Anomalous diffusion limit for a kinetic equation with a thermostatted interface

We consider the limit of solutions of scaled linear kinetic equations with a reflection-transmission-absorption condition at the interface. Both the coefficient describing the probability of absorption and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space heat equation that corresponds to the Kolmogorov equation for a symmetric stable process, which is reflected, or transmitted while crossing the interface and is killed upon the first hitting of the interface. The results of the paper are related to the work in [KOR20], where the case of a non-degenerate probability of absorption has been considered.


Introduction
We study the asymptotic behaviour of a linear kinetic equation on the real line R: with the following boundary condition at x = 0: Here, T denotes the unit one-dimensional torus, understood as the interval [−1/2, 1/2] with identified endpoints, T ± := [k ∈ T : ±k > 0] and R * := R \ {0}.We call the origin o := [y = 0] interface.The parameters γ > 0, T o ≥ 0 are given and the functions ω, p ± , p 0 , defined on T, are assumed to be continuous and non-negative.The scattering operator L k , acting only on the variable k in T, is given by Here R : T 2 → [0, +∞) is C 2 smooth and u belongs to L 1 (T).This equation arises in the kinetic limit of the evolution of the energy density in a stochastically perturbed harmonic chain interacting with a point Langevin thermostat located at y = 0, see [19,20,21,22].The energy density W (t, y, k) at time t is resolved in both the spatial variable y ∈ R and the frequency variable k ∈ T. The function ω is the dispersion relation of the harmonic chain and ω′ (k) is the group velocity of phonons of mode k -theoretical particles that carry the energy due to the chain vibrations of frequency k.The presence of the Langevin thermostat results in the boundary (interface) condition (1.2).The number T o corresponds to the temperature of the thermostat, while p + (k), p − (k) and p 0 (k) are the respective probabilities of transmission, reflection and killing of a mode k phonon at the interface; see [20].They are continuous even functions that satisfy p + (k) + p − (k) + p 0 (k) = 1, k ∈ T. (1.4) With a small risk of ambiguity, in what follows we also write p + := p + (0), p − := p − (0), p 0 := p 0 (0).(1.5)Our main goal is to study the asymptotic behaviour of the solutions of (1.1) on macroscopic space-time scales.More precisely, we are interested in the limits of solutions, when λ → +∞, for the family of rescaled equations W λ (0, y, k) = W 0 (y, k), (y, k) ∈ R × T, (1.6) subject to the boundary condition (1.2), where 1 < α < 2 is a suitably chosen exponent.
In the present paper, we address the situations where p 0 = 0, in fact, we assume that the absorption probability p 0 (k) satisfies, for some κ > 0, p * := lim inf k→0 log |k| −1 κ p 0 (k) ∈ (0, +∞). (1.8) Furthermore, we suppose that the transmission probability does not vanish, that is, (1.9) The dispersion relation ω : T → [0, +∞) is assumed to be even and unimodal, i.e., it possesses exactly one local maximum, at 1/2, and one local minimum, at 0. In addition it is of the C 2 (T * ) class of regularity, where T * := T \ {0}, and there exist one-sided limits of its both derivatives at k = 0.A typical dispersion relation we have in mind is ω(k) = | sin(πk)|, which corresponds to the harmonic chain with the nearest-neighbour interaction.
Concerning the scattering kernel, we assume that it is of the multiplicative form where R 1 , R 2 are two non-negative, even functions belonging to C 2 (T).Without loss of generality, we also suppose that T R j (k) dk = 1, j = 1, 2.
The functions R 1 and R 2 may possibly vanish at some points in T, but we suppose that there exist exponents β j > 0, j = 1, 2, 3, such that S * := lim ) Consistent with the condition (1.13), we further require that dk ∈ (0, +∞). (1.15) We define the probability measure π(dk) := R 2 (k)/ RR 1 (k) dk and we can informally state the main result of the present work; see Theorem 2.3 for a rigorous formulation.
Then, under the hypotheses (1.8)-(1.14)made above, the limit lim λ→+∞ W λ (t, y, k) = W (t, y) exists in a distributional sense on R × T for any t > 0.Moreover, W (t, y) is a unique weak solution, in the sense of Definition 2.2 below, of the equation: with the initial and boundary conditions W (0, y) = W0 (y) and W (t, 0) = T o , respectively.Here, γ is a positive constant depending on the parameters of the model (see (2.9) below), and Γ(•) is the Euler gamma function.
Remark 1.1.The above result should be compared with the result of [19], where the case p 0 = lim k→0 p 0 (k) > 0 has been considered.As we have already mentioned, the informal formulation of the limiting fractional dynamics involves the generator L of the process that is symmetric stable and can be transmitted, reflected or killed when crossing the interface at y = 0.In the situation considered in the present paper we can also view the evolution of W (t, y) as described by equation (1.7), with L corresponding to a symmetric α-stable process that is transmitted or reflected when crossing the interface at y = 0, with the respective probabilities p + and p − .Furthermore, the process is killed upon the first hitting of the interface.
The fractional diffusion limit of a kinetic equation has been the subject of intense investigation in recent years.We refer the interested reader to a review of the existing literature contained in [23].However, there seem to be only few results dealing with a fractional diffusion limit for kinetic equations with a boundary condition.In this context, we mention the papers [1,2,6,7,8,9,19].The case that is somewhat related to ours is considered in [8] and [9].In the first paper, the convergence of scaled solutions to kinetic equations in spatial dimension one, with diffusive reflection condition on the boundary, is investigated.However, this condition is different from ours.Furthermore, the results of [8] do not establish the uniqueness of the limit, stating only that it satisfies a certain fractional diffusive equation with a boundary condition and leaving the question of the uniqueness of solutions for the limiting equation open, see the remark after Theorem 1.2 in [8].In the paper [9], the authors complete the results of [8] and establish an anomalous diffusive limit for a family of solutions of scaled linear kinetic equations in a one-dimensional bounded domain with diffusive boundary conditions.The scattering kernel, defined on R 2 , is also of multiplicative form, as in (1.10), and decays according to an appropriate power law.
Concerning our methods of proof, we rely on the probabilistic interpretation of solutions to kinetic equations.As shown in [19,Proposition 3.2], the solution can be expressed, with the help of an underlying two-dimensional stochastic process {Y (t), K(t)} t≥0 , see Proposition 3.1 below.The component K is the frequency mode of a phonon.If outside the interface [y = 0], it is described by a pure jump, T-valued Markov process corresponding to the operator L k in (1.3).The component Y of the process is the position of the phonon.If the phonon does not cross the interface, it performs a uniform motion between the consecutive scattering events, with velocity −ω ′ (K(t)).On the other hand, if the phonon tries to cross the interface at time t, it will be transmitted, reflected, or killed with the respective probabilities p + (K(t)), p − (K(t)) and p 0 (K(t)).In the case of reflection, the frequency mode of the phonon changes to −K(t).Using this probabilistic interpretation, the asymptotic behaviour of the solutions of (1.6) can be reformulated into the problem of finding the limit of appropriately scaled processes {Y λ (t), K λ (t)} t≥0 , with the parameter λ > 0 corresponding to the ratio between the macro-and microscopic time units.This approach has been successfully implemented in the case of solutions of scaled kinetic equations without interface in [3,17,18].In the case when the killing probability p 0 is strictly positive, as assumed in [19], one can expect that typically the phonon crosses the interface only finitely many times before being killed.Its trajectory can then be obtained by a path transformation (consisting in performing suitable reflections) of the trajectory when no interface is present.This transformation turns out to be continuous in the Skorokhod J 1 topology on the path space, thus the convergence of the scaled processes in the presence of the interface is a consequence of the result for the process in the free space, done, e.g., in [18].This approach cannot be applied in our present situation, since with small p 0 it is not so obvious how to effectively control the number of interface crossings performed by the phonon before it is killed.For this reason, we use a different method to deal with the problem.We define a Lévy-type process Z(t) t≥0 whose scaled limit Z λ (t) t≥0 , as λ → +∞, is related to the respective limit of the position of the phonon by a deterministic time change.To find the limit of Z λ (t) t≥0 , we prove that the associated Dirichlet forms converge, in the sense of the Γ-convergence of forms (see Definition 6.1 below, or [24]), to the Dirichlet form corresponding to the modified α-stable process, described in Remark 1.1.Apart from the limit theory of kinetic equations, we present here some new technical results, which may be of independent interest, for the construction of skew-stable processes, for the characterisation of the fractional Sobolev spaces and for the definition and uniqueness of solutions to non-local parabolic equations.
The paper is organised as follows: after presenting some preliminaries in Section 2, we reformulate the problem of finding the limit of solutions of (1.6), with the boundary condition (1.2) into the problem of finding the limit of the respective stochastic processes.This is done in Section 3. In Section 4, we show how to use the result concerning the limit of stochastic processes to conclude our main result, formulated rigorously inTheorem 2.3.Section 5 is devoted to the proof of the convergence of the scaled Lévy type processes, as stated in Theorem 3.4, assuming the Γ-convergence of their respective Dirichlet forms.The latter is established in Section 6.Some auxiliary facts are proved in Appendix A and B.

2.1.
Precise assumptions on the model.Given an arbitrary set A and two functions f, g : A → [0, +∞], we write f g on A if there is a constant C > 0 such that f (a) ≤ Cg(a), a ∈ A.
We also write f ≈ g on A when f g and g f .In what follows, we always assume the following conventions: A f (a) = 0 and A f (a) = 1, if A = ∅, and ω′ (0) := 0, even though as a rule ω is not differentiable at k = 0.
Let us now denote for any which can be interpreted as expected waiting time for the scattering of a phonon at frequency k.The interplay between scattering and drift will be captured by the function Clearly, we can interpret S(k) as the expected distance travelled by the phonon before scattering.According to our assumptions, S is an odd function, C 1 smooth on [−1/2, 1/2] \ {0}.To simplify some considerations below, we also assume that S is a bijection and hence decreases in (0, 1/2].By (1.12), lim k→0 |k| β 3 |S(k)| = S * and we impose some further restrictions on the model coefficients: From this point on, we shall assume that all the above hypotheses are in force.
• the interface conditions in (1.2) hold, together with the initial condition We highlight that, at least formally, the directional derivative satisfies which justifies the notation in (2.5).It has been shown in [19] that there exists a unique (classical) solution, in the sense of Definition 2.1, to the Cauchy problem (1.1), if the initial distribution W 0 belongs to C To .On this regard, see as well Proposition 3.1 below.
2.3.Weak solution of the limit equation.Recalling the definitions of p ± in (1.9), we introduce the bilinear form and the associated quadratic form Ê[u] := Ê[u, u].We can now define the semi-norm u Ho := Ê1/2 [u] for any Borel function u : R * → R such that the expression is finite and then denote by We proceed to the definition of a weak solution to (1.17) with the interface conditions (1.2).

Definition 2.2. A bounded function
for all t > 0 and test functions F ∈ C ∞ c (R×T).Moreover, the limit W (t, y) is the weak solution, in the sense of Definition 2.2, to the Cauchy problem (1.17) with the initial distribution W0 and the fractional diffusion coefficient (2.9) 3. Probabilistic representation of solutions 3.1.Construction of the position/momentum processes.As usual, N := {1, 2, . ..} and N 0 := {0, 1, 2, . ..}.Let (Ω, F , P) be a probability space carrying the following random objects.We consider a Markov chain K n (k) reporting the consecutive frequencies of the phonon, and the renewal process T n (k) of the scattering times of the phonon frequency (this is Fraktur font . random variables on T, distributed according to the following probability measure: and where {τ n } n∈N 0 is an independent sequence of i.i.d.exponentially distributed random variables with intensity 1 and, we recall, t(k) was defined in (2.1).We introduce the process {T(t, k)} t≥0 as the linear interpolation between the values of T n (k): We then define the continuous-time frequency (momentum) process as Here T −1 is the inverse function of t → T(t, k) and [•] denotes the integer part.
According to (2.2), the phonon position at the time of the n-th scattering of its frequency is In particular, the law of Z n (y, k), for each n ∈ N, is absolutely continuous with respect to the Lebesgue measure on R. We then consider an auxiliary Poisson process {N(t)} t≥0 of intensity 1 that is independent of both {K n } n≥0 and {τ n } n≥0 .We define { Ñ(t)} t≥0 as the linear interpolation between the nodal points of N.
where l := inf{s : N(s) = n} and r := inf{s : N(s) = n + 1}.Our next process, { Z(t, y, k)} t≥0 , is obtained by linearly interpolating between the nodal points of We next define the transmission/reflection/absorption mechanism at the interface o = [y = 0].
To this end, we fix (y, k) ∈ R * × T * and consider the times crosses the interface.Namely, we let n 0 := 0 and then define recursively For the sake of brevity, when there is no danger of confusion, we skip the sequence Z n (y, k) n≥0 from the notation.Similarly, we define the sequence sm { Z(t, y, k)} t≥0 m≥0 of consecutive times when the process Z(t, y, k) crosses the interface o. Namely, we let s0 = 0 and Again, we simplify the notation by omitting the path of the process when there is no danger of confusion.Sometimes, to highlight the dependence of the crossing times on the starting point (which will be relevant in the argument), we may write sm,y,k , or sm,y .We define S(t, k) := Ñ−1 (T −1 (t, k)) and s m := S(s m , k).Then, n m = N(s m ) and sm < s m < sm+1 , P-a.s.
We now consider a sequence {σ m } m≥0 of {−1, 0, 1}-valued random variables, that are independent when conditioned on {K n (k)} n≥0 , such that σ 0 := 1 and Here and below, p ι means p ± if ι = ±1, respectively.Of course, {σ m } m≥1 can be defined by applying the quantile functions for (3.5) to independent i.i.d.uniform random variables.We can finally add the random interface mechanism to the processes considered.Namely, for m ∈ N 0 .The "true" interface position process is then given by We now denote by f := min {m ∈ N : σ m = 0} the interface crossing at which the particle gets absorbed and let s f := S(s f , k).The following probabilistic representation of the solution to (1.1), with interface conditions (1.2), holds.
is the unique classical solution, in the sense of Definition 2.1, to the Cauchy problem (1.1) with initial distribution W 0 .
Proof.The existence and uniqueness of a classic solution and the following representation have already been shown in [19], Section A, and Proposition 3.2.Recalling that Y o (t, y, k) = 0 if t ≥ s y,k,f , we can use that W 0 (0, k) = T o to get that Equation (3.8) then follows immediately from (3.9).

3.2.
Scaling of the process (Y o (t, y, k), K o (t, k)).Equations (1.6) may be considered a special case of (1.1), so we first focus on establishing flexible notation for later use.As before, we fix (y, k) ∈ R * × T * .We let λ > 0 and rescale the clock processes, introduced in the previous section, as follows: We then notice that for large λ, the process S λ (t, k) becomes almost deterministic.More precisely, by (1.15) we have that θ Except for the first deterministic term, the inverse S −1 λ (s, k) = T λ ( Ñλ (s), k) can be represented as a Poisson sum of i.i.d.variables with finite first moment and so, by a standard argument using the strong law of large numbers, it can be shown that: We can now rescale the "position" process in the following way.Let us define { Zλ (t, y, k)} t≥0 as the linear interpolation between the nodal points of Z λ (t, y, k), where We then construct the sequences {s λ m } m≥0 , {s λ m } m≥0 of stopping times for the rescaled processes Z λ (t, y, k), Zλ (t, y, k) as in (3.4), skipping (y, k) from the notation, when unambiguous.Since the law of Z λ (t, y, k) is absolutely continuous, for each y ∈ R there exists a strictly increasing sequence To set the notation for the processes subject to the random mechanism at the interface, let {σ λ m } m∈N be a sequence of {−1, 0, 1}-valued random variables that are independent when conditioned on {K n (k)} n≥0 and such that We then set Here σ λ 0 := 1.Similarly, the continuous trajectory process { Zo λ (t, y, k)} t≥0 can be obtained as in (3.15) but with respect to the stopping times {s λ m } m∈N 0 .
In what follows, we show that it is unlikely that the scaled position process Z λ (t, y, k) crosses the interface after the first jump, when λ becomes large.Indeed, recall from (3.14 The following estimate follows immediately.
Proposition 3.3.For all λ > 0, y ∈ R * and k ∈ T * we have Auxiliary stable Lévy process.To describe the limit, as λ goes to +∞, of processes {Z o λ (t)} t≥0 , we consider the α-stable Lévy process {ζ(t)} t≥0 with the infinitesimal generator  and we also let It is known that t y,f is finite P-a.s. and its law is absolutely continuous with respect to the Lebesgue measure on R (cf.[27,Example 43.22]).We can finally introduce the tentative limit process {ζ o (t, y)} t≥0 , as follows.Recalling the definitions of p ± in (1.5) and p 0 = 0, stemming from (1.8), we take a sequence {σ m } m∈N of i.i.d.{−1, 1}-valued random variables such that for any m ∈ N, the random variable σ m is independent of {ζ(t, y)} t≥0 and P(σ m = ±1) = p ± (the notation of (3.5) is best ignored from now on).Letting σ 0 := 1, we define for m ∈ N 0 , and ζ o (t, y) := 0 if t ≥ t y,f .The process, discussed in detail below, is an interesting take on the question of construction of skew stable Lévy processes, a topic recently discussed in [16].However, in view of limit theorems for kinetic equations, the following is our main motivation.
The proof of the theorem shall be presented in Section 5 below.For simplicity, denote by X the space D[0, +∞) ×C[0, +∞) equipped with the product of the M 1 -topology and the topology of uniform convergence over compact intervals.The definition of the M 1 -topology on D[0, +∞) can be found, e.g., in [ Let us denote K λ (t, k) := K(λt, k) and s λ m := S λ (s λ m , k).We finally define the positionmomentum process If we let η o (t, y) := ζ o θt, y , where θ is defined in (3.11), then the following result is an immediate consequence of Corollary 3.5.
Proof.Invoking Theorem 7.2.3 of [29] and using formula (3.20), we conclude the weak convergence of the processes.Since, for any deterministic time t ≥ 0 we have by virtue of Lemma 6.5.1 in [28], the set of discontinuities of the one-dimensional projection mapping ω → η o (t, y, ω) is of null probability.Using the continuous mapping theorem, see Theorem 2.7 of [5], we conclude the convergence of the one-dimensional distributions.The generalisation to finite-dimensional distributions is trivial.We conclude this section by presenting a probabilistic characterisation of the weak solution of the limit Cauchy problem (1.17) in terms of the process η o (t, y).A proof of this result can be found in the Appendix B below.
is the unique weak solution, in the sense of Definition 2.2, to the Cauchy problem (1.17) with initial condition W0 .

Proof of Theorem 2.3
Since W λ (t, y, k) − T o is the solution of (1.6) with the interface condition (1.2) corresponding to the zero thermostat temperature, we assume without loss of generality that T o = 0. Clearly, the solution W λ is given by (3.8).In what follows, we often write Thanks to Corollary 3.6 for any ε > 0 we can find a sufficiently large M > 1 such that lim sup This fact allows us to restrict, without loss of generality, our attention to the case where T ∞ = 0.By a standard approximation argument, it is further enough to consider an initial condition in particular W 0 (0, k) = 0, for any k in T. If the initial condition W 0 is independent of k, i.e.W 0 (y, k) = W 0 (y), then we can immediately conclude the proof from Proposition 3.1 (with respect to the rescaled processes), equation (3.21) and Corollary 3.6.
The next result enables to replace an arbitrary initial condition W 0 (y, k) by its average over T with respect to the measure π(dk) = R 2 (k)/ RR 1 (k) dk.For notational simplicity, we denote by L 2 π (R × T) the L 2 -space on R × T with respect to the product measure π(dk)dy.
Lemma 4.1.Suppose that W 0 ∈ C c (R × T).Let W0 be the function defined by (1.16).Then, for any ε > 0, there exists δ 0 > 0 such that The above lemma shall be proved at the end of this section.Furthermore, computing π (R×T) and using the equation (1.6) together with the interface conditions (1.2), see the calculations in [4, Section 3], we immediately conclude the following.
We can now finish the proof of Theorem 2.3.We first show the following variant of (2.8), Choose an arbitrary ε > 0. Let δ > 0 be as in Lemma 4.1 and Wλ the solution to the Cauchy problem (1.6) when the initial condition is given at time t = δ by Wλ (δ, y, k) = W0 (y).In particular, we can use Proposition 3.1 to represent Wλ as By virtue of Proposition 4.2 and Lemma 4.1, we have that Choosing δ > 0 sufficiently small, we can then guarantee that As a result, we conclude that for any ε > 0, lim sup and (4.2) immediately follows.The conclusion of Theorem 2.3 with respect to the Lebesgue measure on T, as in (2.8), immediately follows from (4.2) and the fact that F Wλ (t) ∞ ≤ F W 0 ∞ .The claim that W (t, y) is a weak solution of (1.17) follows from Proposition 3.7.
Proof of Lemma 4.1.Fix an arbitrary ε > 0. We show first that there exists δ > 0 such that lim sup where Ŵ o λ (t, y, k) . Indeed, let ρ > 0 be sufficiently small so that lim sup We then consider ρ ′ ∈ (0, ρ) and conclude that for any k in T * , sup where Y λ (t, 0, k) is the analogue of the process Y o λ (t, 0, k), but without the interface (or, equivalently, with p + ≡ 1 in the model).The above equation now implies that for any ρ ′ < ρ, there exists a sufficiently small δ > 0 such that lim sup In particular, it then follows that lim where s λ y,k,1 is the first time the process Y λ (t, y, k) crosses the interface.From the above reasoning, we now claim that there exist ρ, δ > 0 sufficiently small so that lim sup Indeed, the expression inside the integral in (4.9) can be rewritten as Using (4.7), we then choose δ > 0 so that for any k ∈ T * and any |y| ≥ ρ, we have Noticing that the expression inside the integral in (4.9) is uniformly bounded in λ, we can finally use Fatou's lemma to conclude that (4.9) holds.Since W 0 is compactly supported, Equation (4.4), now follows if we prove that for a fixed δ > 0, it holds that lim ρ→0+ lim sup To do so, we first notice that the function is compactly supported, even, non-negative and such that W * 0 (Y o λ (δ, 0, k)) ≤ W * 0 (Y λ (δ, 0, k)).Noticing that a classical argument through stable central limit theorem (see, e.g., [12,Theorem 4.1]) shows that Y λ (δ, 0, k) weakly converges to η(δ) = ζ( θδ), we get that lim sup Equation (4.10) now follows immediately taking the limit, as ρ goes to zero, in the above expression.We can now use (4.5), (4.9) and (4.10) to conclude the proof of (4.4).Using the fact that W 0 is compactly supported and (4.8), we know that for any ε > 0 there exists δ > 0 such that lim sup where Ŵλ (t, y, k) Noticing that the dynamics of the momentum process K λ (t, k) is reversible with respect to the measure π on the torus T and 0 is a simple eigenvalue for the generator L k , we finally have that lim sup and we have concluded the proof.

Proof of Theorem 3.4
To prove Theorem 3.4, we show the convergence of the Markov semigroups corresponding to processes {Z o λ (t, y, k)} t≥0 , see (3.15) .In the first part of the present section, we focus on constructing a Lévy-type process { Ẑo λ (t, y)} t≥0 whose increments, after the first jump, coincide with those of {Z o λ (t, y, k)} t≥0 .
5.1.Construction of the associated Markov process.Let us denote by r(y) the density of S(K 1 ), where S(k) is defined in (2.2) and K 1 is distributed according to R 2 (k)dk (see (3.1)).

Properties of the Markov semigroup corresponding to { Ẑo
λ (t, y)} t≥0 .In order to show Theorem 5.2, we will strongly rely on a convergence property between the corresponding Markov semigroups.The process Ẑo λ (t, y) killed at the interface is Markovian and its transition semigroup is given by (5.11) Here ŝλ y,f := inf{t > 0 : Ẑo λ (t, y) = 0}.We shall also consider the process stopped at ŝλ Let now { ζ(t)} t≥0 be an independent copy of the stable process ζ(t).Similarly to (3.4), we can also consider, for any m ∈ {0, 1, . ..} and z ∈ R * , the m-th consecutive time tz,m that the process {z + ζ(t)} t≥0 crosses the interface.Using the independence of increments for the stable Lévy process, we then rewrite the right-hand side of (5.15) as Then, using the symmetry of the law of a stable process, it follows that for any ε = ±1 and any z in R * , we have that .
The semigroup P o t is symmetric.Fixed u, v in L 2 (R), our aim is to show the following: (5.17) We start by rewriting the left-hand side of (5.17) as where the reversed time process is given by ζ r (s, y) We can now exploit the above lemma with If we assume for the moment that m j=1 ε j = 1, it then obviously follows from (5.20) that ε j , t y,m ≤ t < t y,m+1 dy, (5.21)where t y,m := t m (ζ(•, y)).Suppose now that m j=1 ε j = −1.Since the laws of {ζ(t, y)} t≥0 and {−ζ(t, −y)} t≥0 are identical, we can rewrite (5.20) as ε j , t y,m ≤ t < t y,m+1 dy.As in (5.12), it is now clear that for any u in B b (R), it holds that where P o t u(y) is the semigroup associated with {ζ o (t, y)} t≥0 .We will show in Section 6 below that the following result holds: Theorem 5.5.As λ tends to +∞, the semigroups {P o,λ t } t≥0 , defined in (5.11), strongly converge in L 2 (R), uniformly on compact intervals, to the semigroup {P o t } t≥0 given in (5.13).Thanks to the above result, we can now prove that the Markov process Ẑo λ (t, y) converges to ζ o (t, y).As before, we then denote Ẑλ (t, y) := y + Ẑλ (t) for any y ∈ R * .If we assume for the moment that |ξ| ≥ λ 1/α , we have that where On the other hand, if |ξ| ≤ λ 1/α , we can write from (5.24) that Fixing c > 0, we now notice from the above controls that Ψ λ (ξ) θ * > 0 for all |ξ| ≥ c.Then, it follows that the random variable Ẑλ (t) admits a density f λ (t, •) and belongs to Noticing that the process { Ẑo λ (t, y)} t≥0 can be obtained from { Ẑλ (t, y)} t≥0 by performing a transformation of the trajectory (consisting in transmission-reflection, or moving the phonon to the interface) described in Section 3.2, we have in particular that Since we already know the weak convergence of the laws of Ẑλ (t, y) , it in turn implies the tightness of the laws of the random variables Ẑo λ (t, y), as λ → +∞.To conclude the proof, it is enough to show that for any y ∈ R * and any φ ∈ C ∞ c (R), we have Let ε > 0 be arbitrary.Choosing h ∈ (0, t] sufficiently small, we can guarantee that where t y,1 , ŝλ y,1 represents the first time the processes crosses the interface o (cf.Equation (3.4)).Recalling the definition of the semigroups P o,λ t , P o t in (5.12) and (5.13), we then have that By the above reasoning and (5.12), it now follows that Thanks to Proposition 5.5 and [15, Theorem 46.2], we know that P o,λ t−h φ and f λ (h, •) strongly converge in L 2 (R), as λ → +∞, to P o t−h φ and f (h, •), the density of ζ(h), respectively.Therefore, To deal with the above limit we use the following lemma, whose proof is presented in Section 5.3.2.

5.3.2.
Proof of Lemma 5.6.We are going to show (5.34)only for f in C ∞ c (R).The general statement of Lemma 5.6 then follows from a density argument.Fixed M > 0, let us consider an even, smooth function φ M : R → [0, 1] such that We can then define ψ M := 1 − φ M .Using (5.28), we now show that for any |y| ≤ M/2, it holds that Clearly, a similar reasoning holds for ζ o (t, y) as well.Fixed ε > 0, we now choose M > 0 and λ 0 > 1 large enough so that for any λ ≥ λ 0 , |y| ≤ M/2, it holds that Using again that M is large enough so that suppf ⊆ [−M/2, M/2], we then conclude that The above estimate and Proposition 5.5 finally imply that and Equation (5.34) immediately follows.

5.3.3.
Weak convergence in J 1 -topology.By the previous argument, it is enough to prove tightness of the laws of { Ẑo λ (t, y)} t≥0 over D[0, +∞) with the J 1 -topology.We shall make use of the following notation.Given a function f in D[0, t * ], t * > 0 and a < b in [0, t * ] we denote ω (a, b, f ) := sup{|f (t) − f (s)| : a ≤ s < t < b}.We can then define the D-modulus of f of step δ > 0 as: where I δ is composed of all the partitions such that t j − t j−1 ≥ δ, for any j = 1, . . ., N. It is not difficult to check that the laws of { Ẑλ (t, y)} t≥0 are tight over D[0, +∞) with the J 1 -topology.Theorem 13.2 in [5] then implies that for any t * > 0 and any ε > 0, it holds that (5.39)Moreover, using nested partitions of the time interval [0, t * ], one can easily show that Thanks to the above control and (5.28), we therefore conclude that (5.38) and (5.39) hold for Ẑo λ (t, y) as well and the tightness of the latter, as λ → +∞, immediately follows.
6. Proof of Theorem 5.5 We are going to show here the strong L 2 -convergence of the semigroups P o,λ t , defined in (5.11) and associated with the Markov processes Ẑo λ (t, y), to the semigroup {P o t } t≥0 , given in (5.13) and corresponding to the process ζ o (t, y).The main tool used in the proof is the notion of the Γ-convergence of the Dirichlet forms corresponding to the semigroups.Before going into the actual proof, we recall some basic facts on the subject.6.1.Basic notions on Sobolev spaces and Dirichlet forms.Given β in (0, 2), we define the following form: for any Borel function u : R → R. We admit the possibility that the right-hand side of (6.1) equals infinity.Then, H β/2 (R) := {u ∈ L 2 (R) : E[u] < ∞} is a Hilbert space when endowed with the following norm We have H α/2 (R) ⊂ C b (R) when, as in our case, α > 1 (see, e.g., [14], equation (1.4.33)) and thus u(0) is well defined.Moreover, C ∞ c (R) is dense in H α/2 (R) and it is not difficult to check that the form E is regular on H α/2 (R) in the sense of [14,Example 1.4.1].
Recalling that R * := R {0}, we also consider the family 0 (R * ) can be equivalently characterised as the set of functions u in H α/2 (R) such that u(0) = 0.For a proof of this fact, see Corollary 6.4 below.It easily follows that the form E is regular on H α/2 0 (R * ).By (1.2), we also notice that the form Ê, given in (2.6), is comparable with E on the space H α/2 0 (R * ).In particular, it is regular and thus, it is a Dirichlet form on H α/2 0 (R * ).To prove Proposition 5.5, we will show that the corresponding Dirichlet forms, defined below, converge in a suitable sense.More precisely (cf.[11,Definition 4.1], or [24, Section 1]), Definition 6.1.Let {E λ } λ>0 be a family of Dirichlet forms on L 2 (R) endowed with their natural domains: Then, E λ is called Γ-convergent to a Dirichlet form E ∞ , as λ → +∞, if for any u ∈ L 2 (R) the following conditions are satisfied: i) for any family {u λ } λ>0 weakly convergent to u in L 2 (R), it holds that ii) there exists a family {v λ } λ>0 strongly convergent to u in L 2 (R) such that The notion of Γ-convergence is particularly useful for our purposes since it naturally implies (cf.[24, Corollary 2.6.1]) the strong convergence of the corresponding semigroups on L 2 (R).
We conclude this subsection presenting some properties of the Sobolev spaces we have just introduced.The proofs of the results formulated below can be found in Appendix A. Let us start with the following variant of the Hardy-type inequality of Dyda [13].Proposition 6.2.For β = 1, there exists a positive constant We then obtain a characterisation of H We stress here that even though the constant function 1 is not in L 2 (R), it is still possible to use the symmetry of the operator P o t , arguing by approximation.Therefore, It is then easy to verify that, for any u Fixed y in R * , let { ζ(t)} t≥0 be the symmetric α-stable Lévy process starting at y but killed at hitting 0. Clearly, its transition semigroup { Pt } t≥0 , given as in (5.13), is made of symmetric Markov contractions on L 2 (R).In particular, its corresponding Dirichlet form equals: for any function u belonging to its natural domain D( Ẽ) = H α/2 0 (R * ).For a proof see, e.g., [10,Section 3.3.3].We consider as well Ẽt [u] defined as in (6.11) with respect to Pt .The same calculations that lead to (6.12) can be performed again to show that Ẽt Recalling the construction of the process ζ o (t, y) in (3.18), we now write that We then denote by P m the family of sets {ε 1 , . . ., ε m } ⊆ {−1, 1} m such that m j=1 ε j = 1 and by P c m its complement.Let s m := which ends the proof of the proposition.6.3.Proof of Theorem 6.5.In the present section, we are going to show that the Dirichlet forms { Êλ } λ>0 , defined in (6.7), are Γ-convergent, when λ tends to +∞, to the Dirichlet form r * Ê given in (2.6)-(5.8).Thanks to Proposition 6.6, Theorem 6.5 will then follow immediately.
Recalling the meaning of the Γ-convergence in Definition 6.1, condition ii) easily follows choosing the trivial family {v λ } λ>0 of functions given by v λ = u and using the Lebesgue dominated convergence theorem, together with (1.9) and (5.9).We now focus on showing condition i).Let {u λ } λ>0 be a family of functions weakly convergent to u in L 2 (R).We start by noticing that if lim inf λ→+∞ Êλ [u λ ] = +∞, then condition i) clearly holds.We can then suppose without loss of generality that lim inf λ→+∞ Êλ [u λ ] < +∞.(6.16)Let {λ n } n∈N be a sequence in (0, +∞) such that λ n → +∞ and where we denoted u n := u λn , Ên := Êλn .We will use the following lemma, whose proof is presented in Section 6.4.Lemma 6.8.Let {u n } n≥1 be a bounded sequence in L 2 (R) such that lim n→∞ Ên [u n ] < ∞.Then, there exists a subsequence {u n k } k≥1 that is a.s.convergent to u.
Fatou's lemma and (5.9) then imply that Proof.Thanks to (1.9), we know that p * := inf k∈T p + (k) > 0. Since lim n→∞ Ên [u n ] < ∞, we then conclude from (6.7) that where rλ is given by (5.6).The right-hand side is then of the same order of magnitude as where Ψ n := Ψ λn was defined in (5.24).Thus, for any n so large that K ≤ λ 1/α n , the estimates (6.21), (5.25) and (5.27) imply that and the conclusion of Lemma 6.10 follows.where for any fixed L > 0, we have On the other hand, recalling that {u n } n≥1 is uniformly bounded in L 2 (R) and φδ belongs to the Schwartz class, we conclude that for any ε > 0 it is possible to choose L := L(ε), sufficiently large, such that uniformly in n.From (6.24)-(6.26),we can then conclude that (6.23) follows.From the compactness of {u n φ δ } n≥1 , we can now choose a subsequence that is a.e.convergent on [−δ, δ].
Using the Cantor diagonal argument, we then find a subsequence of {u n } n≥1 that converges a.e. on R.
6.5.Proof of Lemma 6.9.First observe that since the sequence {u n } n≥1 is bounded in L 2 (R), it is enough to show that for any ρ small enough, we have: We will actually prove an analogue of (6.27) with the integral over [0, 1], as the argument in the case of [−1, 0] is similar.The proof is divided into three separate steps.
Step 2. We then show that for any λ n > 1, the function u n can be decomposed as: u n (y) = u (1)  n (y) + u (2)  n (y), (6.31) and the proof of the corollary is concluded.
[19]Clearly, the constant function T o belongs to C To .Furthermore, F ∈ C To if and only if F −T o ∈ C 0 .This allows us to reduce the proofs of some results below to the case T o = 0. Following[19], we now recall the definition of solution to equation (1.1) with the interface conditions (1.2).Definition 2.1.A bounded, continuous function W : R + × R * × T * → R is called a solution to equation (1.1) with the interface conditions (1.2) if all the following conditions are satisfied:• for any ι, ι ′ in {−, +}, the restriction of W to R + ×R ι ×T ι ′ can be extended to a bounded, continuous function on R+ × Rι × Tι ′ ; • for any (t, y, k) ∈ R + × R * × T * fixed, the function s → W Statement of the main result.Once we have formulated the definition of a solution, we are ready to formulate rigorously our main result.It reads as follows:Theorem 2.3.Suppose that the assumptions (1.8), (1.9) and (2.3) are in force.Let T ∞ ∈ R and W 0 in C To be such that W0 − T ∞ ∈ L 2 (R) and W0 − T o ∈ H o , with W0 defined in(1.16).Let W λ (t, y, k) be the solution, in the sense of Definition 2.1, to the Cauchy problem (1.6).Then, By a straightforward modification of (3.4), we let t 0 := 0 and denote by t m ζ(t, y) * .For any y = 0, we denote ζ(t, y) := y + ζ(t).t≥0m≥1 the consecutive times when the process ζ(t, y) crosses the interface o.We often abbreviate t m,y := t m ζ(t, y) 28, Section 12].It then immediately follows from Theorem 3.4, Proposition 3.2 that: Corollary 3.5.As λ → +∞, the processes Zo λ (t, y, k), S λ (t, k) t≥0 converge both in finite distributions and weakly, over X , to ζ o (t, y), θt t≥0 .
In particular, s + 1 = p + .By a direct calculation, it is possible to show the following: Lemma 6.7.The sequence {s m } m∈N tends to 1/2.Moreover, if p + > 1/2, then {s m } m∈N is strictly decreasing, if p + = 1/2, then it holds that s m = 1/2 for any m ≥ 1 and if p + < 1/2, then {s 2m−1 } m∈N increases while {s 2m } m∈N decreases.If we suppose now that u belongs to D(E o ), the closure of C ∞ c (R * ) with respect to the form E o , then(6.15)implies that u belongs to D( Ẽ) = H ) = D(E o ) and there exists a positive constant C such that C {ε 1 ,...,εm}∈Pm m j=1 p ε j .* ) and implies in particular that H * 6.4.Proof of Lemma 6.8.Fix δ > 0 and let us consider the function φ δ given in (5.36).We claim that the sequence {u n φ δ } n≥1 is strongly compact in L 2 (R).Using Pego Criterion, see [26, Theorem 3], it is enough to show that lim K→+∞ |y|>K