ADVECTION-DIFFUSION EQUATION ON A HALF-LINE WITH BOUNDARY L´EVY NOISE

. In this paper we study a one-dimensional linear advection-diﬀusion equation on a half-line driven by a L´evy boundary noise. The problem is motivated by the study of contaminant transport models under random sources (P. P. Wang and C. Zheng, Ground water, 43 (2005), [34]). We determine the closed form formulae for mild solutions of this equation with Dirichlet and Neumann noise and study approximations of these solutions by classical solutions obtained with the help of Wong–Zakai approximations of the driving L´evy process.

1. Introduction, classical solutions and formulation of the problem. This paper is motivated by a physically important model of a contaminant transport in a one-dimensional semi-infinite pipe with a constant flow velocity and diffusion. In the classical setting, the contaminant concentration u " upt, xq satisfies the advection-diffusion equation $ ' & ' % B t upt, xq " B xx upt, xq´νB x upt, xq, t ą 0, x ą 0, up0, xq " 0, Bupt, 0q " gptq, t ě 0.
(1.1) with zero initial concentration and continuously differentiable source process g " gptq, which affects the concentration of the contaminant at the boundary point x " 0. We assume that the diffusion coefficient equals 1, the flow velocity ν P R.
The boundary conditions are treated in a unified way with the help of the boundary operator B, namely we set for the Dirichlet and Neumann problems respectively. In the Dirichlet case, the source g prescribes the concentration of the contaminant at the boundary, in the Neumann setting it determines the transfer rate through the boundary. If the input 638 LENA-SUSANNE HARTMANN AND ILYA PAVLYUKEVICH function g P C 1 b pR`, Rq, the solution (1.1) is well known in the closed form, see e.g. [9,28].
In realistic models, the assumption that g is smooth and deterministic is too restrictive. For instance, in [34,10,20] the authors allow g to be a random source of contamination in an open channel flow. In particular, the contaminant can be released at random time instants, so that g can consist of a random train of deltaspikes or be a Brownian noise.
In the present paper we tackle two problems. First, we will solve equation (1.1) with a general boundary Lévy noise g " 9 Z, including Brownian motion or Poisson process. We will determine the mild solution of (1.1) as process with values in the fractional Sobolev space H θ pR`q, find its explicit form as a convolution integral w.r.t. the driving Lévy process and determine its law in the large time limit. Second, we study the so-called Wong-Zakai approximations of solutions, namely we consider absolutely continuous approximations of the driving process Z and study convergence of classical solutions to the mild solution of the original equation in the non-standard M 1 -Skorokhod topology.
More details on the theory of PDEs with boundary noise can be found e.g. in [13,12,5,8]. PDEs with the Lévy noise on the boundary were considered in [27], and more recently in [16] and [7]. In the deterministic case, controllability of the onedimensional heat equation on the half line was studied in [22], whereas [2,1,14,21] considered the one-dimensional heat equation on the half line with white Gaussian noise on the boundary.
Eventually, we mention the works [18,34,20] for applications of the mathematical model (1.1) to hydrology, [23] for a discussion on the proper choice of boundary conditions from the physical point of view, and [17] for a bibliography on transport of chemicals through soil.

2.
Solutions of the advection-diffusion equation on a half line with boundary Lévy noise. On a filtered probability space pΩ, F , F, Pq, let Z " pZptqq tě0 be a Lévy process with the characteristic function Ee iλZptq " e tΦpλq , λ P R, where the Gaussian variance, drift and the Lévy measure satisfy σ 2 ě 0, a P R, and µpt0uq " 0, ş pz 2^1 q µpdzq ă 8. We solve the advection-diffusion equation Zptq, t ě 0, Define the Dirichlet map operator D B : R Þ Ñ C 2 b pR`, Rq by the relation D B a " φ, where φ is a unique bounded solution of the ordinary differential equation A straightforward calculation yields that Assume for a moment that we are in the classical setting (1.1) and the input g " 9 Z is a smooth function, g P C 1 pRq. Consider the non-homogeneous equation We claim that upt, xq "ũpt, xq`´D B pgptqq¯pxq. Indeed, the direct substitution yields B t upt, xq " B tũ pt, xq`´D B p 9 gptqq¯pxq " B xxũ pt, xq´νB xũ pt, xq`´p1`νqD B pgptqq¯pxq " B xx upt, xq´νB x upt, xq, and the initial and boundary conditions of (1.1) are also satisfied: up0, xq "ũp0, xq`´D B pgp0qq¯pxq " 0, The solution to the problem (2.3) is found with the help of the convolution formula (Duhamel's principle). Let S B be the C 0 -semigroup of the operator A B " B xx´ν B x on the domain DpA B q. Thenũ is found explicitly as Using the C 0 -continuity of S B we note that so that the integration by parts gives Together with (2.4) this gives upt, xq " 1`νq Id´A B¯SB pt´sqD B pgpsqqpxqds. (2.5) The formula (2.5) allows us to work with the following definition. The latter definition presupposes that the integral on the right hand side exists. The constriction of an integral of Hilbert-valued deterministic integrand w.r.t. a Lévy process is standard, see e.g. [11,29].
The semigroup S B has a well-known explicit representation in terms of the Green function of the heat equation, see [9,28]: x`yq 2 4t¯, Hence straightforward integration allows us to simplify G D px, tq :"´p1`νq Id´A D¯SD ptqe´x " t¯, (2.7) which yields the closed form solution for u. The first result of this paper is formulated in the following Theorem which essentially determines the appropriate Hilbert space H " H θ pR`q. 2) with a Dirichlet boundary condition has a mild solution in H θ pR`q for θ ă´3 2 which has the explicit form upt, xq " dZpsq.
(N) The equation (2.2) with a Neumann boundary condition has a mild solution in H θ pR`q for θ ă´1 2 which has the explicit form t´s¯¯d Zpsq.
In all cases the mild solution is unique and the paths t Þ Ñ upt,¨q are càdlàg in H θ pR`q a.s. Moreover, for any x ą 0, the paths t Þ Ñ upt, xq are continuous in R.
The proof of the Theorem is given in Section 5. Sample paths of solutions u driven by an α-stable Lévy subordinator and a symmetric α-stable Lévy process are presented in Fig. 1 and Fig. 2. Note that negative jumps of the noise may cause negative values of the solution. This explains why Lévy subordinators should be used to model contaminant concentrations.

Limiting probability distribution of the contaminant concentration.
The explicit form of the solution allows us to calculate the stationary contaminant distribution in the large time limit.
To determine the limiting distribution of u in the stationary regime, we consider the equation (2.2) on the time interval r´τ, 0s, τ ą 0, driven by a shifted Lévy process Z τ " pZpt´τ qq těτ . Let u τ " u τ pt, xq, t P r´τ, 0s be its solution. For x ą 0, we consider the limit in law   Figure 3. The scales cpxq of the limiting distribution in the Dirichlet case for ν "˘1, 0 (left), and the Neumann case for ν " 1 (right); α " 0.9, c " 1.
In physically meaningful models, the process Z does not take negative values, i.e. is a Lévy subordinator with the Laplace transform with b ě 0 and the jump measure satisfying µpt0uq " 0, ş 8 0 pz^1q µpdzq ă 8. In this case, upt, xq ě 0 a.s. and its Laplace transform is It is instructive to calculate the limiting law in the following particular case.
Let Z be an α-stable subordinator with Ψpλq "´cλ α , c ą 0 being the scale parameter and α P p0, 1q the stability index. Then In other words, the limiting concentration upxq at the location x ą 0 has a spectrally positive α-stable distribution with the scale The straightforward integration allows to determine the limiting scale c D pxq in case of the Dirichlet boundary noise as where K ν is the modified Bessel function of the second kind. Taking into account the asymptotic expansion we get that for large values of x and α P p0, 1q In the Neumann case, it is clear that c N pxq "`8 for ν ě 0. For ν ă 0, the result of numerical integration is presented in Fig. 3 (right).
It is interesting to note that the integral (3.1) diverges for ν " 0 and α P p0, 2{3s in the Dirichlet case. The same critical value α " 2{3 was discovered in [25] in the analysis of limiting distributions of Lévy driven transport dynamics. Approximations of irregular trajectories of random processes by smooth paths are well known in the literature under the name of Wong-Zakai approximations [36,37]. In particular for dynamical systems driven by Brownian motion there is a number of results in both finite and infinite dimensional settings which state that the approximations converge in the uniform topology to the solution of the Stratonovich equation, see, e.g. [6,31,33].
Commonly used examples of absolutely continuous approximations of a Lévy process Z are polygonal approximations, Z n ptq " Z´k´1 n¯`n´Z´k n¯´Z´k´1 n¯¯´t´k´1 n¯, k ě 1, n ě 1, (4.1) red noise approximations or short memory averaging Zpsq ds, n ě 1.
A common feature of these approximations is that they approximate a continuous process Z (i.e. a Brownian motion with drift) in the uniform topology. If Z has jumps, these jumps are approximated continuously and in a monotonous way. Such type of approximations can be very well described with the help of the so-called M 1 -Skorokhod topology. Let V be a separable Banach space with the norm }¨}. In this paper, we will mainly deal with V " R for approximations of the Lévy process Z and V " H θ pR`q for approximations of solutions of the equation (2.2). For a fixed time T ą 0, the space of V -valued càdlàg functions is denoted by Dpr0, T s, V q. Each f P Dpr0, T s, V q may have at most countably many discontinuities.
For two elements v 1 , v 2 P V we define a segment rrv 1 , v 2 ss as a straight line between v 1 and v 2 : In order to define the so-called (strong) M 1 metric on Dpr0, T s; V q, we define for each f P Dpr0, T s, V q the extended graph of f by Γpf q :" tpt, vq P r0, T sˆV : v P rrf pt´q, f ptqssu, where f p0´q :" f p0q. A total order relation on Γpf q is given by A parametric representation of the extended graph of f is a continuous, nondecreasing, surjective function pr, uq : r0, 1s Ñ Γpf q, pr, uqp0q " p0, f p0qq, pr, uqp1q " pT, f pT qq.
Let Πpf q denote the set of all parametric representations of f .
The mapping d M is called strong M 1 metric on Dpr0, T s, V q. This topology was introduced by Skorokhod in his seminal paper [30]. The extensive analysis of M 1topology in the finite dimensional setting can be found in [35]. For a generalization to Banach and Hilbert spaces, see [24]. The second main result of this paper is the following theorem Finally we note that away of the boundary x " 0, the solution pt, xq Þ Ñ upt, xq is a smooth function. Thus the following theorem holds.  More precisely, let SpRq be the Schwartz space of rapidly decreasing functions and let S 1 pRq be its dual space. Let L 2 pRq be the Hilbert space of equivalence classes of square-integrable functions f : R Ñ C with scalar product xf, gy 2 " ş R f pxqgpxq dx and the associated norm }f } 2 2 :" ş R |f pxq| 2 dx. On SpRq or S 1 pRq respectively we define the Fourier transform F such that for ϕ P SpRq pFϕqpξq :" p2πq´1 {2 ż R e´i xξ ϕpxq dx and for T P S 1 pRq, FT is the functional on S 1 pRq, such that FT pϕq " T pFϕq for every ϕ P SpRq.
For θ P R, H θ pRq denotes the fractional Sobolev space, namely a separable Hilbert space H θ pRq :" tf P S 1 pRq : p1`ξ 2 q θ{2 pFf qpξq P L 2 pRqu with the norm }f } θ,2 :" , see e.g. [15]. In the present paper we will work in the restriction of H θ pRq to R`, denoted by H θ pR`q. We equip this space with the norm }g} H θ pR`q :" inf r g| R`" g }r g} θ,2 .
To define the associated scalar product in H θ pR`q, note that for every f P H θ pR`q there is a unique extension ext f to R, such that }f } H θ pR`q " } ext f } θ,2 and such that the relation xf, gy H θ pR`q :" xext f, ext gy θ,2 defines a scalar product on H θ pR`q, see the Appendix for details. Completeness and separability of H θ pR`q then follow from the completeness and separability of H θ pRq. Furthermore, we write DpR`q for the space of infinitely differentiable functions f : R`Ñ C with compact support in p0, 8q. We need this space to give meaning to the boundary condition of operator A. First note that if Au P H θ pR`q then u P H θ`2 pR`q (see the following Lemma 5.1 and its proof). Since the Sobolev spaces are spaces of equivalence classes of functions, the meaning of the boundary conditions up0q " 0 and B x up0q " 0 for u P H θ`2 pR`q may not be obvious, whereas for u P DpR`q these conditions are well defined. So in what follows, the expressions up0q " 0 and B x up0q " 0 will be understood in the sense of closures of DpR`q in H θ`2 pR`q. This relies on the important fact that if θ ă 1 2 , then DpR`q is dense in H θ pR`q, see the appendix for details. Lemma 5.1. (i) For θ ă´3 2 , the operator A " A D with domain DpAq " tu P H θ pR`q : Au P H θ pR`q, up0q " 0u generates a C 0 -semigroup in H θ pR`q. (ii) For θ ă´1 2 , the operator A " A N with domain DpAq " tu P H θ pR`q : Au P H θ pR`q, B x up0q " 0u generates a C 0 -semigroup in H θ pR`q.
Proof. In the following we write H for H θ pR`q. We use the Hille-Yosida Theorem for contractive C 0 -semigroups (see [26], Let f P H. We define h :" F´1´p F extpf qqpξq λ`ξ 2`i νξā nd g :" h| R`. Then, because of the properties of the Fourier transform, Noting that we also get g P H θ`2 pR`q " DpAq and thus

Proof of Theorem 4.2.
Similarly to the convergence in the uniform topology and in the standard Skorokhod metric J 1 , convergence of a sequence of functions in the metric d M can be described by quantifying the oscillation of the functions. For v, v 1 , v 2 P V the distance from v to the segment rrv 1 , v 2 ss is defined by Define for f P Dpr0, T s; V q and δ ą 0 the oscillation function by Let T ą 0, and let tZ n u ně1 be a sequence of absolutely continuous functions, such that Z n Ñ Z a. PpM pu n , δq ě εq " 0.
1. Neumann case (θ ă´1 2 ): First note that we can extend the solutions upt,¨q and u n pt,¨q to R, simply by extending G N ps, xq defined in (2.7) to a function G N : R`ˆR Ñ R`in the following way: ξq.
If we can show that (i) and (ii) hold in H θ pRq for these explicit extensions, then the result follows easily for H θ pR`q. Linearity of the integral allows us to split both u and u n into two parts and consider each separately. So, for (i), we see that for every t P r0, T s }u n pt,¨q´upt,¨q} H θ pR`q ď }u n 1 pt,¨q´u 1 pt,¨q} H θ pR`q`} u n 2 pt,¨q´u 2 pt,¨q} H θ pR`q , where u 1 pt, xq " Since tZ n u ně1 converge to Z in M 1 on r0, T s and Z is stochastically continuous, it follows that Z n ptq Ñ Zptq for any t P r0, T s in probability, so that the first summand vanishes in probability as n Ñ 8.
Finally we note that the estimate (6.1) and the boundedness of jumps of Z imply that for any q ą 1 and n large enough there are C 1 , C 2 ą 0 such that Hence, the dominated convergence theorem yields To estimate the difference }u n 2 pt,¨q´u 2 pt,¨q} 2 H θ pR`q , we integrate by parts again. Note that for 0 ď s ă t ď T and x P R, |x|´νpt´sq T . Since x Þ Ñ erfcp|x|q is integrable, so is x Þ Ñ erfc´| x|´νpt´sq 2 ? pt´sq¯, and we can estimate the Fourier transform in the following way: where CpT q is a constant that only depends on T . It follows, thaťˇˇ" Furthermore, for the derivative of the error function we see: .
Obviously, x Þ Ñ e´p is integrable for every s P r0, tq, so we can estimate t´s , for some constant C 1 , that does not depend on ξ or s. For the second term we make a simple substitution to see ? t´s e y 2 dy ď C 2 ?
t´s , with C 2 being another constant. Eventually this yields for some C 3 ą 0 }u n 2 pt,¨q´u 2 pt,¨q} 2 Since ş t 0 pt´sq´p {2 ds ă 8 for any 1 ă p ă 2 we can use the Hölder inequality to get the anticipated convergence. Now, we turn to condition (ii). We will only look at the first summand of G N here. The term containing u n 2 can be treated similarly. For any 0 ď t 1 ď t ď t 2 ď T , |t 2´t1 | ď δ and any α P r0, 1s we estimate |Z n ptq´αZ n pt 1 q´p1´αqZ n pt 2 q|`αI 1`p 1´αqI 2`α I 3`p 1´αqI 4 .
Because of the M 1 -convergence of Z n , for any ε ą 0 there is α, such that for n Ñ 8 |Z n ptq´αZ n pt 1 q´p1´αqZ n pt 2 q| ă ε.

LENA-SUSANNE HARTMANN AND ILYA PAVLYUKEVICH
We estimate the first integral as The terms I 2 , I 3 , I 4 are estimated analogously.

Dirichlet case (θ ă´3 2 ):
In the Dirichlet case the Fourier transform of G D has the explicit form Obviously, the only difference to the first summand in the Neumann case is the factor´pν´2iξq. But since in this case θ ă´3 2 , we only have to note, that the term |pν´2iξq| 2 p1`ξ 2 q θ (6.2) plays the same role as p1`ξ 2 q θ for θ ă´1 2 in the Neumann case. Consequently, the proof in the Dirichlet case essentially repeats the steps of the Neumann case.
7. Proof of the Theorem 4.3. We consider the case of Neumann boundary conditions.
It is easy to see that for x ą 0 and t ą 0 the function t¯, t ą 0, is absolutely continuous, and its time derivative equals and is also continuous, 9 G N p0, xq " 0, sup tě0 | 9 Gpt, xq| ď M pxq ă 8. Thus integration by parts yields upt, xq " G N pt´s, xqZ n psq ds.

8.
Appendix. Sobolev spaces have been studied in numerous books, e.g. in [19,15,32]. However, a lot of the results presented there are either valid for θ ą 0, or they are proved in a much broader generality and the proofs rely heavily on the more complex theory of function spaces. The aim of this appendix is to direct proofs specifically for the properties of H θ pR`q that we needed for this article.
For our main result (Theorem 2.2) we used that H θ pR`q is a separable Hilbert space. For θ ą 0 this result follows, e.g., from [15], p. 73, Proposition 3.39. In [32], Theorem 4.5.5, an explicit extension operator is given for a more general class of function spaces. In the following lemma we will prove the the minimality of the norm of the extension directly. (i) For every f P H θ pR`q, there is a unique extension ext f to R such that }f } H θ pR`q " } ext f } θ,2 . (ii) The operator ext : H θ pR`q Ñ H θ pRq is bounded and linear. Proof. We start by showing the existence of the extension for fixed f P H θ pR`q. Let E be the subset of H θ pRq, containing all extensions of f to R. Let For g, h P E we have by the parallelogram law that Since for every ϕ P DpR`q g`h 2¯p ϕq " 1 2 pgpϕq`hpϕqq " 1 2 pf pϕq`f pϕqq " f pϕq, g`h 2 P E and } g`h 2 } 2 θ,2 ě δ 2 . So we get }g´h} 2 θ,2 ď 2}g} 2 θ,2`2 }h} 2 θ,2´4 δ 2 .
That means, pg n q is a Cauchy sequence and since H θ pRq is complete, there is r f P H θ pRq with r f " lim nÑ8 g n in H θ pRq. Since H θ pRq is a subspace of S 1 pRq, this especially implies, that for ϕ P DpR`q, Thus r f P E and, due to the continuity of the norm, } r f } θ,2 " lim nÑ8 }g n } θ,2 " δ. The uniqueness (in the sense of equivalent classes) follows easily from (8.1). Let g P E be another element with }g} θ,2 " δ. Then we only have to prove the linearity of the operator. Boundedness directly follows from the definition.
To show linearity we define M :" tf P H θ pRq : f | R`" 0u and its orthogonal complement M K " tg P H θ pRq : xg, f y θ,2 " 0 for all f P M u. First we show, that for every h P H θ pR`q, extphq P M K .
Lemma 8.1 and the properties of H θ pRq imply that H θ pR`q is a complete, separable Hilbert space with }f } 2 H θ pR`q " xf, f y H θ pR`q " xext f, ext f y θ,2 . Eventually we show denseness of test functions in H θ pR`q for negative θ. The idea of the proof follows the argument of Lemma 1.11.1 in [19].
Proof. Denote by D 0 pRq the subspace of all ϕ P DpRq with ϕ " 0 in a neighbourhood of 0, that is: D 0 pRq :" tϕ P DpRq : Dr ą 0 such that @x P B r p0q ϕpxq " 0u.
Obviously it is enough to show that D 0 pRq Ă H θ pRq is dense if θ ă 1 2 , because then for any h P H θ pR`q there is a sequence td n u nPN Ă D 0 pRq that approximates the extension of h in H θ pRq. Since for every n P N, d n " 0 in a neighbourhood of zero, the restriction d n | R`i s in DpR`q and td n | R`unPN approximates h in H θ pR`q.
Let N : H θ pRq Ñ C be a continuous linear functional. A consequence of the Hahn-Banach Theorem states, that we only have to show, that if N vanishes on D 0 pRq, it also vanishes on the whole space H θ pRq (see [4], Corollary 1.8 and Remark 5). According to the Riesz representation theorem there is a unique element h N P H θ pRq, such that for every u P H θ pRq N puq " xu, h N y H θ pRq " ż R p1`|ξ| 2 q θ Fph N qpξqFpuqpξq dξ.
Let now N puq " 0 for all u P D 0 pRq. We can interpret the function f pξq " Fp1`|ξ| 2 q θ Fph N qpξq as an element of S 1 pRq and get by the definition of F on S 1 pRq that pf, uq " pp1`|ξ| 2 q θ Fph N q, Fuq " N puq " 0 @u P D 0 pRq. This, however, means that supp f " t0u, from which follows (see [15], Theorem 2.31), that f " ÿ jďm c j D j δ 0 for some m P N 0 and c j P C, j " 0, . . . , m, where δ 0 is the δ-distribution.
For θ ă 1 2 this is only possible, if c j " 0 for j " 0, . . . , m, and so h N " 0 and therefore N " 0 on H θ pRq.