Identification of the Point Sources in Some Stochastic Wave Equations

and Applied Analysis 3 define the stochastic integral ∫t 0 ∫ R λ(s, x)δ(x − y)F(ds dx), where δ(x − y) is the Dirac delta function. To this end, we will use smooth functions to approximate the Dirac delta function. Let φ be a smooth function with compact support K = [−1, 1] d. Set φ n (x − y) = n d φ(n(x − y)). It is clear that φ n converges to δ(x − y) in the sense of distribution as n → ∞. For each n, the stochastic integral


Introduction
Assume that there are a certain number of objects in a certain area of ocean or other media.The total number of the objects and the location of each object are unknown.We need to identify the total number and the precise locations of the objects.The objects are also assumed to emit (point source) sound waves and we are able to measure these sound waves received in some known locations.The objective is to use these measurements for our identification problem.
This type of problem has been studied mathematically in the framework of inverse problems for partial differential equations (wave equations).The sound travels according to the following second-order wave equation with point sources: where  ∈ R  ,  ∈ (0, ),  ∈ N, and  1 , . . .,   are some given points in R  , () is the Dirac delta function, and   ,  = 1, 2, . . ., , are some known constants.The solution ((, ),  ∈ , 0 ≤  ≤ ) is supposed to be known for some space points  and for some interval [0, ].The total number  and each location   of point sources are estimated from ((, ),  ∈ , 0 ≤  ≤ ).We refer to, for example, [1] and in particular the references therein for some recent study in this area.This theory has found substantial applications in determining the heat sources in heat conduction, the magnetic sources in brain, the earthquake sources of seismic waves, and so on.
In practice, the sound wave travels inevitably under some influence of noises.Voluntarily or involuntarily, the point sources themselves may also emit noises to avoid being detected.Thus, we are led to the following stochastic wave equations: where  ∈ R  ,  ∈ (0,),  ∈ N, and  1 , . . .,   are some given points in R  , () is the Dirac delta function, V 0 and V 1 () are two given deterministic functions, and Ḃ  , Abstract and Applied Analysis  = 0, 1, 2, . . ., , are independent Gaussian noises which are white in time and correlated in space.
When  1 = ⋅ ⋅ ⋅ =   = 0, the stochastic wave equation (2) has been studied since long time.Let us mention the first lecture note [2] and a recent lecture note [3].Many properties such as the sample path Hölder continuity of the solution are obtained (see [4,5] and the references therein).
However, when   ,  = 1, . . ., , are not all zero, then (2) is highly singular because of the presence of the Dirac delta functions multiplied by the Gaussian noises.Such equation has not been studied yet.The first objective of this paper is to give the definition of the solution to such an equation and to show the existence of uniqueness of the solution under some appropriate conditions.This will be done in Section 2. Since the case when  1 = ⋅ ⋅ ⋅ =   = 0 has been well studied, we will now assume  0 = 0 to simplify our presentation.However, since our objective is the identification of the  and point source positions  1 , . . .,   , we will not go to spend too much effort here.For this reason, we restrict ourselves to only one space dimension case.We will present higher space dimension case in another project.
To well explain our approach of identification, we will further restrict our model.We will consider the special case of (2) where the coefficients   (, , ) =   () are independent of  and  and V 0 = V 0 = 0. Namely, we will concentrate on the stochastic wave equation of the form In this case, we will write down the explicit expression of the solution.It will be done in Section 3. In this section, we also obtain some properties of the solution which will be useful in the later section of the paper.Now, we assume that, in (3), the total number of point sources  and the positions   ,  = 1, . . ., , are unknown.However, we are able to observe the sound signal received at some given known locations  1 , . . .,   , continuously in the time interval [0, ].Namely, we assume that ((,  1 ), . . ., (,   ), 0 ≤  ≤ ) are known.We would like to use this information to identify  and   ,  = 1, . . ., .In Section 4, we will develop a new approach to obtain some statistical estimators N and ŷ1 , ŷ2 , . . ., ŷ to estimate the total number  and the locations  1 ,  2 , . . .,   of the point sources.The approach combines the reciprocity gap functional approach from the theory of partial differential equations with theory from stochastic processes.We show the almost sure convergence of our estimators N and ŷ1 , ŷ2 , . . ., ŷ to the true parameters  and  1 ,  2 , . . .,   .
If we assume that the noise is spatially homogeneous, that is,   (, ) =   ( − ), then there exists a nonnegative tempered measure   which is the Fourier transform of   ()d.With this notation, we can also write From the general theory of stochastic integral (see, e.g., [3]), we see that if ((, ), 0 ≤  ≤ ,  ∈ R  ) is a real valued F  -adapted process such that then the stochastic integral ∫  0 ∫ R  (, )  (, d) is welldefined and Now, let  be a given fixed point in R  and let ((, ), 0 ≤  ≤ ,  ∈ R  ) be a real valued F  -adapted process.We want to define the stochastic integral ∫  0 ∫ R  (, )( − )(d d), where ( − ) is the Dirac delta function.To this end, we will use smooth functions to approximate the Dirac delta function.
Let  be a smooth function with compact support  = [−1, 1]  .Set   ( − ) =   (( − )).It is clear that   converges to ( − ) in the sense of distribution as  → ∞.For each , the stochastic integral is well-defined.We want to know whether   has a limit in  2 (Ω, F, ) or not.First, we have the following computations: Passing the difference   ( − ) −   ( − ) to that of  and , we have ×  ()  () d d d ×  ()  () d d d 1 can be written as Now, we let (, ) be a continuous function of  and .We also assume that there is a  0 > 0 such that sup We write  1 = Thus, we have  1 converging to 0.
In the same way, we can show that  2 converges to 0 under the same conditions.Theorem 1.Let Ḃ  (, ) be a Gaussian noise which is white in time and correlated in space with covariance   (, ).Assume that (, ) is a continuous function of  and .Let ((, ), 0 ≤  ≤ ,  ∈ R  ) be an F  -adapted processes such that conditions (14) hold.Then, the stochastic integral Proof.The above argument shows the existence of the stochastic integral ∫  0 ∫ R  (, )( − )  (d, d).We have ×  ()  () d d d. (18) Now, the Fatou lemma yields (17).
We also need to bound general moments of the stochastic integral ∫  0 ∫ R  (, )( − )  (d, d).We have the following Burkholder type inequality.
Theorem 2. Let Ḃ  (, ) be a Gaussian noise which is white in time and correlated in space with covariance   (, ).Assume that (, ) is a continuous function of  and .Let ((, ), 0 ≤  ≤ ,  ∈ R  ) be an F  -adapted processes such that Then, the stochastic integral ∫  0 ∫ R  (, )( − )  (d, d) exists and Now, we turn to consider the existence and uniqueness of the solution to the stochastic wave equation ( 2).We will follow the idea of mild solution.Since the Green function is more sophisticated to study in the dimension higher than 1, we will only study the one space dimensional wave equation in this paper.Higher dimension case needs much more care.Assume that V 0 is bounded continuous functions in R and V 0 is bounded continuously differentiable functions in R with bounded derivative.Then, there is a unique solution to (21).
Proof.Since the solution of wave equation has the past-light cone property (see [6], p. 63), we can study the solution on a bounded domain.To simplify the presentation, we assume  = 0.
Let us define B  as the set of all mappings  : [0, ] × R × Ω → R such that (, , ) is continuous in (, ) ∈ [0, ]×R for almost all  ∈ Ω and sup It is clear that B  is a Banach space with the norm We will prove the existence and uniqueness of the solutions in for any (, ) ∈ [0, ] × R. We also define  0 = 0. First, we show the well-posedness of the above stochastic integral for every  = 1, 2, . ... as  → ∞.Now, letting  tend to infinity on both sides of (26), we see that  satisfies (23).The uniqueness can be proved in similar way.Thus, we complete the proof of this theorem.
Let  be large enough; for example,  >   .We denote  0 = 0, +1 = .On [0, ], ℎ() has its second-order distributional derivative by For  ∈  2 [0, ] with (0) = 0, we can define linear operator R(), which is called the reciprocity gap functional as follows: By the integration by parts formula, we have For any function independent of the unknown parameters, we know from the definition that R() is also independent of the unknown parameters.Namely, R() is observable.
To obtain our estimators for the parameters, we take Then, where   = sin(  /2).
Remark 7. One can find, from the proof, that vector  is the solution of linear equations  0 =  0 .

Estimations for Point Sources from Discrete Time Observations
By Theorem 5, we know that the parameters we want to estimate are contained in the eigenvalues of Hermite matrix  −1 0  1 .We assume in this section that the wave signals are observed at the location  ∈ R but at discrete time instants 0 =  0 <  1 < ⋅ ⋅ ⋅ <   = .We denote Δ =  +1 −   ( = 0, 1, . . .,  − 1).

Convergence of Estimations
In this section, we will show that the estimations obtained in the previous section converge to the true values a.s. as time space Δ tends to zero.
This will be fulfilled by the following two lemmas.