If time were a graph, what would evolution equations look like?

Linear evolution equations are considered usually for the time variable being defined on an interval where typically initial conditions or time-periodicity of solutions are required to single out certain solutions. Here we would like to make a point of allowing time to be defined on a metric graph or network where on the branching points coupling conditions are imposed such that time can have ramifications and even loops. This not only generalizes the classical setting and allows for more freedom in the modeling of coupled and interacting systems of evolution equations, but it also provides a unified framework for initial value and time-periodic problems. For these time-graph Cauchy problems questions of well-posedness and regularity of solutions for parabolic problems are studied along with the question of which time-graph Cauchy problems cannot be reduced to an iteratively solvable sequence of Cauchy problems on intervals. Based on two different approaches -- an application of the Kalton--Weis theorem on the sum of closed operators and an explicit computation of a Green's function -- we present the main well-posedness and regularity results. We further study some qualitative properties of solutions. While we mainly focus on parabolic problems we also explain how other Cauchy problems can be studied along the same lines. This is exemplified by discussing coupled systems with constraints that are non-local in time akin to periodicity.


Introduction
Time has classically been considered as a linear phenomenon, especially in western cultures. This has been clearly mirrored in the physical description of the world, all the way from ancient Greek philosophy to modern partial differential equations of mathematical physics. Many real world phenomena can bemore or less naively -modeled as abstract Cauchy problems such as the heat, transport or Schrödinger equation, which are classically considered with domain for the time variable t in a finite interval [0, a] or a half-line [0, ∞), and there cannot be a unique solution until an initial condition ψ(0) = g is imposed.
Here, for simplicity one may have in mind a sectorial operator −A in a Hilbert space X. It is folklore that, unlike western cultures, many eastern cultures regard time as a cyclic or spiral-like phenomenon. This does not necessarily lead to mathematical clashes: indeed, if the time variable t is cyclic and hence lives in a torus S 1 or the full real line R, then looking for solutions of (1.1) amounts to inquire existence of periodic solutions.
In each case the time domain is an oriented one-dimensional manifold, thus there is a clear direction at each point in time and a well-defined time before and after it. Going beyond this, there are different perceptions of time expressed for instance in the multiverse interpretation of quantum mechanics or in the discussions on closed time-like curves in general relativity. Here, we would like to invite the reader to participate in a thought experiment and to assume time not to consist of a one-dimensional manifold, but rather of a metric graph or network. Such ramified structures consist -roughly speaking -of intervals glued together at their endpoints and allow for more freedom in the modeling of evolutionary systems in real and some possibly hypothetical applications. The purpose of this note is to widen the scope of classical evolution equations and to show how graphs can be used to model time evolution. The main idea and recurrent motive is to consider initial conditions as boundary conditions in time: we will make this more precise in the following.
We notice in passing that there do exist classical settings where the notion of one-dimensional time is generalized: In the context of analytic semigroups time is allowed to be in a sector of the complex plane as sketched in Figure 1.(d). This has a plethora of pleasant mathematical consequences, but it is not evident how to make sense of it physically. Instead, we reckon that allowing time to live on network-like structures may have a practical interpretation as will be discussed in terms of examples. This means there is no freedom left for initial conditions, but one is free to choose any fixed jump condition ψ(0) − ψ(a) = g, and the solution can be expressed (provided 1 − e a A is invertible) as (1.5) ψ 0 (t) = e tA (1 − e a A ) −1 g which solves (1.4) on (0, a). Note that the notions of mild, strong and classical solutions defined edge-wise carry over to the setting of metric graphs in time. The regularity of ψ 0 given in (1.5) clearly depends on the regularity of (1 − e a A ) −1 g and therefore on g, as well as on the mapping properties of (1 − e a A ) −1 .
Considering only the first equation in (1.2), this can be solved -instead of using the variation of constants formula -by means of operator theory by finding realizations of ∂ t with initial condition ψ(0) = 0 such that the sum of closed operators ∂ t − A is invertible. For L p -spaces in time this approach succeeded where the essential ingredient is the theorem of Kalton and Weis on the sum of closed operators. Similarly, equation (1.3) can be solved by considering a periodic realization of the time derivative.
1.2. Time-graph Cauchy problem. Consider again evolution equations whose time domain are intervals, like in Figure 1: both under initial and periodicity conditions they can be split into a part with force, but homogeneous boundary condition in time, and a part without force and inhomogeneous boundary condition in time. We therefore consider finitely many inhomogeneous evolution equations (∂ t − A i )ψ i = f i on (0, a i ) for each i = 1, . . . , n, on time intervals of length a i > 0, i = 1, . . . , n, where we assume that A i are generators of analytic semigroups in Hilbert spaces X i , f i ∈ L 2 (0, a i ; X i ) are given and the coupling is defined by where B is a bounded operator in X 1 ⊕ . . . ⊕ X n which encodes the geometry of the graph by means of transmission conditions, and g i ∈ X i are given 'inhomogeneous boundary conditions in time' in analogy to the fixed jump conditions for the periodic case. This class of time-graph Cauchy problems comprises the classical settings, where the classical initial value problem corresponds to B = 0, and the time-periodic problem is given by B = 1 with g i = 0 for i = 1, . . . , n.
We present two strategies to solve this problem: First, when all g i = 0, one can apply the Kalton-Weis theorem on the sums of closed operators for suitable time and space operators. Second, going beyond this, explicit formulae in terms of semigroups and transmission conditions as in (1.5) can be derived by a Green's function Ansatz interpreting the system ∂ t − A i as a system of vector valued ordinary differential equation in time where inhomogeneous boundary conditions in time are included.

1.3.
First examples, results and outlook. As a next step towards more non-standard examples, one can modify the time-periodic situation. One may want to extend the scope in order to look for "spiral-like" solutions: instead of pure periodicity, we may for instance impose a phase shift after each time-period a > 0, i.e., ψ(x, t + a) = αψ(x, t) for some α ∈ C, which corresponds to B = α · 1 and g = 0. Or having two or possibly even more phase shifts this can be modeled using graphs (see Figure 2), and this corresponds to B = diag(α 1 , α 2 ) and g = 0. To illustrate various features of time-graphs one can consider the graphs depicted in Figure 3. Building on the initial example of time-periodicity, one can take its state at a certain time as input to a new system. This would correspond to the tadpole-like graph in Figure 3.(a) with matching of the type where ψ 1 lives on the loop and ψ 2 lives on the adjacent interval.   More generally, basic building blocks are the joining and the splitting of two systems -as depicted in Figure 3.(b)-3.(c) -which can be used to describe a system which splits into two non-interacting dynamics or two systems which interact after some time by means of some superposition. These blocks can be assembled to form graphs with loops, see Time-graph Cauchy problems can be understood as a system of Cauchy problems on intervals with possibly non-local constraints such as periodicity, fixed jump conditions, or certain symmetries. Since the map B = (B ij ) 1≤i,j≤n is a block operator with B ij : X j → X i , one can rewrite (1.6) as that is, a Cauchy problem is assigned on on each interval and their 'jump conditions' are interdependent. If B jj = 0 the Cauchy problem on (0, a j ) is non-local and resembles periodicity, and for B jj = 0 the Cauchy problem on (0, a j ) is an initial value problem.
Note that time-graphs with oriented loops can also be used to model control loops and other control problems. One can think also of signals that after a certain time are processed differently as illustrated in Figure 3.(i). This means that a system changes its character after a certain time. For instance a heat equation is followed after a certain time by a transport process that after a certain time turns again into a heat equation: thus modeling time delays in a diffusive process. Note that couplings at the vertices of a time-graph can also be frequency dependent, and thus frequency dependent dynamics can be modeled, too. Moreover, there are some more non-standard situations where time-graphs come into play. A tree graph as depicted in Figure 3.(f) can serve as an illustration for the multiverse interpretation of quantum mechanics, where it is assumed that, in contrast to a probabilistic interpretation, each possible state is actually attained, but each in one separated universe. Figure 3.(g) and 3.(h) give some possibilities how one may represent time travel -independent of its actual physical possibility -using time-graphs, see also Section 9 below.
Our main result states the well-posedness of such time-graph models, under some compatibility assumption on the matrix B, which encodes the transmission conditions in time, and the 'spatial' operators operators A i . In particular, a generalized variation of constants formula is obtained, allowing us to derive additional mapping properties.
The question of whether the time-graph Cauchy problem reduces to a sequence of Cauchy problems on intervals which can be solved iteratively is traced back to the block structure of B, and it is pointed out that loops which are reflected by the transmission conditions B prevent such iterative solvability and therefore in such situations one indeed needs tools for global solvability such as for the case of periodicity. The methods developed for the case of parabolic problems can be adapted also for some non-parabolic problems such as Schrödinger equations, wave equations, or even coupled dynamics of different types as first and second order Cauchy problems as illustrated in Figure 3 (j).
1.4. Organization of the paper. In the subsequent Section 2 we recapitulate key elements of the classical theory of evolution equations, some of which are necessary in order to develop our approach to time-graphs. Thereafter, in Section 3, the notion of networks and function spaces thereon are made precise. In Section 4 the Banach space-valued time derivative operator on graphs with couplings and the spatial operator are studied. In Section 5 the time-graph problem for the case g = 0 is tackled, using the Kalton-Weis sum theorem on commuting operators applied to the time derivative and the spatial operator, where some compatibility assumptions on the boundary conditions are required. Section 6 follows a more direct approach computing the Green's function for the time-graph problem explicitly. This gives our main result on the solvability of the time-graph Cauchy problem for g in a trace space under less restrictive compatibility conditions. Section 7 addresses the question under which condition solutions to time-graph problems can be reduced to Cauchy problems on intervals. In Sections 8 we discuss a few examples, focusing on specific instances of time graphs and broaching extensions to classes of non-parabolic evolution equations, including Schrödinger, wave and mixed-order equations.
Some of the suggested settings may look mostly motivated by science-fictional or hypothetical physical scenarios, as they may allow for loss of causality: In Section 9 we discuss these and further related aspects by commenting on tentative interpretations of evolution supported on network-type time structures.

Classical Cauchy problems
Many of the methods applied here make use of classical results on evolution equation theory and initial value problems. It is well-established that the initial value problem with A being a closed linear operator on a Banach space X has for all g ∈ X a unique mild solution if and only if A generates a C 0 -semigroup on X, cf. [ABHN10, Thm. 3.1.12], where at least f ∈ L 1 (0, a; X) is admissible, a > 0. If X is a Hilbert space and g = 0, the stronger condition of maximal L 2 -regularity amounts to requiring that there is for all f ∈ L 2 (0, a; X) a unique solution ψ of (2.1) in the maximal L 2 -regularity space, i.e., ψ ∈ L 2 (0, a; D(A)) ∩ {ψ ∈ W 1,2 (0, a; X) : ψ(0) = 0}, such that ψ L 2 (0,a;D(A)) + ψ W 1,2 (0,a;X) ≤ C f L 2 (0,a;X) for a constant C > 0 independent of f . Maximal L 2 -regularity holds if and only if the semigroup generated by A on X is analytic semigroup. This is related to the notion of sectorial operators: considering sectors in the complex plane recall that a closed densely defined linear operator B is sectorial of For Banach spaces X of class UMD maximal L p -regularity can be characterized using the notions of R-sectoriality and H ∞ -calculus, where one implication follows from the Dore-Venni-type sum theorem of Kalton and Weis on commuting operators [KW01, Thm. 6.3], cited here in Theorem 2.1 below. The key idea in the original Dore-Venni Theorem and its generalizations is to look at equations evolution equation on a Banach space X as stationary equations on a Bochner space of X-valued functions.

The operator
In the following we will seldom use this result in its full generality, as we mostly restrict to the case of Hilbert spaces; we refer the interested reader to the classic monograph [DHP03] where all these notions are introduced. This theorem is formulated for a Banach space X, but if X is a Hilbert space, then the notions of R-sectoriality and sectoriality agree. We recall that whenever −A is sectorial, the solution ψ(t) := e tA g lies in D(A) for all t > 0 and all initial data g ∈ X; and that moreover ψ lies for all p ∈ (1, ∞) in the maximal L p -regularity space whenever the initial data belong to the trace space, i.e., g ∈ (X, D(A)) 1−1/p,p , given by the real interpolation functor (·, ·) θ,p , cf. [PS16, § 3.4].
The Ansatz using the Kalton and Weis sum theorem has been applied successfully by Arendt and Bu in [AB02,AB04a,AB04b]. In particular, the fact that both time domains R and S 1 are groups has allowed them to apply methods of harmonic analysis and to deliver a comprehensive theory of Cauchy problems with periodic in-time boundary conditions. For a similar approach where the stationary part, i.e., k = 0, is treated separately, and in particular applications to quasi-and semi-linear problems see also the works of Kyed and co-authors, cf. [KS17,EK17,CK18]. Existence of periodic solutions for (linear or even nonlinear) hyperbolic equations is well-known for a large class of problems, cf. the comprehensive monograph [Vej82].

Finite metric graphs
3.1. Finite graphs. A graph is a 4-tuple where V denotes the set of vertices, I the set of internal edges and E the set of external edges, with E ∩ I = ∅. We refer to elements of the set E ∪ I collectively as edges. In order to fix an orientation, one distinguishes incoming E − and outgoing E + external edges, where E = E − ∪ E + and E − ∩ E + = ∅.
The structure of the graph is given by the boundary map ∂. On one hand, it assigns to each internal edge i ∈ I an ordered pair of vertices is called its initial vertex and ∂ + (i) its terminal vertex. On the other hand, each incoming external edge e − ∈ E − and each outgoing external edge e + ∈ E + is associated by means of ∂(e − ) = ∂ − (e − ) and ∂(e + ) = ∂ + (e + ) with a single vertex (its initial and terminal vertex, respectively). A graph is called balanced if |E − | = |E + |. We will see that orientations do play a role only when we study evolution equations that are of first (or, more generally, odd) order in time; in the case of even time order equations, orientations are only imposed for the sake of a consistent parametrization.
The structure of the network is given by the |V| × |E ∪ I|-outgoing and ingoing incidence matrices I + := (ι + ve ) and I − := (ι − ve ) defined by The matrix I := (ι ve ) defined by I := I + − I − is the signed incidence matrix of G. This encodes the structure of the graph and allows one to define directions on G, and a directed graph is the graph, where one can move only along the prescribed direction while for the non-directed graph G one can move into both directions. A graph is called finite if |V| + |I| + |E| < ∞ and a finite graph is called compact if E = ∅.

Function spaces on metric graphs.
A graph G is endowed with the following metric structure. Each internal edge i ∈ I is associated with an interval [0, a i ], with a i > 0, such that its initial vertex corresponds to 0 and its terminal vertex to a i . Each external edge e ∈ E − and e ∈ E + is associated to a half line [0, ∞) and (−∞, 0], respectively, such that ∂(e) corresponds to 0. The numbers a i are called lengths of the internal edges i ∈ I and they are collected into the vector The couple consisting of a finite graph endowed with a metric structure is called a metric graph (G, a).
The metric on the non-directed metric graph (G, a) is defined via minimal path lengths along connected vertices, while for the directed metric graph minimal path lengths along connected vertices is computed taking into account the directions. Let for each j ∈ I ∪ E be X j a complex Banach space with norm · X j , then any collection of functions can be identified with a map ψ : where the notation for elements in is shortened to t and ψ, and occasionally we write slightly redundantly ψ j (t) = ψ j (t j ). The metric graph (G, a) is identified with a quotient of j∈I∪E I j , and therefore t ∈ (G, a) is identified with t = t j ∈ I j for some j ∈ E ∪ I. Similarly, maps ψ defined as in (3.2) can be identified with maps on (G, a), where on the vertices in general a set of values can be attained.
Equipping each edge of the oriented or non-oriented metric graph with the one-dimensional vectorvalued Bochner-Lebesgue measure, one obtains a measure spaces. One defines where dt j refers to integration with respect to the Bochner-Lebesgue measure on I j . We set and introduce, with a slight abuse of notation, several related spaces: For p ∈ (1, ∞) the space defines a Banach space, and indeed a Hilbert space provided p = 2 and X j are Hilbert spaces; the canonical norm and inner product are given by respectively. The corresponding Sobolev spaces are defined for p ∈ [1, ∞) and m ∈ N by

Operators on metric graphs
As a first step to study the motivating problem, i.e., the derivative operator with transmission conditions on graphs is analysed.
4.1. Derivative operators on graphs. One considers the n-th derivative operators D n on graphs formally given by where one can define minimal and maximal operators in L p (G, a; X) by These are closed linear operators. Ifp = 2 and each X j is a Hilbert space, one has (D min n ) * = (−1) n D max n , and hence D min n is symmetric if n is even and skew-symmetric if n is odd. In this article, the focus lies on the first and second derivative operator, for which we use the notation D t := D 1 and D tt := D 2 .

4.2.
Accretive coupling conditions for the first derivative. When considering the first derivative operator, it is assumed that G is balanced, i.e., there are as many outgoing as incoming external edges. From now on, let X j be Hilbert spaces. On L 2 (G, a; X) a class of m-accretive realizations D b.c t of D t defined by boundary conditions is presented, i.e., D min . Integrating by parts yields the following Lagrange identity for the first derivative operator One introduces the space of boundary values, where using that the graph is balanced, i.e., |E + | = |E − |, We define the vectors of boundary values ψ − ∈ K and ψ + ∈ K by where for a fixed bijection i.e., one orders the outgoing and incoming edges into pairs, and defines ∂ + (e) := ∂ + (e + ), ∂ − (e) := ∂ − (p(e + )), where e = (e + , p(e + )) ∈ E p .
Hence, one obtains For any subspace M ⊂ K 2 one can define a realization by  Proof. If M ⊂ K 2 is closed, then ψ n → ψ and ψ ′ n → ϕ in L 2 (G, a; X) for ψ n ∈ D(D t (M)) imply first due to the closedness of D max t that ψ ∈ D(D max t ) and ϕ = ψ ′ . Second, due to the boundedness of the trace operator one has in K 2 that ψ n → ψ ∈ M.
Here, the following type of boundary conditions is considered. Let B ∈ L(K) a bounded operator in K. Note that B is a block operator matrix given with respect to the decomposition of K := i∈I∪E − X i , i.e., For such B ∈ L(K) we consider the boundary conditions defined by One defines the operator Under additional assumptions these boundary conditions force the numerical range to lie in a left halfplain of the complex plain.
Lemma 4.2 (Adjoint operator and numerical range). Let B ∈ L(K), then D t (B) is closed and its Hilbert space adjoint in L 2 (G, a; X) is given by and there holds Note that Moreover, for ψ ∈ D(D t (B)) one obtains by integration by parts A similar proof yields the claimed identity for Re D t (B) * ψ, ψ .
Proof. It is a direct consequence of Lemma 4.2 that if B L(K) ≤ 1, then Recall that one has for the operator norm in K that B 2 for an explanation of this notation.
4.3. Spatial operators. As before we assume that X j are Hilbert spaces. For each edge j ∈ I ∪ E let A j be a given operator in X j with D(A j ) ⊂ X j . We consider the abstract time-graph-Cauchy problem Note that the operators A j in X j induce operators in L 2 (I j ; X j ) which with a slight abuse of notation are also denoted by A j and D(A j ) = L 2 (I j ; D(A j )). Using this we define the operator A E in L 2 (G, a; X): it acts on functions supported on the time branches by and with this the Cauchy problem (4.5) can be formulated as maximal regularity problem Moreover, the operators A j induce an operator A V in the space of boundary values K that acts on functions supported on the vertices by The following lemma is straightforward.
Then for the induced operators A E in L 2 (G, a; X), A V in K, and A V 2 in K 2 the following holds: holds as an equality of sets, i.e., without counting multiplicities; , then minus the induced operators are sectorial of angle 5. The Kalton and Weis sum theorem and the parabolic operator 5.1. Solvability of the inhomogeneous problem with homogeneous boundary conditions. Having specified time-derivative and spatial operators, one can now define the parabolic operator The Kalton-Weis sum theorem, formulated here in Theorem 2.1, can now be applied to D t (B) and A E using Corollary 4.5 and assuming that the A E is sectorial and commuting with D t (B). This gives the wellposendess for the time-graph Cauchy problem with homogeneous initial conditions and inhomogeneous right hand-side.
Proposition 5.1. Let G be balanced and B be a contraction in K. Let X j be for all j ∈ I ∪ E Hilbert spaces and −A j sectorial operators of angle φ −A i < π/2 on X j . Assume that D t (B) and A E are resolvent commuting. Then the operator P (B) is closed.
. Remark 5.2. A criterion to assure the that the operators D t (B) and A E commute is that (A V − λ) −1 and B commute for λ ∈ ρ(A V ).

5.2.
Trace spaces and the parabolic operator. The approach using the Kalton-Weis result on commuting operators allowed us to find a simple way how to check solvability for the time-graph Cauchy problem with homogeneous boundary data. However, the condition that D t (B) and A E commute seems too strict since (4.5) makes sense without, and in fact closedness of the parabolic operator can be assured under weaker assumptions.
For notational simplicity we assume from now on that there are no external edges, i.e., the time-graph is assumed to be compact. Considering the maximal parabolic operator where E = ∅, one defines the corresponding trace space where [·, ·] θ for θ ∈ (0, 1) denotes the complex interpolation functor. Recall that for sectorial −A one has the continuous embedding  Lemma 5.5 (Closedness of the parabolic operator). Let each −A j be sectorial of angle smaller than π/2, and let B ∈ L(K) such that B ∈ L(K) is compatible with K A . Then P (B) is a closed operator on L 2 (G, a; X).
Proof. One shows first that P max is closed. Note that P max decouples the edges and hence it is sufficient to prove closedness for a graph consisting of a single interval [0, a]. Consider the operator P (B) = P 0,δ for B = 0 on [−δ, a i ] for δ > 0. This is closed and to trace back this property to P max one considers continuous extension and restriction operators where the extension can be realized for instance by even reflection and then multiplying by a cut-off function with value one on [−δ/2, a] and zero in a neighborhood of −δ. Then P max = R • P 0,δ • E and closedness can be proved straightforward. Now, let ϕ n ∈ D(P (B)) with ϕ n → ϕ and P (B)ϕ n → ψ in L 2 (G, a; X).
Then by closedness of P max and since P (B) is a restriction of P max , ϕ ∈ D(P max ) and ψ = P max ϕ. Using (5.1), it follows that ϕ n ± → ϕ ± , and hence ϕ ∈ D(P (B)).

The parabolic operator and the Green's functions approach
The operator theoretical consideration of the parabolic operator gives information on the solvability for homogeneous boundary data. However, it does not provide a solution formula, and it does not include the case of in-homogeneous boundary data. To address these issues we supplement our findings by computing explicitly the Green's function for (4.5).
6.1. Green's function for the parabolic problem. Now, we are in the position to collect suitable assumptions for the time-graph Cauchy problem; we stress that the following are more general than the ones in Proposition 5.1, where here E = ∅ has been assumed for notational simplicity only.
Assumption 6.1. Let E = ∅ and B ∈ L(K). Let X j be a Hilbert space and −A j a sectorial operator of angle φ −A j < π/2 on X j for each j ∈ I.
In the following a solution formula is derived generalizing the variation of constants formula from semigroup theory. Note that square integral maps In this sense, the Green's function for zero initial conditions, i.e., for B = 0, is Since each operator −A j is sectorial, it generates an analytic C 0 -semigroup; in particular, e t j A j is a well-defined bounded linear operator on X j for each t j in the time branch I j . In the following we will adopt for t = {t j } j∈I j , the notation and hence e tA ∈ L(K) is a diagonal block operator matrix in K.
Proposition 6.2 (Inhomogeneous problem with homogeneous boundary conditions). Under the Assumption 6.1, let B be compatible with K A , and let (1 − B e aA )| K A be boundedly invertible in K A . Then P (B) is boundedly invertible, i.e., for each f ∈ L 2 (G, a; X) there exists a unique solution to (4.5) in D(D t (B)) ∩ D(A E ) which is given by with r 0 (t, s; A E ) given by (6.1) and for some vector c i ∈ X i that is "inherited" from the final state in the preceding edges. Indeed, the boundary condition can be used to determine c i . Since recalling that a i denotes the length of the edge i, the condition ψ − = Bψ + gives Hence we obtain the vector-valued identity for c = {c i } i∈I Recall that the Green's function is the resolvent operator's integral kernel, i.e., a function r(t, s) := r(t, s; B, A E ) such that it defines a left and right inverse of P (B), i.e., (a) ϕ(t) = (D t (B) − A E ) G r(t, s)ϕ(s)ds for ϕ ∈ L 2 (G, a; X), First, note that on each edge j ∈ I, respectively, where one applies the classical variation of constants formula and the properties of the semigroups e t j A j . Hence ψ = ψ 0 Here, ψ 1 f is the correction term for the variation of constants term ψ 0 f assuring that that the boundary conditions are satisfied. Secondly, one has to prove that ψ satisfies the boundary conditions, and indeed hence ψ − − Bψ + = 0. We conclude that G r(·, s; B, A) · ds is the right inverse of P (B).
Because the adjoint kernels (6.5) {r 0 (t, s; A E ) * } j,l := e (s j −t j )A * j if j = l and t j < s j , 0 otherwise, t j , s l ∈ j∈I∪E I j .
consist of the Green's function for the time-reversed problems To conclude, note that 1 − e A * a B * is invertible if and only if so is its adjoint 1 − B e aA . We have thus proven that the adjoint of G r(·, s; B, A) · ds is the right inverse of P (B) * , hence taking adjoints G r(·, s; B, A) · ds P (B) = 1 D(P (B)) , hence G r(·, s; B, A) ds is also the left inverse of P (B).
Remark 6.4. Two sufficient conditions for invertibility of 1 − Be aA are that each A i are m-dissipative and B is a strict contraction; or else that each A i +ǫ is m-dissipative for some ǫ > 0 and B is a contraction.
For general B the resolvent of D t (B) can be obtained applying Proposition 6.2 for A j := λ1 X j , λ ∈ C which induces an operator A E = λ E .
The inverse of the parabolic operator P (B) can be seen as being given by a functional calculus where the spectral parameter in Corollary 6.5 is replaced by the operator A. This is akin to the case of classical semigroups, where the solution operator of the ordinary differential equation (∂ t − λ)ψ = f, ψ(0) = ψ 0 , is considered, and semigroup theory -interpreted as functional calculus for the exponential functionsallows one to "replace λ by some generator A".
6.2. Inhomogeneous boundary conditions. We have so fare implicitly focused on the case of 0boundary conditions imposed on sources of the time-graph, i.e., at the initial endpoints of those time branches that have no predecessors. This is clearly a relevant limitation and would e.g. lead to identically vanishing solutions as soon as f ≡ 0. Initial conditions can be introduced by interpreting them as inhomogeneous boundary conditions with respect to time. Thus one considers the problem (6.6) for given g ∈ K and f ∈ L 2 (G, a; X). For B = 0 this corresponds to the usual initial condition ψ(0) = ψ 0 . The solution to this problem can be computed using the Green's function, where -as for ordinary differential equations with inhomogeneous boundary conditions -the Lagrange-identity (4.1) plays an important role. We start with a heuristic argument. Integration by parts yields G [(∂ s r(t, s; A E ))ψ(s) + r(t, s; A E )∂ s ψ(s)] ds = [r(t, ·;

Due to the properties of the Green's function
Hence G (∂ s r(t, s; A E ))ψ(s) + r(t, s; A E )∂ s ψ(s)ds = where one uses that assuming that ψ solves the Cauchy problem. Hence where ψ 0 is given more explicitly by  Theorem 6.6. Let Assumption 6.1 be fulfilled and let B be compatible with K A . If (1 − B e aA )| K A is boundedly invertible in K A , then for any g ∈ K A and f ∈ L 2 (G, a; X) there is a unique solution ψ ∈ W 1,2 (G; a; X) ∩ L 2 (G, a; D(A E )).
to (6.6). The solution is given by where the kernel r(·, ·; B, A) is given in (6.2), and In particular there exists a constant C independent of f and g such that D t ψ L 2 (G,a;X) + A E ψ L 2 (G,a;X) ≤ C( f L 2 (G,a;X) + g K A ). To prove uniqueness assume that there is another solution ψ ′ 0 to (6.6) in the solution space, and consider the difference ψ δ := ψ ′ 0 − ψ 0 which solves due to the linearity of the equation

Proof. Note that
Because of the former equation, there exists g ′ ∈ K A such that ψ δ = e tA g ′ , while the latter implies (1 − Be aA )g ′ = 0, and by the invertibility of 1 − Be aA it follows that ψ δ ≡ 0. Hence the inhomogeneous boundary value problem is uniquely solvable with ψ 0 in the maximal L 2 -regularity class, and the rest of the statement follows from Proposition 6.2.
(a) The solution formula given in Theorem 6.6 is a generalization of the well-known variation of constants formula. Considering only one interval [0, a] with boundary conditions u(0) = 0, i.e. B = 0, we find that ψ 0 (t) = e tA ψ 0 and r(t, s; B, A E ) = r 0 (t, s; A E ). 6.3. Mapping properties. Assume that X j = L 2 (S j , µ j ) for some measure space (S j , Σ j , µ j ) are spaces of complex valued functions, and denote by X j,R the cone of real valued functions. This induces spaces K R and K 2 R . Proposition 6.8. Let the assumptions of Theorem 6.6 be satisfied and let X j = L 2 (S j , µ j ) be Hilbert spaces of complex valued functions.
(a) If B and the operator families (e t j A j ) t j ∈I j leave K R invariant, then the solution ψ in Theorem 6.6 is real for real data g and f . Proof. We have shown in Proposition 6.2 that the Green's function is given by r 0 (·, ·; A) + r 1 (·, ·; B, A). It is apparent that the claimed properties for the solution to (6.6) are proved as soon as corresponding properties hold for both r 0 (·, ·; A), r 1 (·, ·; B, A), and ψ 0 where the corresponding properties of r 0 are covered by the classical theory. Now, r 1 can be studied using its factorization into operators that also enjoy the corresponding properties. For the mapping properties of ψ 0 analogous arguments apply.
6.4. Maximal L p -regularity. For notational and mathematical simplicity, we have focused on the Hilbert space case and on maximal L 2 -regularity. In the case of evolution equations on R + , under the assumptions of Proposition 6.8.(c) the semigroup e tA extrapolates to a C 0 -semigroup on all L p -spaces, p ∈ [1, ∞); this semigroup is additionally analytic on L p , p ∈ [1, ∞), if e tA satisfies Gaussian estimates. By a celebrated result in [HP97] this implies in turn L p -maximal regularity for p ∈ (1, ∞), but our theory does not seem to allow us to discuss kernel estimates. However, the solution formulae (6.2) and (6.7) suggest a straightforward generalization to the general case of maximal L p -regularity in Banach spaces. To this end, let X j be Banach spaces and p ∈ (1, ∞). Consider the trace space and collect the following assumptions.
Assumption 6.9. Assume that E = ∅, X j be Banach spaces of class UMD, and p ∈ (1, ∞). Suppose that −A j are R-sectorial operators in X j of angle smaller than π/2, and that B ∈ L(K).
Proposition 6.10 (Maximal L p -regularity). Let the Assumption 6.9 be fulfilled, B be compatible with K A,p . If 1 − B e aA is boundedly invertible in K A,p , then for any g ∈ K A,p and f ∈ L p (G, a; X) there is a unique solution ψ ∈ W 1,p (G, a; X) ∩ L p (G, a; D(A)).
to (6.6), where the solution is given by the same formulae as in Theorem 6.6 and Proof. The unperturbed part of the Green's function r 0 (·, ·; A) defines a bounded operator It remains to verify that the correction terms have the same mapping properties. Using that G e (a−s)A f (s)ds ∈ K A,p which follows from the classical variation of constants formula and maximal L p -regularity of the initial value problem, it follows that G r 1 (t, s; B, A)f (s)d is in the maximal L p -regularity space. The proofs of Proposition 6.2 and Theorem 6.6 now carry over to the present situation.
Remark 6.11 (Transference principle). Transference principles which relate maximal L p -regularity for the initial value problem to the maximal L p -regularity problem with time-periodicity on the real line are well-established. Here, let E = ∅ and P (B) for B = 0 have maximal L p -regularity, then P (B) has maximal L p -regularity for any B ∈ L(K) satisfying the assumption of Proposition 6.10.
6.5. Regularity and other notions of solutions. So far we have focused on solvability in maximal regularity regularity spaces since these fit into a suitable functional analytic framework. Note that the the solution formula from Theorem 6.6 can be made sense of even under milder assumptions. Consider the case where E = ∅, X j are Banach spaces, the operators A i generate C 0 -semigroups in X i , and B ∈ L(K). Classical result from semigroup theory carry over as long as sufficient compatibility of B is assumed. For instance having more regular data, improves the regularity of solutions.

Mild solutions. Under the assumptions that
(1 − B e aA ) −1 ∈ L(K), g ∈ K, and f ∈ L 1 (G, a X) the function defined by (6.7) is a mild solution on each edge, i.e., ψ j ∈ C(I j ; X j ) for each j ∈ I, cf.
respectively, implies that the solution is classical on each edge, i.e., continuously differentiable with respect to time, cf. [EN00, Cor. VI.7.6] and [EN00, Cor. VI.7.8]. Of course there are many refinements of the classical semigroup theory which one can carry over to time-graphs by assuming sufficient compatibility between B and the inverse of 1 − B e aA .

Iterative solvability
A time-graph Cauchy problem is iteratively solvable if it reduces to a finite sequence of initial value problems. This is made precise in the following definition.
Then we say that the (4.5) is iteratively solvable as a sequence of Cauchy problems on intervals. If B jj = 0 for all j = 1, . . . , n, then (4.5) is iteratively solvable as a sequence of initial value problems.
Iterative solvability can be traced back to the block structure of B. Proof. To prove (a) we start assuming that B is block tri-diagonal. Then Hence ψ n is the solution to the Cauchy problem on i n , and ψ j for j = 1, . . . , n − 1 solves the Cauchy problem on i j with B ji ψ i (a i ) =: ϕ j (ψ j+1 , . . . , ψ n , g).
Conversely, if (4.5) is iteratively solvable as a sequence of Cauchy problems on intervals, then there exists an ordering of the edges such that B n,j = 0 for j = n, and since ϕ j depends only on ψ j+1 , . . . , ψ n and g one concludes that B ij = 0 for j > i.  Proof. To prove (b): By assumption m ≥ 2 and B 1,2 , . . . , B (m−1),m = 0 and B m,1 = 0. In particular for any permutation of the edges π one has B π(1)π(2) = 0 and B π(m)π(1) = 0, and therefore B cannot be block tri-diagonal and the claim follows from Proposition 7.2 (a).
To prove (a) use using Proposition 7.2 (b) and note that if m = 1, then B 11 = 0, and if m ≥ 2 this follows already from (b). (For example, the shift on a loop satisfies (7.1) with respect to the derivative operator with periodic boundary conditions.) It seems that there are very few graphs whose automorphism group is a Lie group: in most cases, the automorphism group is finite. Having this, for any G ∈ Γ,Ĝ commutes with the solution operator given in Theorem 6.6. Thus the symmetry is reflected by the solution.

Examples and applications
The case of a loop with phase shift have been discussed already in the introduction as small modification of the classical periodic case. Also, the tadpole-like graph has been discussed there. We now discuss some other cases depicted in Figure 3. 8.1. Splitting of systems. Take the graph consisting of three internal edges as in Figure 3.(b), and consider for sectorial spatial operators −A i in Hilbert spaces X i , i = 1, 2, 3, the problem ∂ t ψ i − A i ψ i = f i , i = 1, 2, 3, with homogeneous boundary conditions ψ 1 (0) = g 1 , ψ 2 (0) = B 21 ψ 1 (a 1 ), ψ 3 (0) = B 31 ψ 1 (a 1 ). This corresponds to (6.6) for is invertible for any B 21 , B 32 , cf. Remark 7.3. Therefore by Theorem 6.6 a unique solution to this problem exists for all g ∈ K A ; in particular g = (g 1 , 0, 0) T ∈ K A means that g 1 ∈ [X 1 , D(A 1 )] 1/2 . The tree graph given in Figure 3.(f) results from an iteration of such as splitting procedure, where as above any splitting condition as long as it is compatible with the trace space is admissible. That such splitting problems can be solved iteratively as sequence of initial value problems is straight forward, it can also be seen more formally by applying Proposition 7.2. 8.2. Superposition of systems. Analogously, take the graph consisting of three internal edges as in Figure 3.(c), and consider for sectorial spatial operators −A i in Hilbert spaces X i , i = 1, 2, 3, the problem ∂ t ψ i − A i ψ i = f i , i = 1, 2, 3, with homogeneous boundary conditions ψ 1 (0) = g 1 , ψ 2 (0) = g 2 , ψ 3 (0) = B 31 ψ 1 (a 1 ) + B 32 ψ 2 (a 2 ), for i = 1, 2, 3. This corresponds to (6.6) for Assuming that B is compatible with K A one observes that 1 − Be a A is invertible in K A , cf. Remark 7.3. So, Theorem 6.6 is applicable again, and there is a unique solution to this problem if g 1 ∈ [X 1 , D(A 1 )] 1/2 and g 2 ∈ [X 2 , D(A 2 )] 1/2 . One observes that this is iteratively solvable as a sequence of initial value problems, too.
Note that 1 − Be aA is invertible if and only if 1 − e a 1 A 1 is invertible. So, solvability is assured if for instance e a 1 A 1 < 1, i.e., the semigroup generated by A 1 is contractive and exponentially decaying. This tadpole graph system can be interpreted as a time-periodic system where the output is used as initial data for a new system. In the notion introduced in Section 7, this means the problem can be solved iteratively, first solving a time-periodic problem and then an initial value problem, the data of which depend on the solution obtained in the first step.
8.4. Frequency dependent couplings. Frequency-dependent transition conditions between time branches may also be considered. Assume for simplicity that A 1 = A 2 are positive self-adjoint operators with discrete spectrum k 1 ≤ k 2 ≤ . . . where (ψ n ) is an orthonormal basis of eigenfunctions: this induces a map also in D(A 1/2 ). Then is an admissible transmission condition where the first row induces an initial condition on ψ 1 (0) and the second is the frequency shift. One may also consider a projection P I onto certain frequency ranges I ⊂ N, P I ψ := n∈I a n ψ n and one could have a splitting of the system This is iteratively solvable as sequence of initial value problems, and hence it is well-posed. 8.5. Lions maximal L 2 -regularity problem for non-autonomous Cauchy problems. Lions' maximal regularity problem for non-autonomous Cauchy problems considers for f ∈ L 2 (0, T ; X) for a Hilbert space X and asks if the solution satisfies ψ ∈ W 1,2 (0, T ; X); this would in turn imply that also A(·)ψ ∈ L 2 (0, T ; X). This problem has a long history and there are many remarkable works on this. More precisely, depending on A(·) there are counterexamples as well as criteria which assure an affirmative answer, we refer the interested reader to [HM00] for an early study of maximal regularity for non-autonomous problems, and to [ADF17] for further information and updated references.
For the particular case of A(·) being a step function with matching trace spaces This has already been used by [EML16] as a first step to consider A(·) which are of bounded variation. The time-graph approach does not give any additional information at this point but it underlines the role of the compatibility assumption for the trace spaces. Using our approach we directly see that this can be studied by means of which of course is iteratively solvable.
8.6. Outline on non-parabolic Cauchy problems. So far, the focus has been on parabolic Cauchy problems, but the Green's function Ansatz makes sense also in some non-parabolic situations.
8.6.1. Schrödinger equation. Let us study the Schrödinger-type problem Provided that 1 − B e iaA is invertible in K, the solution map given by ψ 0 in Theorem 6.6 is well-defined for all g ∈ K and defines a mild solution. (Notice that invertiblility in K is sufficient since here we merely aim at mild solutions.) Here, using that all operators commute, which is in turn equivalent to B 2 = 2B cos(aA). If in particular B is invertible, then the above condition amounts to B = 2 cos(aA). Accordingly, there is a unitary solution operator for fixed jump condition, i.e., B = 1, if and only if σ(aA) ⊂ π/3 + 2πZ. So, the classical case B = 0 is not the only case of a unitary solution operator. Note that time inversion is still possible even for non-unitary solution operator, but the time-reversed dynamics differs from the time-forward dynamics and going forth and back is not necessarily equal to stay at one time.
8.6.2. Second order Cauchy problem. The above setting can be generalized to different kind of evolution equations. Let B 1 , B 2 be bounded linear operators on K 2 and consider the second order Cauchy problem The idea is to decompose this into (∂ 2 t − A)ψ = (D t (B 1 ) − i|A| 1/2 )(D t (B 2 ) + i|A| 1/2 )ψ, where A is skew-symmetric. Hence one has to solve two first order problems iteratively, where assuming in addition to the assumptions of Theorem 6.6 that A j for j ∈ I are self-adjoint and boundedly invertible. Then for any g 1 , g 2 ∈ K there is a unique solution to (8.2). 8.7. Outline on mixed order systems. It is also possible to discuss evolution equations whose nature is different on each time branch; in particular, it is possible to define an operator on T which agrees with a first derivatives on a subset of T and with a second derivative on the remaining time branches. Defining appropriate transition conditions is however less obvious: a thorough discussion of "well-behaved" transition conditions can be found in [HM13]. Following these lines one can use the Kalton-Weis approach to solve a Cauchy problem of the type Focusing on this example we consider operators A 1 , A 2 in Hilbert spaces X 1 , X 2 and couplings defined by for an orthogonal projection P in K := X 1 ⊕ X 2 , P ⊥ = 1 − P and L ∈ L(K) with LP ⊥ = P ⊥ L. With these couplings the operator T P,L defined on L 2 (0, a 1 ; X 1 ) ⊕ L 2 (0, a 2 ; X 2 ) by T P,L ψ 1 = −ψ ′ 1 , T P,L ψ 2 = ψ ′′ 2 , D(T P,L ) := {ψ 1 ∈ W 1,2 (0, a 1 ; X 1 ), ψ 2 ∈ W 2,2 (0, a 1 ; X 1 ) : (P + L)ψ + P ⊥ ψ ′ = 0} is m-dissipative if −L is dissipative, see [HM13,Theorem 4.1], and similarly to Corollary 4.5 one concludes that it has a bounded H ∞ -calculus of angle π/2. If the spatial operator −A is sectorial and commutes with the boundary conditions, the one can apply the Kalton-Weis theorem to obtain well-posedness in a maximal regularity space. A Green's function approach could be pursued as well on the lines of the Green's function from [HM13, Proposition 6.6].

A tentative interpretation of time-graphs
Several convenient properties of semigroups depend decisively on the order structure of the underlying set, so it is conceivable to relax the standard approach to time evolution and study semigroup-like operator families that are indexed on posets different from R + .
One particularly simple case is that of a tree-like time structure. More precisely, we allow for a time that looks like a rooted tree, see Figure 3.(f). This seems to be conceptually very close to H. Everett's many worlds interpretation of quantum mechanics [DG15], but our mathematical theory is not restricted to Schrödinger-type equations, see Section 6, and a precise analysis of similarities and differences with Everett's interpretation goes far beyond the scope of this note. In a very simplified synopsis the many worlds interpretation claims in order to conciliate probabilistic and deterministic interpretations of quantum mechanics that time splits at each point in time and each possible state is actually attained in one of the parallel universes.
Imagining parallel-universes, one would have no evidence of these in the case of a tree graph. Only if there is some interaction between different 'time-branches', then one can know of the other now nonparallel but interacting universes. This leads to an interpretation of time-travel in terms of time-graphs, and it seems that time-graphs are a convenient way of picturing to oneself time-travel independently of its actual physical meaning.
Note that in the case of tree graphs, an evolutionary system can be solved iteratively starting with the first edge where some initial condition is imposed, one determines the state at the end of the edge which is the used as new initial conditions for the next level of the tree etc. Interesting problems arise when oriented loops are allowed, as outlined in Section 7. Then the time-evolution can no longer be described iteratively and there is no clear direction distinguishing 'before' and 'after'. This is one of the problems occurring in the scientific interpretation of closed time-blike curves in general relativity, cf. [HE73], and much earlier this has spurred the imagination of science fiction authors. Just to mention a few, there are H.G. Wells 'novel The time machine (1895) as well as The Man in the High Castle (1962) and further works written by Philip K. Dick between the 1950's and the 70's, whose main theme are interacting parallel universes. Variations of these themes and particular the so-called grandfather paradox -preventing one's one birth during a journey back in time or violating causality in some other way -are at the origin of several mainstream movies, like the classic Back to the Future (1985) or the more recent Looper (2012), in whose plot a person is sent back from 2074 to 2044 so that he can meet and be killed by his younger self 1 . A different topic are time-loops, illustrated for instance in the movies Groundhog Day (1993) or Miss Peregrine's Home for Peculiar Children (2016), which tell the stories of a manrespectively, a group of children and their guardian -trapped in a time loop around February 2, 1992 -respectively, September 3, 1943 -, before eventually managing to escape: in mathematical terms, this means that a certain function is not periodic (the main characters' days are not identical), but rather only satisfies a certain identity condition at different instants. This suggests a much more down-to-earth interpretation of evolution on branching time structures: namely, it conveniently allows us to formalize the requirement that solutions at different time instants respect certain algebraic relations.
The question of whether such fictional situations can be reconciled with our deep-rooted perception of time as linear seems for us related to the problem of representing a time-graph Cauchy problem as a sequence of initial value problems which can be solved one after another. As we have pointed out in Section 7 this is closely related to the existence of loops, and in this case the solution operator acts in a truly global (in time) fashion, as the solution at some point depends on all other times including future-like times. 9.1. Time travel, multiverses and the grandfather paradox. A time-travel scenario can be depicted as in Figure 3.(g) with a link between some point in the future and a point which might be in the past. Considering such a graph one can e.g. impose the transmission conditions ψ 1 (0) = g 1 , ψ 2 (0) = ψ 1 (a 1 ) + ψ 4 (a 4 ), ψ 3 (0) = ψ 2 (a 2 ), ψ 4 (0) = ψ 2 (a 2 ) that correspond to Therefore, this is not solvable as sequence of Cauchy problems on intervals, but well-posed in the sense of Theorem 6.6 under suitable assumptions on the spatial operators A i . In a sense, time-travel occurs in this deterministic world, but there is no free will to cause a grandfather paradox: the system is forced to be contradiction-free. This resembles the case of time-periodic solutions. Given a solution ψ to ∂ t ψ − Aψ = f, ψ(0) = ψ(1), one can compare this to the solution to ∂ t ϕ − Aϕ = f, ϕ(0) = ψ(0).
Due to the uniqueness of solutions the system comes back to its original state, i.e., ψ = ϕ and ϕ(1) = ψ(1) = ψ(0). Living in a time-periodic world, would thus be locally like living in a time-interval world with initial conditions, but nevertheless globally it is time-periodic. Similarly, in the scenarios of This is solvable as sequence of initial value problems, because B 14 = 0 the loop in the graph is not reflected by boundary conditions. A grandfather paradox does not occur because we actually have a sequence of initial value problems, and this seems the way we represent time-travel in our thoughts when watching a science fiction movie: A time traveler reaches a past where the initial conditions are the actual state of the past plus the time traveler. This mixer gives new initial conditions which lead to new events which actually do not affect the time from which the time traveler comes from. Traveling back to the present does not lead to any contradictions since one has a simple superposition of two time branches so that even changes due to the altered time branch can be incorporated. This becomes more transparent when as in Figure 4.(c) -compared to Figure 4.(b) -an auxiliary edge is inserted.  Representing such plot properly would in addition need a dynamical graph the development of which depends on the solution; one would also need to incorporate an end-condition stating that if the solution reaches a certain state, then the time evolution proceeds as a usual time axis. This would be a nonlinear feature. Typically, such an end-condition consists in the main character's reaching a certain goal or a key insight into the meaning of life.
Summarizing, it seems that our thinking is preassigned to represent time as linear with well-specified past and future, and even when imagining science fictional scenarios of time travel, time-loops and parallel universes we search for an ordering asking 'what happened first?', 'what happened then?', '. . . and then?' etc. So, in our language every science fictional scenario needs a representation as a sequence of iteratively solvable sequence of initial value problems. The other way round, thinking of a properly time-periodic movie would be quite repetitive.