Hölder Continuity of Solutions of SPDEs with Reflection

Article

Abstract

In this paper, we obtain the Hölder continuity of the solutions of SPDEs with reflection, which have singular drifts (random measures).

Keywords

Parabolic obstacle problem Stochastic partial differential equations with reflection Random measure Garsia’s lemma 

Mathematics Subject Classification (2010)

60H15 60F10 60F05 

1 Introduction and Framework

Consider the following stochastic partial differential equation (SPDE) with reflection: Here \(\dot{W}\) denotes the space-time white noise defined on a complete probability space \((\varOmega, \mathcal{F},\{\mathcal{F}_{t}\}_{t\geq0 },P)\), \(\mathcal{F}_{t}=\sigma(W(s,x): x\in[0,1], 0\leq s\leq t)\); u0 is a non-negative continuous function on [0,1], which vanishes at 0 and 1; η(x,t) is a random measure which is a part of the solution pair (u,η); \(\frac{\partial^{2}}{\partial x^{2}}\) denotes the Laplacian operator on [0,1] equipped with the Dirichlet boundary condition. The coefficients f and σ are measurable mappings from \(\mathbb{R}\) into \(\mathbb{R}\). The following definition is taken from [5, 9].

Definition 1.1

A pair (u,η) is said to be a solution of (1.1) if
  1. (i)

    u is a continuous random field on \(\mathbb{R}_{+}\times[0,1]\), u(t,x) is \(\mathcal{F}_{t}\) measurable and u(t,x)≥0 a.s.

     
  2. (ii)
    η is a random measure on \(\mathbb{R}_{+} \times(0,1)\) such that
    1. (a)

      η({t}×(0,1))=0, for all t≥0.

       
    2. (b)

      \(\int_{0}^{t}\int_{0}^{1} x(1-x)\eta(ds,dx)<\infty\), for all t≥0.

       
    3. (c)
      η is adapted in the sense that for any measurable mapping ψ:
      $$\int_0^t\int_0^1\psi(s,x)\eta(ds,dx)\ \mbox{is}\ \mathcal {F}_t\mbox{-measurable}. $$
       
     
  3. (iii)
    (u,η) solves the parabolic SPDE in the following sense ((⋅,⋅) denotes the scalar product in L2[0,1]): \(\forall t\in\mathbb{R}_{+}, \phi\in C^{2}_{0}([0,1])\) with ϕ(0)=ϕ(1)=0, where u(t):=u(t,⋅).
     
  4. (iv)

    Qu=0, where \(Q= \mathbb{R}_{+}\times(0,1)\).

     

This equation was first studied by Nualart and Pardoux in [9] when σ(⋅)=1, and by Donati-Martin and Pardoux in [5] for a general diffusion coefficient σ without obtaining the uniqueness and by Xu and Zhang in [12] for general σ with also the proof of the uniqueness. Various properties of the solution of (1.1) were studied later in [4, 6, 8, 13, 14]. SPDEs with reflection can also be used to model the evolution of random interfaces near a hard wall. It was proved by Funaki and Olla in [7] that the fluctuations of a ∇ϕ interface model near a hard wall converge in law to the stationary solution of a SPDE with reflection. We also mention that the random contact set {(t,x);u(t,x)=0} of the solution u was investigated in [4] when σ=1. For stochastic Cahn–Hilliard equations with reflection, see [13].

We assume throughout the paper that the mappings
$$f,\sigma:\mathbb{R} \rightarrow\mathbb{R} $$
are Lipschitz continuous:
The purpose of this paper is to obtain the Hölder continuity of the solution of the SPDE with reflection. This is not trivial because of the singularity introduced by the random measure term in the equation. When the noise is additive (i.e., σ=1), the question of Hölder continuity was investigated in [4], where the Hölder continuity with respect to the space variable was obtained. Regarding the time variable, the authors in [4] only established the following lower bound:
$$u(t,x)-u(s,x)\geq-\gamma(t-s)^{\alpha}, \quad t\geq s\geq0. $$
In this paper, we obtain the Hölder continuity of the solution of the SPDE with reflection with respect to both the time and the space variables for general equations with multiplicative noise. Our method is a careful refinement of the approach in [4]. The idea is to consider the penalized approximating equations and prove uniform moment estimates for the solutions of the approximating equations. For regularity of solutions of other type SPDEs, we refer the readers also to [2, 10].

Let Gt(x,y) be the heat kernel associated with the Laplacian operator \(\frac{\partial^{2}}{\partial x^{2}}\) on [0,1] equipped with the Dirichlet boundary condition. If we let \(v(t,x)=\int_{0}^{1}G_{t}(x,y)u_{0}(y)\,dy\), then u(t,x)−v(t,x) will solve a similar SPDE with reflection as (1.1), but with initial data (function) 0. Since the Hölder continuity of v(t,x) is well understood, the study of the regularity of u(t,x) is reduced to the study of u(t,x)−v(t,x). Therefore, without loss of generality, in the paper we will assume u0=0 in (1.1).

The organization of this paper is as follows: In Sect. 2, we prepare some results on the penalized approximating equations. In Sect. 3, we establish the Hölder continuity.

2 The Approximating Equations

For ε>0, set
$$ g_{\varepsilon}(u)=\frac{\arctan([u\wedge 0]^2)}{\varepsilon} . $$
(2.1)
Consider the following penalized SPDE: or equivalently in the mild form (see [11]): Here, Gt(x,y) is the heat kernel. Fix T>0 and let QT=[0,T]×[0,1]. For vC(QT), set ∥v=max0≤sT, 0≤x≤1|v(s,x)|. It was shown in [5] (see also [4]) that, for p≥1,
$$ \lim_{\varepsilon\rightarrow0}E\bigl [\big\|u^{\varepsilon}-u\big\|_{\infty}^p \bigr]=0, $$
(2.4)
where u is the solution to (1.1).

Notice that the function \(g_{\varepsilon}(u)=\frac{u^{-}}{\varepsilon }\) was used in [6]. Our choice of gε does not change the limit of uε, but makes gε differentiable.

Set Let vε(t,x)=uε(t,x)−Nε(t,x). Then it is easy to see that vε satisfies the following random PDE:

Lemma 2.1

For anyα<1, ε>0, there exists a random variableCε(ω) such thatMoreover, for anyp≥1, we have
$$\sup_{\varepsilon}E\bigl[C_{\varepsilon}^p\bigr]<\infty. $$

Proof

First of all, we recall the following properties of the heat kernel Gt(x,y) from [1]: for s,t∈[0,T] with st and x,y∈[0,1], We claim that, for any p≥2, where Cp is a constant independent of ε. Set It suffices to show that both terms \(I_{1}^{\varepsilon}\) and \(I_{2}^{\varepsilon}\) satisfy (2.11). The term \(I_{2}^{\varepsilon}(t,x)\) is the more complicated of the two, so we only consider it. By Burkholder’s inequality and Hölder’s inequality, for p≥2, stT, we have where (2.8) and the fact that σ is Lipschitz continuous were used. Similarly, in view of (2.9) and (2.10) it follows that Putting together (2.14) and (2.15), we prove (2.11) for \(I^{\varepsilon}_{2}\). Since the constant Cp in (2.11) is independent of ε, applying a version of the Garsia’s lemma proved in [3, Proposition A.1 and Corollary A.3], it follows that, for any p>8, there exists a random variable ηp,ε(ω) such that and Since p can be chosen to be arbitrarily large, the lemma follows. □

3 Hölder Continuity

Recall the following lemma from [4].

Lemma 3.1

LetVC1,2(QT) andψ,FC(QT) withψ≤0. Suppose thatVsolves the equationwith Dirichlet or Neumann boundary conditions. Then the following estimate holds:
$$\|V\|_{\infty}\leq\|F\|_{\infty}. $$

We need the following lemma in the proof of the main result.

Lemma 3.2

LetfCα,β(QT) satisfying
$$ \bigl|f(t,x)-f(s,y)\bigr|\leq C_f\bigl(|t-s|^{\alpha}+|x-y|^{\beta} \bigr). $$
(3.2)
Then, forρ1>0,ρ2>0, there exists\(f^{\rho_{1}, \rho_{2}}\in C^{\infty}(Q_{T})\)such thatwhereCα,βis a constant only depending onα,β.

Proof

First we extend the definition of f to \(\mathbb{R}^{2}\) by setting where p(s,y) denotes the point in QT that is nearest (in Euclidean norm) to (s,y). In particular, \(\bar{f}(s,y) = f(s,y)\) if (s,y)∈QT. Then it is easy to see that \(\bar{f}\) satisfies (3.2) with the same constant Cf. Denote by Pu(x,y), u>0 the Gaussian heat kernel:
$$P_u(x,y)=\frac{1}{\sqrt{2\pi u}}e^{-\frac{(x-y)^2}{2u}}. $$
For ρ1>0,ρ2>0, define \(f^{\rho_{1}, \rho_{2}}\) by
$$ f^{\rho_1, \rho_2}(t,x)=\int_{\mathbb{R}}\int _{\mathbb{R}}P_{\rho_1^2}(t,s)P_{\rho_2^2}(x,y) \bar{f}(s,y)\,ds\,dy. $$
(3.5)
We will show that \(f^{\rho_{1}, \rho_{2}}\) has the required properties. Since Pu(⋅,⋅) is a probability density, we have which is what we need. Note that
$$\int_{\mathbb{R}}P_{\rho_1^2}(t,s)(t-s)\,ds=0. $$
Differentiating \(f^{\rho_{1}, \rho_{2}}\) with respect to t we get Similar calculations yield the estimate for \(\|\frac{\partial f^{\rho _{1}, \rho_{2}}}{\partial x}\|_{\infty}\). The proof is complete. □

The following theorem is the main result of the paper.

Theorem 3.3

Letube the solution of the SPDE with reflection (1.1) withu0=0, and fixT>0. Then for anyα<1 andp>1, we have the following moment estimate: In particular, uadmits a version that is Hölder\((\frac{1}{4}-, \frac{1}{2}-)\)on [0,T]×[0,1].

Proof

Fix any α<1 and let Nε(t,x) be defined as in Sect. 2. For ρ1>0 and ρ2>0, define the smooth function \(N^{\varepsilon,\rho_{1},\rho_{2}}(t,x)\) as \(f^{\rho_{1},\rho_{2}}(t,x)\) in (3.5) replacing f by Nε.

Let \(v^{\varepsilon,\rho_{1},\rho_{2}}\) be the solution of the following random PDE: Since \(g_{\varepsilon}^{\prime}(u)\leq0\), applying Lemma 3.1 (or [9, Lemma]) we conclude that
$$ \bigl \|v^{\varepsilon,\rho_1,\rho_2}-v^{\varepsilon }\bigr\|_{\infty}\leq \bigl\|N^{\varepsilon,\rho_1,\rho_2}-N^{\varepsilon}\bigr\|_{\infty}. $$
(3.10)
In view of Lemma 3.2, it follows from (3.10) that
$$ \bigl\|v^{\varepsilon,\rho_1,\rho_2}-v^{\varepsilon }\bigr\|_{\infty}\leq C_{\alpha,\beta}C_{\varepsilon}(\omega)\bigl[\rho_1^{\frac {\alpha}{4}}+ \rho_2^{\frac{\alpha}{2}}\bigr], $$
(3.11)
where Cε(ω) is the random variable appeared in (2.7). Introduce the following random PDEs: Formally differentiating \(v^{\varepsilon,\rho_{1},\rho_{2}}(t,x)\) we see that \(m^{\varepsilon,\rho_{1},\rho_{2}}(t,x)=\frac{\partial v^{\varepsilon,\rho_{1},\rho_{2}}}{\partial t}(t,x)\) and \(w^{\varepsilon ,\rho_{1},\rho_{2}}(t,x)=\frac{\partial v^{\varepsilon,\rho_{1},\rho _{2}}}{\partial x}(t,x)\). Notice \(g_{\varepsilon}^{\prime}\leq0\), apply Lemmas 3.1 and 3.2 to obtain and Setting ρ1=|ts|, ρ2=|xy|, it follows from (3.11), (3.14), and (3.15) that Thus, This yields, for p≥1, By the Fatou lemma, we obtain from (3.18) that Applying a variant of Garsia’s lemma (see [3, Proposition A.1 and Corollary A.3]) we conclude that Since p can be chosen to be arbitrarily large and α to be as close to 1 as one wants, we see that u is \((\frac{1}{4}-, \frac {1}{2}-)\) Hölder. The proof is complete. □

References

  1. 1.
    Bally, V., Millet, A., Sanz-Solé, M.: Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations. Ann. Probab. 23(1), 178–222 (1995) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Dalang, R.C., Sanz-Solé, M.: Hölder–Sobolev regularity of the solution to the stochastic wave equation in dimension three. Mem. Am. Math. Soc. 199, 931 (2009) Google Scholar
  3. 3.
    Dalang, R.C., Khoshnevisan, D., Nualart, E.: Hitting properties for systems of non-linear stochastic heat equations with additive noise. ALEA Lat. Am. J. Probab. Math. Stat. 3, 231–271 (2007) MathSciNetMATHGoogle Scholar
  4. 4.
    Dalang, R.C., Mueller, C., Zambotti, L.: Hitting properties of parabolic SPDE’s with reflection. Ann. Probab. 34(4), 1423–1450 (2006) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Donati-Martin, C., Pardoux, E.: White noise driven SPDEs with reflection. Probab. Theory Relat. Fields 95, 1–24 (1993) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Donati-Martin, C., Pardoux, E.: EDPS réfléchies et calcul de Malliavin [SPDEs with reflection and Malliavin calculus]. Bull. Sci. Math. 121(5), 405–422 (1997) MathSciNetMATHGoogle Scholar
  7. 7.
    Funaki, T., Olla, S.: Fluctuations for ∇ϕ interface model on a wall. Stoch. Process. Appl. 94(1), 1–27 (2001) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Haussmann, U.G., Pardoux, E.: Stochastic variational inequalities of parabolic type. Appl. Math. Optim. 20, 163–192 (1989) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Nualart, D., Pardoux, E.: White noise driven by quasilinear SPDEs with reflection. Probab. Theory Relat. Fields 93, 77–89 (1992) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Sanz-Solé, M., Vuillermot, P.A.: Equivalence and Hölder–Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations. Ann. Inst. Henri Poincaré Probab. Stat. 39(4), 703–742 (2003) MATHCrossRefGoogle Scholar
  11. 11.
    Walsh, J.B.: An introduction to stochastic partial differential equations. In: Hennequin, P.L. (ed.) Ecole d’été de Probabilité de St Flour. Lect. Notes Math., vol. 1180. Springer, Berlin (1986) Google Scholar
  12. 12.
    Xu, T., Zhang, T.: White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles. Stoch. Process. Appl. 119(10), 3453–3470 (2009) MATHCrossRefGoogle Scholar
  13. 13.
    Zambotti, L.: A reflected stochastic heat equation as symmetric dynamics with respect to the 3-d Bessel bridge. J. Funct. Anal. 180, 195–209 (2001) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Zhang, T.: White noise driven SPDEs with reflection: strong Feller properties and Harnack inequalities. Potential Anal. 33(2), 137–151 (2010) MathSciNetCrossRefGoogle Scholar

Copyright information

© School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut de MathématiquesEcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.School of MathematicsUniversity of ManchesterManchesterUK

Personalised recommendations