx(0)‖Lγω + ‖uκxx(Y ) − uxx(Y )‖Lγω=: I1 + I2 + I3. (2.7)Claim 4. I1 ≤ CδM1 + Cδ´ 1δ(r)r2dr, where δ := |Y |p and M1 was given in (2.3).Proof. The proof is by induction. When κ = 0, note that u0xx satisfies the following homogeneous equation:du0xx = aij Diju0xx dt + σ ik Diu0xx dwkt in Q 3/4.From interior estimates, we have‖u0xx(X) − u0xx(Y )‖Lγω ≤ C M1|X − Y |p, ∀ X, Y ∈ Q 1/4. (2.8)When κ ≥ 1, denote hι = uι − uι−1, for ι = 1, 2, . . . , κ, then hι satisfiesdhι = aijhιi j dt + σ ikhιi dwkt in Q ι.From Claim 2, we have for −ρ2(κ+1) ≤ t ≤ 0 and |x| ≤ ρκ+1,‖hιxx(x, t) − hιxx(0,0)‖Lγω ≤ Cρκ−ι(ρι−1). (2.9)Using (2.8) and (2.9), we can obtain the estimateI1 ≤ ‖uκ−1xx (Y ) − uκ−1xx (0)‖Lγω + ‖hκxx(Y ) − hκxx(0)‖Lγω≤ ‖u0xx(Y ) − u0xx(0)‖Lγω +κ∑ι=1‖hιxx(Y ) − hιxx(0)‖Lγω≤ CδM1 + Cδ1ˆδ(r)r2dr.Claim 4 is proved. K. Du, J. Liu / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 371–375 375Claim 5. Ii ≤ C´ δ0(r)r dr, for i = 2, 3.Proof. The estimate of I2 is a refinement of convergence in Claim 3. In fact, by Claim 2 we have the precise estimateI2 = ‖uκxx(0) − uxx(0)‖Lγω ≤∑j≥κ|(u j − u j+1)xx|0;Q j+2 ≤ Cρκˆ0(r)rdr, (2.10)where C = C(n, λ, γ ). We can obtain a similar estimate for I3 by shifting the centre of domains. To sum up, Lemma 2.1 is proved. Having proved Lemma 2.1 we are in a position to derive the global estimate of solutions to (1.1) and complete the proof of Theorem 1.2.Outline of proof of Theorem 1.2. The proof is by an argument of frozen coefficients. Denote Qr,τ = Br × (0, τ ), and letMτx,r(u) = sup0≤t≤τ(  Br(x)E |u(t, y)|γ dy)1/γMτr (u) = supx∈RnMτx,r(u).By multiplying cut-off functions and applying Lemma 2.1 we can get|uxx|(α,α/2);Qρ/2,τ ≤ C(Mτ0,ρ(u) + | f |α;Qρ,τ + |g|1+α;Qρ,τ), (2.11)for some sufficiently small ρ > 0. The derivation of (2.11) involves a rather delicate computation, which makes use of interpolation inequalities in Hölder spaces (see [4, Lemma 6.35] or [6, Theorem 3.2.1]). Since the centre of domains can shift to any point x ∈ Rn , we obtain|u|(2+α,α/2);Qτ ≤ C(Mτρ(u) + | f |α;Qτ + |g|1+α;Qτ), (2.12)where C = C(n, λ, γ , α).To estimate Mτρ(u), applying Itô’s formula, and using Hölder and Sobolev–Gagliargo–Nirenberg inequalities, we can getMτρ(u) ≤ C1τ (Mτρ(u) + |uxx|0;Qρ,τ + | f |0;Qτ + |g|0;Qτ ),where C1 = C1(n, λ, γ ). Letting τ = (2CC1 + C1)−1, by virtue of (2.12) we obtain|u|(2+α,α/2);Qτ ≤ C0(| f |α;Qτ + |g|1+α;Qτ ), (2.13)where C0 = C0(n, λ, γ , α).Finally, the proof of (1.5) and Theorem 1.2 is completed by induction. References[1] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992.[2] R. Dalang, D. Khoshnevisan, E. Nualart, Hitting probabilities for systems of non-linear stochastic heat equations with additive noise, ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007) 231–271.[3] K. Du, J. Liu, On the Cauchy problem for stochastic parabolic equations in Hölder spaces, submitted for publication, available at arXiv:1511.02573.[4] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001.[5] N.V. Krylov, A W n2 -theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Relat. Fields 98 (3) (1994) 389–421.[6] N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996.[7] N.V. Krylov, On Lp -theory of stochastic partial differential equations in the whole space, SIAM J. Math. Anal. 27 (2) (1996) 313–340.[8] N.V. Krylov, An analytic approach to SPDEs, in: Stochastic Partial Differential Equations: Six Perspectives, in: Mathematical Surveys and Monographs, vol. 64, 1999, pp. 185–242.[9] N.V. Krylov, B. Rozovsky, On the Cauchy problem for linear stochastic partial differential equations, Izv. Math. 11 (6) (1977) 1267–1284.[10] R. Mikulevicius, On the Cauchy problem for parabolic SPDEs in Holder classes, Ann. Probab. (2000) 74–103.[11] E. Pardoux, Équations aux dérivées partielles stochastiques non linéaires monotones, Thèse, Univ. Paris-Sud, Orsay, 1975.[12] B. Rozovsky, On stochastic partial differential equations, Sb. Math. 25 (2) (1975) 295–322.[13] B. Rozovsky, Stochastic Evolution Systems, Springer, 1990.[14] X.-J. Wang, Schauder estimates for elliptic and parabolic equations, Chin. Ann. Math., Ser. B 27 (6) (2006) 637–642.[15] M. Zakai, On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeitstheor. Verw. Geb. 11 (3) (1969) 230–243.

A Schauder estimate for stochastic PDEs

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A Schauder estimate for stochastic PDEs

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