Space-time coupled evolution equations and their stochastic solutions

We consider a class of space-time coupled evolution equations (CEEs), obtained by a subordination of the heat operator. Our CEEs reformulate and extend known governing equations of non-Markovian processes arising as scaling limits of continuous time random walks, with widespread applications. In particular we allow for initial conditions imposed on the past, general spatial operators on Euclidean domains and a forcing term. We prove existence, uniqueness and stochastic representation for solutions.


Introduction
We study the space-time coupled evolution equation (CEE) H ν u(t, x) = −f (t, x), in (0, T ] × Ω, where f and φ are given data and H ν u(t, x) = ∞ 0 e rL u(t − r, x) − u(t, x) ν(r) dr, t > 0, (1.2) so that −H ν = (∂ t − L) ν is the subordination of the heat operator (∂ t − L) by an infinite Lévy measure ν. Here the Markovian semigroup {e rL } r≥0 acts on the space variable Ω ⊂ R d , and we denote the associated stochastic process by B = r → B r . As our main result, we prove the stochastic representation for the solution u(t, x) to (1.1) to be (1.3) where S ν is the Lévy subordinator with Lévy measure ν, S ν is independent of B x , with x denoting the starting point of B, τ 0 (t) = inf{r > 0 : t − S ν r < 0} is the inverse of S ν and τ Ω (x) is the life time of B x , x ∈ Ω. Let [29].) It is important to observe that is a non-Markovian process that is trapped when t → τ 0 (t) is constant. Then there is a clear intuition for the initial condition on the "past" in problem (1.1), as the time parameters of φ are weighted according to the overshoot S ν τ0(t) − t, which is the waiting/trapping time of Y t (caused by the time change τ 0 ). As an example, let φ(t, x) = 1 {t<−1} 1 {x>0} (f = 0 and Ω = R), then the solution is u(t, x) = P [Y x t will not move for at least 1 time-unit & Y x t is positive] , and thus the modeller's choice of φ allows to gain control over the length of the traps in combination with the spatial position of the process. This implies that imposing initial conditions on the past in (1.1) results in a "finer" probabilistic model when compared to only imposing standard initial conditions at 0 (i.e. φ independent of time, so that in the example above u(t, x) = P[Y x t is positive]).
Select now time independent initial data φ(t) = φ 0 , f = 0, d = 1 and let r → B r be a Lévy process with density p r (·). Notating Φ(dy, dr) = p r (y)ν(r) dy dr, we can now write H ν u(t, x) as x) Φ(dy, dr) + R φ 0 (x − y)Φ(dy, (t, ∞)), and the CEE (1.1) is a particular case of [24, Theorem 4.1, eq. (4.1)]. In [24], problem (1.1) appears in Fourier-Laplace space as Space-time CEEs and their stochastic solutions random walks (OCTRWs). The overshoot is reflected in the time change of B living above t, in the sense that S ν τ0(t) > t [6,Theorem III.4]. Recall again that Y is trapped when τ 0 is constant, like the fractional-kinetic process t → B τ0(t) [33,Chapter 2.4]. But the duration of a waiting time induced by τ 0 equals the length of the last discontinuity of S ν τ0 , mirrored in the coupling of space (B S ν ) and time (τ 0 ). In particular, if the subordination is performed by an α-stable subordinator S α , then Y scales like B, because S α τ0(t) = tS α τ0 (1) .
This implies that if B is a Brownian motion, the order of the mean square displacements (MSDs) of Y t and B t are both t, in contrast with the fractional kinetic process, whose MSD is of order t α [33,Chapter 2.4]. The related literature known to us deals with variations of the CEE (1.1) in Fourier-Laplace space, mostly motivated by central limit theorems for coupled OCTRWs. See [30,36] for multidimensional extensions of OCTRW limits, [31] for explicit densities in certain fractional cases, and [41,25] for alternatives to the first derivative in time. Due to their peculiar properties OCTRWs are popular models appearing for instance in physics and finance [40,42,43,19,39,26]. Worth mentioning that the OCTRW limit first appeared in [24] as the overshooting counterpart of CTRW limits studied in [5,3,32], which result in different CEEs. In this latter case, the counterpart of (1.1) expects the solution to be the subordination of B by S ν τ0(t)− , for S ν s− the left continuous modification of S ν s . We could not treat this case, as our method relies on Dynkin formula, and we could not recover a suitable version for the left continuous process S ν s− . However, this case is treated in the general setting of space-time Feller semigroups in [4], as discussed below. Note that, although related, problem (1.1) is different from [35, problem (1.1)], as the latter does not impose initial conditions, and in turn it does not describe anomalous diffusion.
We present two main results. The first one is Theorem 3.5, where we prove wellposedness and stochastic representation for solutions. We call these generalised solutions, and they are (carefully chosen) pointwise limits of potentials of the space-time process (1.4) killed outside (0, T ] × Ω. The second main result is Theorem 4.8, where we prove that (1.3) is a weak solution for (1.1) for weak data and e L self-adjoint. We could not prove uniqueness of weak solutions, which appears to be a subtle problem already for the simpler (uncoupled) Marchaud-Caputo EE [1]. We mention that we assume existence of densities for the processes B and S ν to work with weak data, but this assumption could be dropped by working with smoother data, as we discuss in Remark 3.6-(ii).
To the best of our knowledge, the novel contribution of this article is the following. A general probabilistic method to treat wellposedness and stochastic representation for the CEE (1.1) when it features: initial conditions on the past, general spatial operators on Euclidean domains and a forcing term. Moreover, our proof method tightly follows [17] and [38], which treat the rather different uncoupled EEs of Caputo/Marchaudtype. Therefore proposing a unified method for a large class of fractional/nonlocal EEs with initial conditions on the past. Besides the introduction of initial conditions on the past for CEEs, this work appears to be the first one that formulates and solves the governing equation of Y in differential form, without relying on Fourier-Laplace transform techniques. This was also part of the contribution of [4], which treats different CEEs, as mentioned above. We remark that [4] and our work share the idea of considering the generator of the respective space-time coupled process killed when the process in the time variable crosses a barrier (t − S ν in (1.4) crossing 0 in this work). Then the potential of this space-time killed process is a bounded operator and so one can invert its generator (compare the proofs of [4, Theorem 4.1] and Lemma 3.3, respectively). Unfortunately, as mentioned above, the rest of our strategy does not appear to be compatible with the left continuous modifications of the processes in [4] and thus we do not know whether one can impose initial conditions on the past for the processes treated in [4].

Space-time CEEs and their stochastic solutions
The article is organised as follows; Section 2 introduces general notation, our assumptions and the main semigroup results used to treat the operator H ν ; Section 3 proves Theorem 3.5 and presents some concrete fundamental solutions to (1.1); Section 4 proves Theorem 4.8.

Notation and subordinated heat operators
We denote by R d , N, 1, a.e., a ∨ b and a ∧ b the d-dimensional Euclidean space, the positive integers, the indicator function, the statement almost everywhere with respect to Lebesgue measure, the maximum and the minimum between a, b ∈ R, respectively. We denote by Γ(β) the Gamma function for β ∈ (−1, 0) ∪ (0, ∞), and we recall the standard identity Γ(β + 1) = Γ(β)β. We denote by E the topological closure of a set E. If E is a locally compact space, then we write C(E) for the real-valued continuous functions on E. We write C ∞ (E) for the Banach space of functions in C(E) vanishing at infinity with the supremum norm [10, page 1]. This means that for every f ∈ C ∞ (E) and > 0 there exists a compact set K ⊂ E such that sup x∈E\K |f (x)| ≤ , moreover we canonically extend f to E ∪ {∂} by f (∂) = 0 for ∂ a cemetery state and if E is not compact we set E ∪ {∂} to be the one-point compactification of E, otherwise ∂ is an isolated point. We denote by B(E) the Banach space of real-valued bounded Borel measurable functions on E with the supremum norm. We mostly work with the space-time Banach spaces for some Ω ⊂ R d and any T > 0, with the convention that we extend the functions in C ∞ ((0, T ] × Ω) to zero on {0} × Ω and for f in any of the above three spaces we write T ]} for any T ≥ 0, and C 1 c (0, T ) to be the space of continuously differentiable functions in C(R) with compact support in (0, T ). For two sets of real-valued functions F and G we define For a sequence of functions {f n } n≥1 and a function f , we write f n → f bpw (bpw a.e.) if f n converges to f pointwise (a.e.) as n → ∞, and the supremum (essential supremum) norms of all f n 's are uniformly bounded in n. We denote by L 1 (Ω), L 2 (Ω) and L ∞ (Ω) the standard Banach spaces of Lebesgue integrable, square-integrable and essentially bounded real-valued functions on Ω, respectively. We denote by · X the norm of a Banach space X.
The notation we use for an E-valued stochastic process started at x ∈ E is X x = {X x s } s≥0 = s → X x s . Note that the symbol t will often be used to denote the starting point of a stochastic process with state-space E ⊂ R. By a strongly continuous contraction semigroup e G we mean a collection of bounded linear operators e sG : X → X, s ≥ 0, where X is a Banach space, such that e (s+r)G = e sG e rG for every s, r > 0, e 0G is the identity operator and lim s↓0 e sG f = f in X and sup s e sG f X ≤ f X for every f ∈ X. The generator of e G is defined as the pair (G, Dom(G)), where Dom(G) := {f ∈ X : Gf := lim s↓0 s −1 (e sG f − f ) exists in X}. We say that a set C ⊂ Dom(G) is a core for (G, Dom(G)) if the generator equals the closure of the restriction of G to C. Recall that Dom(G) is dense in X. For a given λ ≥ 0 we define the resolvent of e G by (λ−G) −1 := ∞ 0 e −λs e sG ds, and recall from [18, Theorem 1.1] that for λ > 0, (λ − G) −1 : X → Dom(G) is a bijection and it solves the abstract resolvent equation and if (−G) −1 : X → X is bounded, then the above statement holds for λ = 0 [18, Theorem 1.1']. Also, for any f ∈ Dom(G) and C = Gf X ∨ 2f X we have e sG f − f X ≤ C(s ∧ 1) for all s ≥ 0. By a Feller semigroup we mean a strongly continuous contraction semigroup e G on any of the Banach spaces C ∞ (E) defined above such that for each s > 0, e sG f ≥ 0 if f ≥ 0. Feller semigroups are in one-to-one correspondence with Feller processes, where a Feller process is a time-homogeneous sub-Markov process {X s } s≥0 such that , and we always work with such modification. For further discussions on these terminologies and notation we refer to [10].
We will use the following assumption for the spatial semigroup e L .

(H1)
The operator e rL allows a density with respect to Lebesgue measure for each In the examples below we say that a stochastic process s →  (v) Any subordination of a Feller process by a Lévy subordinator which itself satisfies (H1), which is a straightforward consequence of [23,Theorem 4.3.5]. This case includes the spectral fractional Laplacian (−∆ Ω ) β [9,8].
(vi) We mention the articles [12,20] and references therein for related discussions about some jump-type generators with symmetric and non-symmetric kernels.
For our notion of weak solution in Section 4 we will use a stronger assumption for the spatial semigroup. In this assumption below we could allow Ω = R d , but we do not as it would affect the clarity of the exposition, as we would have to consider extra cases in several steps in Section 4.
(H1 ) the set Ω is a bounded open subset of R d , and e L is a Feller semigroup on X = C ∞ (Ω) or X = C ∞ (Ω) such that assumption (H1) holds and e L is self-adjoint, in the sense that for each r > 0 (i) Assumption (H1 ) holds for several processes obtained by killing a Feller process on R d upon exiting a regular bounded domain Ω. This is for example the case of the Dirichlet Laplacian −∆ Ω , the regional fractional Laplacian (−∆) β Ω and the spectral fractional Laplacian (−∆ Ω ) β , β ∈ (0, 1). These killed semigroups are Feller,

Subordinators and subordinated heat operators
In this section we define the simple space-time process r → (t−r, B x r ) with state-space (−∞, T ] × Ω (and suitable killed/absorbed versions) and then we subordinate it to obtain the space-time process (1.4) (and the respective subordinated killed/absorbed versions). Basic semigroup theory then leads us to Theorem 2.9, which gives a description of these processes in terms of generators of space-time Feller semigroups. We will always assume the following. (i) Recall that for each r > 0, the random variable S ν r allows a density p ν r (with respect to the Lebesgue measure) [34,Theorem 27.7]. Also, τ 0 (t) = inf{r > 0 : S ν r ≥ t} almost surely for every t > 0 as S ν is increasing [6, Chapter III.2], and for every (ii) To obtain the stable subordinator case select ν(r) := r −1−α /|Γ(−α)|, r > 0, α ∈ (0, 1), then S ν = S α is the α-stable subordinator, characterised by the Laplace transforms E[e −kS α r ] = e −rk α , for r, k > 0. Denote its densities by p α r , r > 0, and recall that E[τ 0 (t)] = t α /Γ(α + 1) [8,Example 5.8]. We refer to [8,Chapter 5.2.2] for other examples of subordination kernels ν.
We define three semigroups that correspond to three different space-time valued processes related to the heat operator −∂ t + L. Namely the "free" process s → (t − s, B x s ), the "absorbed at 0" process s → ((t − s) ∨ 0, B x s ), and the "killed at 0" process s → ((t − s), B x s ) for t > s and ∂ otherwise. It is straightforward to prove that such semigroups are Feller and we omit the proof.  We now define three semigroups that respectively correspond to subordinating the three semigroups in Definition 2.6 by an the independent Lévy subordinator S ν .
Definition 2.7. For appropriate functions u, we define for r > 0
(ii) If u is independent of time, then The next theorem shows that the operators in Definition 2.7 define Feller semigroups, it gives a pointwise representation for the generators on "nice" cores, and finally it connects the domains of the generators of e rH ν 0 and e rH ν 0 ,kill . These statements serve various purposes, but let us outline our main line of thinking. Our strategy is to reduce (1.1) to (3.1) with an appropriate forcing term, as suggested by the simple Lemma 4.6 (here we use the generators pointwise representation). Hence we solve problem (3.1) in the framework of abstract resolvent equations (Theorem 3.5). To do so, we use Theorem 2.9-(iv) to reduce problem (3.1) to the 0 initial condition version, easily solved by inverting H ν,kill 0 (Lemma 3.3). Moreover, Theorem 2.9 allows us to access Dynkin formula. Theorem 2.9. Assume (H0). Then, with the notation of Definitions 2.1, 2.6 and 2.7: (i) The operators e rH ν , r ≥ 0 form a Feller semigroup on C ∞ ((−∞, T ] × Ω). We denote the generator of the semigroup by (H ν , Dom(H ν )).
Moreover, Dom(H) is a core for (H ν , Dom(H ν )), and for g ∈ Dom(H)   Consider the resolvent representation for g for a given λ > 0 and g λ ∈ C ∞ ([0, T ] × Ω) given by (i) Let us stress that Theorem 2.9-(iv), although unsurprising, is a vital technical ingredient for this work. This is because it allows to obtain uniqueness of our notion of "solution in the domain of the generator" to (3.1) for non-zero initial data (see the proof of Lemma 3.3-(i)). This notion of solution is our building block for weak solutions to (1.1) in Section 4.
(ii) To see that H ν u is well defined pointwise for u ∈ Dom(H) simply use the general bound in (2.1) along with (H0).
for each x ∈ Ω, where we used B x s = B x s∧τΩ(x) and its independence with respect to S ν s . We will later use the following simple lemma.
Proof. This is straightforward, because as r ↓ 0, uniformly in both t and x.

Generalised solutions
We prove existence, uniqueness and stochastic representation for generalised solutions to the 'Caputo-type' problem under assumptions (H0) and (H1). In particular, we will obtain the probabilistic representation 2) for the solution to (3.1).
(ii) Problem (3.1) corresponds to problem (1.1) for time independent initial condition φ(t) = φ 0 , in a similar way as Caputo and Marchaud evolution equations are related in [38]. By the appropriate choice of forcing term g, problem (3.1) formally rewrites as problem (1.1), which we make rigorous in Section 4.
We first assume some compatibility condition on the forcing term and the initial data in order to construct the following kind of strong solution by inverting H ν 0 .   (ii) Moreover, the solution in the domain of the generator allows the stochastic representation (3.2).

Proof. i) We first claim that
is the unique solution to the abstract evolution equation To conclude, it is now enough to show thatũ = u − φ 0 is a solution to (3.4) if and only if u is a solution to (3.3). For the 'only if' direction, define u :=ũ + φ 0 .
, and this proves that u can be written as (3.2).
We now give another definition of solution as the pointwise limit of solutions in the domain of the generator. This allows us to drop the compatibility condition on the data in Lemma 3.3.
Then, by DCT, F (g n ) → F (g) as n → ∞, using the dominating function  (ii) We assumed (H1) in Theorem 3.5 to treat L ∞ data. But if we were to assume continuous data (or the closure of C ∞ ((0, T ] × Ω) with respect to bpw convergence [10, page 1]), then Theorem 3.5 would still hold by changing "bpw a.e." with "bpw" in Definition 3.4. In this case we would also not need the subordinator S ν to allow a density. These conditions would still allow to define and establish existence of weak solutions on appropriate spaces with the same strategy of Section 4. But we considered such technical treatment beyond the scope of this work, which aims to present a clear and concrete treatment of the new formulation of the CEE (1.1) with initial conditions on the past.
We now show that the fundamental solution that defines (1.3) allows a density with respect to Lebesgue measure. Then we conclude this section with several examples of concrete densities for solutions to (1.1). Lemma 3.7. Assume (H0). Then for each t > 0, the random variable S ν τ0(t) − t allows a density supported on (0, ∞), and we can write the density for almost every r ∈ (0, ∞) as Proof. This follows, for example, by performing the proof of [17,Proposition 3.13] in the simpler setting without the spatial process.
The inhomogeneous term is treated similarly and we omit the proof.
Proof. This is a straightforward application of DCT given Lemma 3.8 and E[τ 0 (T )] < ∞.

Weak solutions
In this section we prove that the stochastic representation (1.3) is a weak solution for problem (1.1), under the stronger assumption (H1 ) on the spatial semigroup e L . We recall that Ω is open and bounded under (H1 ), we introduce the notation and we define the adjoint operator of H ν as For our notion of weak solution we need the pairing u, H ν, * ϕ to be well defined for some test function ϕ (see Definition 4.7), and we want to allow constant-in-time data φ, so that the solution u will be in L ∞ ((−∞, T ) × Ω). Moreover, recalling that a generalised solution u is characterised by the existence of a sequence u n → u bpw a.e., we want to be able to show u n , H ν, * ϕ → u, H ν, * ϕ . And so, to guarantee a well defined pairing and access dominated convergence arguments, we now prove that H ν, * ϕ ∈ L 1 ((−∞, T ) × Ω).
where we used [17,Lemma 4.3] in the second inequality. Considering II, which is finite using (H0) and that Ω is bounded.   Proof. Let k > 0 such that ϕ(t) = 0 for every t ≤ k. Note that for each (t, x) ∈ (k, T ] × Ω we have the bound for a.e. r > 0 and so H ν u (defined in (2.6)) is well defined for each (t, x) ∈ (k, T ] × Ω and bounded on the same set. And so the left hand side in (4.1) is well defined recalling that ϕ ∈ L 1 (R×Ω).
To conclude we compute where we use DCT in the second identity, for the third identity we use (2.2), Fubini's Theorem and ϕ(t + r) = 0 for t ≥ T − r, and for the fourth identity we use DCT, thanks to Lemma 4.2 and u ∈ L ∞ ((−∞, T ) × Ω).
Our approximation procedure, in the proof of Theorem 4.8, will be carried out using the following assumption on the approximating data.
Proof. Recalling (2.6), simply compute for t > 0, x ∈ Ω We are now ready to define our weak solution for problem (1.1) and to show that the stochastic representation (1.3) is indeed a weak solution.  Proof. We assume that e L acts on C ∞ (Ω) (the proof for e L acting on C ∞ (Ω) is essentially identical 1 , and we omit it). Note that in each step we may redefine the notation u,ũ, u n ,ũ n , f, f n , φ and φ n . Also, we assume in the first two steps that φ satisfies (H2).