# Hölder Continuity of Solutions of SPDEs with Reflection

## 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

*u*

_{0}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

*u*,

*η*) is said to be a solution of (1.1) if

- (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. - (ii)
*η*is a random measure on \(\mathbb{R}_{+} \times(0,1)\) such that- (a)
*η*({*t*}×(0,1))=0, for all*t*≥0. - (b)
\(\int_{0}^{t}\int_{0}^{1} x(1-x)\eta(ds,dx)<\infty\), for all

*t*≥0. - (c)
*η*is adapted in the sense that for any measurable mapping*ψ*:$$\int_0^t\int_0^1\psi(s,x)\eta(ds,dx)\ \mbox{is}\ \mathcal {F}_t\mbox{-measurable}. $$

- (a)
- (iii)(
*u*,*η*) solves the parabolic SPDE in the following sense ((⋅,⋅) denotes the scalar product in*L*^{2}[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*,⋅). - (iv)
∫

_{ Q }*u**dη*=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].

*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:

Let *G* _{ t }(*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 *u* _{0}=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

*ε*>0, set

*G*

_{ t }(

*x*,

*y*) is the heat kernel. Fix

*T*>0 and let

*Q*

_{ T }=[0,

*T*]×[0,1]. For

*v*∈

*C*(

*Q*

_{ T }), set ∥

*v*∥

_{∞}=max

_{0≤s≤T, 0≤x≤1}|

*v*(

*s*,

*x*)|. It was shown in [5] (see also [4]) that, for

*p*≥1,

*u*is the solution to (1.1).

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

*v*

^{ ε }(

*t*,

*x*)=

*u*

^{ ε }(

*t*,

*x*)−

*N*

^{ ε }(

*t*,

*x*). Then it is easy to see that

*v*

^{ ε }satisfies the following random PDE:

## Lemma 2.1

*For any*

*α*<1,

*ε*>0,

*there exists a random variable*

*C*

_{ ε }(

*ω*)

*such that*

*Moreover*,

*for any*

*p*≥1,

*we have*

## Proof

*G*

_{ t }(

*x*,

*y*) from [1]: for

*s*,

*t*∈[0,

*T*] with

*s*≤

*t*and

*x*,

*y*∈[0,1],

*p*≥2,

*C*

_{ p }is a constant independent of

*ε*. Set

*p*≥2,

*s*≤

*t*≤

*T*, we have

*σ*is Lipschitz continuous were used. Similarly, in view of (2.9) and (2.10) it follows that

*C*

_{ p }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

*η*

_{ p,ε }(

*ω*) such that

*p*can be chosen to be arbitrarily large, the lemma follows. □

## 3 Hölder Continuity

Recall the following lemma from [4].

## Lemma 3.1

*Let*

*V*∈

*C*

^{1,2}(

*Q*

_{ T })

*and*

*ψ*,

*F*∈

*C*(

*Q*

_{ T })

*with*

*ψ*≤0.

*Suppose that*

*V*

*solves the equation*

*with Dirichlet or Neumann boundary conditions*.

*Then the following estimate holds*:

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

## Lemma 3.2

*Let*

*f*∈

*C*

^{ α,β }(

*Q*

_{ T })

*satisfying*

*Then*,

*for*

*ρ*

_{1}>0,

*ρ*

_{2}>0,

*there exists*\(f^{\rho_{1}, \rho_{2}}\in C^{\infty}(Q_{T})\)

*such that*

*where*

*C*

_{ α,β }

*is a constant only depending on*

*α*,

*β*.

## Proof

*f*to \(\mathbb{R}^{2}\) by setting

*p*(

*s*,

*y*) denotes the point in

*Q*

_{ T }that is nearest (in Euclidean norm) to (

*s*,

*y*). In particular, \(\bar{f}(s,y) = f(s,y)\) if (

*s*,

*y*)∈

*Q*

_{ T }. Then it is easy to see that \(\bar{f}\) satisfies (3.2) with the same constant

*C*

_{ f }. Denote by

*P*

_{ u }(

*x*,

*y*),

*u*>0 the Gaussian heat kernel:

*ρ*

_{1}>0,

*ρ*

_{2}>0, define \(f^{\rho_{1}, \rho_{2}}\) by

*P*

_{ u }(⋅,⋅) is a probability density, we have

*t*we get

The following theorem is the main result of the paper.

## Theorem 3.3

*Let*

*u*

*be the solution of the SPDE with reflection*(1.1)

*with*

*u*

_{0}=0,

*and fix*

*T*>0.

*Then for any*

*α*<1

*and*

*p*>1,

*we have the following moment estimate*:

*In particular*,

*u*

*admits a version that is Hölder*\((\frac{1}{4}-, \frac{1}{2}-)\)

*on*[0,

*T*]×[0,1].

## Proof

Fix any *α*<1 and let *N* ^{ ε }(*t*,*x*) be defined as in Sect. 2. For *ρ* _{1}>0 and *ρ* _{2}>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 *N* ^{ ε }.

*C*

_{ ε }(

*ω*) is the random variable appeared in (2.7). Introduce the following random PDEs:

*ρ*

_{1}=|

*t*−

*s*|,

*ρ*

_{2}=|

*x*−

*y*|, it follows from (3.11), (3.14), and (3.15) that

*p*≥1,

*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. □

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