In this paper we study the Cauchy problem for a one-dimensional stochastic wave equation driven by a general stochastic measure. We consider solution of this problem in the mild sense (see (12) below). Our goal is to show the convergence of solutions of equations using the approximation of stochastic measures by partial sums of Fourier series and Fejèr sums.

Existence and uniqueness of the solution of our equation are obtained in [2]. The wave equation driven by stochastic measure defined on subsets of the spatial variable was considered in [1]. Similar equations driven by random stable noises are studied in [9, 10], where the properties of generalized solutions are investigated.

Convergence of solutions of stochastic wave equation by using the approximation of stochastic integrator was studied in [3, 4]. Mild solutions of equations driven by the Gaussian random field in dimension three were considered in these papers.

Approximation of stochastic measures may be obtained by using Fourier and Fourier–Haar series, the corresponding results are given in [15, 17]. The partial sums of the resulting series generate random functions of sets, which are signed measures for each fixed ω∈Ω . The resulting equations can be solved as a nonstochastic equation for each ω. The results of our paper imply that we obtain in this way an approximation of the solution of (12).

Papers [15, 17] contain examples of applying Fourier series of stochastic measures to the convergence of solutions of the stochastic heat equation. A similar application of the Fourier transform is given in [18]. Continuous dependence of solutions of wave equation upon the data was studied in [1, 2]. In this paper we obtain a continuous dependence upon the values of stochastic integrator of the equation.

The paper is organized as follows. In Section 2 we recall the basic facts about stochastic measures and Fourier series. Important auxiliary lemma concerning convergence of stochastic integrals is proved in Section 3. The formulation of the Cauchy problem and theorem about approximation of the solution by using Fourier partial sums are given in Section 4. The similar approximation that uses Fejèr sums is obtained in Section 5. Section 6 contains one example with comments about the convergence rate.

Let X be an arbitrary set and let B(X) be a σ-algebra of subsets of X . Let L0(Ω,F,P) be the set of (equivalence classes of) all real-valued random variables defined on a complete probability space (Ω,F,P) . The convergence in L0(Ω,F,P) is understood to be in probability.

We do not assume positivity or moment existence for μ. In other words, μ is a vector measure with values in L0 .

For a deterministic measurable function g:X→R and SM μ, an integral of the form ∫Xgdμ is defined and studied in [6, Chapter 7], see also [11, Chapter 1]. In particular, every bounded measurable g is integrable with respect to any μ. Moreover, an analogue of the Lebesgue dominated convergence theorem holds for this integral (see [6, Theorem 7.1.1], [11, Corollary 1.2]).

Important examples of SMs are orthogonal stochastic measures, α-stable random measures defined on a σ-algebra for α∈(0,1)∪(1,2] (see [19, Chapter 3]). Conditions under which a process with independent increments generates an SM may be found in [6, Chapter 7 and 8].

Let the random series ∑n≥1ξn converge unconditionally in probability, and mn be real signed measures on  B , |mn(A)|≤1 . Set μ(A)=∑n≥1ξnmn(A) . Convergence of this series in probability follows from [21, Theorem V.4.2], and μ is a SM by [5, Theorem 8.6].

Many examples of the SMs on the Borel subsets of [0,T] may be given by the Wiener-type integral (1) μ(A)=∫[0,T]1A(t)dXt.

We note the following cases of processes Xt in (1) that generate SM.

We will give another example. Let ζ be an arbitrary SM defined on Borel subsets of [a,b]⊂R , function h:[0,T]×[a,b]→R be such that h(0,y)=0 , and (2) |h(t,y)−h(s,x)|≤L(|t−s|+|y−x|γ),γ>1/2,L∈R. 1/2,\hspace{1em}L\in \mathbb{R}.\]]]> Then h(·,y) is absolutely continuous for each y, |∂h(t,y)∂t|≤L a. e., and we can define SM (3) μ(A)=∫[a,b]dζ(y)∫A∂h(t,y)∂tdt,A∈B([0,T]), see details in [16, Section 3]. Note that Theorem 1 of [16] implies that the process (4) μt=μ((0,t])=∫[a,b]h(t,y)dζ(y),t∈[0,T], has a continuous version. In this case the process Xt=μt in (1) defines an SM.

Let B be a Borel σ-algebra on (0,1] . For arbitrary SM μ on B we consider the Fourier series in the following sense.

Denote (5) ξk=∫(0,1]exp{−2πikt}dμ(t):=∫(0,1]cos(2πkt)dμ(t)−i∫(0,1]sin(2πkt)dμ(t),k∈Z.

Stochastic integrals on the right hand side of (5) are defined for any μ, since the integral functions are bounded. Thus the Fourier series is well defined for every SM on B .

We will also consider Fejèr sums for SM μ: S˜j(t)=1j+1∑0≤k≤jSk(t).

For necessary information concerning the classical Fourier series, we refer to [22]. In the sequel, C and C(ω) will denote nonrandom and random constants respectively whose exact value is not essential.

Put Δkn=((k−1)2−n,k2−n],n≥0,1≤k≤2n.

Let the function g(z,s):Z×[0,1]→R be such that ∀z∈Z:g(z,·) is continuous on [0,1] . Here Z=Z0×[0,1],Z0 is an arbitrary set, z=(z0,t) . Denote g(n)(z,s)=g(z,0)1{0}(s)+∑1≤k≤2ng(z,(k−1)2−n∧t)1Δkn(s).

From [14, Lemma 3] it follows that the random function η(z,t)=∫(0,t]g(z,s)dμ(s),z∈Z, has a version (7) η˜(z,t)=∫(0,t]g(0)(z,s)dμ(s)+∑n≥1(∫(0,t]g(n)(z,s)dμ(s)−∫(0,t]g(n−1)(z,s)dμ(s)), such that for all ε>0,ω∈Ω,z∈Z 0,\hspace{0.1667em}\omega \in \varOmega ,\hspace{0.1667em}z\in Z$]]> (8) |η˜(z,t)|≤|g(z,0)μ((0,t])|+{∑n≥12nε∑1≤k≤2n|g(z,k2−n∧t)−g(z,(k−1)2−n∧t)|2}12×{∑n≥12−nε∑1≤k≤2n|μ(Δkn∩(0,t])|2}12. We note that the series with the values of SM in (8) converges a. s. (see [13, Lemma 3.1]). Lemma 1.

Let Z be an arbitrary set, g,gj:Z×[0,1]→R , and the following conditions hold

(i) supz∈Z,t∈[0,1]|gj(z,t)−g(z,t)|→0 , j→∞ ;

(ii) for some constants Lg>0 0$]]>, β(g)>1/2 1/2$]]> supz∈Z,j≥1|gj(z,t)−gj(z,s)|≤Lg|t−s|β(g)t,s∈[0,1];

(iii) for some random constant Cμ(ω) |μ((0,t])|≤Cμ(ω),t∈(0,1]. Then for versions (7) of the processes ηj(z,t)=∫(0,t]gj(z,s)dμ(s),η(z,t)=∫(0,t]g(z,s)dμ(s) the following holds: supz∈Z,t∈[0,1]|ηj(z,t)−η(z,t)|→0a.s.,j→∞.

Consider the Cauchy problem for a one-dimensional stochastic wave equation (11) ∂2u(t,x)∂t2=a2∂2u(t,x)∂x2+f(t,x,u(t,x))+σ(t,x)μ˙(t),u(0,x)=u0(x),∂u(0,x)∂t=v0(x), where (t,x)∈[0,1]×R , a>0 0$]]>, μ is an SM defined on the Borel σ-algebra B((0,1]) .

The solution of equation (11) is understood in the mild sense, (12) u(t,x)=12(u0(x+at)−u0(x−at))+12a∫x−atx+atv0(y)dy+12a∫0tds∫x−a(t−s)x+a(t−s)f(s,y,u(s,y))dy+12a∫(0,t]dμ(s)∫x−a(t−s)x+a(t−s)σ(s,y)dy. The integrals of random functions with respect to dx are taken for each fixed ω∈Ω . We impose the following assumptions.

A1. Functions u0(y)=u0(y,ω):R×Ω→R and v0(y)=v0(y,ω):R×Ω→R are measurable and bounded for every fixed ω∈Ω .

A2. The function f(s,y,v):[0,1]×R×R→R is measurable and bounded.

A3. The function f(s,y,v) is uniformly Lipschitz in y,v∈R : |f(s,y1,v1)−f(s,y2,v2)|≤Lf(|y1−y2|+|v1−v2|).

A4. The function σ(s,y):[0,1]×R→R is measurable and bounded.

A5. The function σ(s,y) is Hölder continuous: |σ(s1,y1)−σ(s2,y2)|≤Lσ(|s1−s2|β(σ)+|y1−y2|β(σ)),1/2<β(σ)≤1.

A6. For some random constant Cμ(ω) |μ((0,t])|≤Cμ(ω),t∈(0,1] .

From A1–A5 it follows that equation (12) has a unique solution (see Theorem 2.1 [2]). The Hölder continuity condition was imposed on u0 in [2], but was not used in proof of existence and uniqueness of the solution.

Note that the processes Xt in examples 2–4 of SMs and μt in (4) are continuous, therefore A6 is fulfilled in these cases.

Consider the following equations: (13) uj(t,x)=12(u0(x+at)−u0(x−at))+12a∫x−atx+atv0(y)dy+12a∫0tds∫x−a(t−s)x+a(t−s)f(s,y,uj(s,y))dy+12a∫(0,t]Sj(s)ds∫x−a(t−s)x+a(t−s)σ(s,y)dy.