Some Dirichlet forms on graphs as traces of one-dimensional diffusions

We compute explicitly traces of one-dimensional diffusion processes. The obtained trace forms can be regarded as Dirichlet forms on graphs. Then we discuss conditions ensuring the trace forms to be conservative. Finally, the obtained results are applied to the one-dimensional diffusion related to the Bessel’s process of order ν.


Introduction
Throughout this paper we are concerned with computation of traces of one-dimensional diffusions generated by the Feller operator d . Further details about the form are given in the next section. Given a diffusion of the above type, a positive measure µ with support V ⊂ I and a linear operator J : dom E (s) → L 2 (V, µ) we shall first compute the trace of E (s) with respect to the measure µ by means of the method elaborated in [BBST19]. We shall demonstrate in particular that the obtained trace form in L 2 (F, µ) is in fact a graph Dirichlet form if the measure µ is discrete.
Once the computation has been performed we shall turn our attention to study conservativeness property, i.e. conservation of total mass, for the trace Dirichlet form. We shall show that, for fixed E (s) , conservativeness depends strongly on the measure µ and its support V .
The motivation rests on two facts: first to put the particular case for the Bessel's process analyzed in [BM20] in a general framework. Second, the significance of conservativeness property both in analysis and in probability. In fact in analysis conservativeness is equivalent to existence and uniqueness of solution of the heat equation with bounded initial data. Whereas in probability conservativeness implies that, almost surely, the related stochastic process starting at any point will have an infinite lifetime.
At this stage we mention that there is a huge literature concerned with conservative Dirichlet forms. Regarding the subject we refer the reader to [Stu94,AG12,MUW12] [Gim16, Gim17,KL12].
The paper is organized as follows. In section 2 we introduce some necessary definitions and notations concerning Dirichlet forms related to one-dimensional diffusions as well as Feller's classical properties of boundaries. Section 3 is devoted to compute the trace of Dirichlet form on discrete sets as well as on composite of continuous and discrete sets. In Section 4 we study conservativeness property for traces of Dirichlet forms on discrete sets. In this respect we shall give necessary and sufficient conditions ensuring the trace form to be conservative. Thereby we extend [BM20,Theorem 3.7] to this general framework. The obtained theoretical results will be applied to the one-dimensional diffusion related to the Bessel's process of order ν, in the last section.

Framework and basic notations
We start by introducing some notations. Let I := (r 1 , r 2 ) where −∞ ≤ r 1 < r 2 ≤ ∞. Let us consider a continuous strictly increasing function s : I → R (a scaling function for short ). Thus s has the following representation where σ > 0 and σ ∈ L 1 loc (I). Obviously ds(x) = σ(x) dx.
Let us designate by AC loc (I) the space of locally absolutely continuous function on I and by AC s (I) the space of s-absolutely continuous functions on I, i.e. the set of functions u : I → R such that there exists an absolutely continuous function φ with u = φ • s. Let us consider a speed measure m with full support I defined by where ρ > 0 and ρ ∈ L 1 loc (I).
We designate by D (s) := u : I → R : u ∈ AC s (I), Let us define a quadratic form E with domain D ⊂ L 2 (I, ρ dx) by Proof. Since every u ∈ D is s-absolutely continuous, it is a composition of continuous functions, hence continuous.
It is well known that E is a regular strongly local Dirichlet form in L 2 (I, ρ dx) (hence in particular closed and densely defined). Moreover, the positive self-adjoint operator associated with the form E via Kato's representation theorem, which we denote by L, is given by (see [Lin04] and [FOT11, Chap. 1]) , with boundary conditions at r 1 and r 2 Lu =Lu for all u ∈ D(L).
Remark 2.2. The second-order ordinary differential operator ([CF12, p.63-64]) with real-valued functions a > 0 and b can be converted into Feller's canonical form d dm d ds Hence, by formal computation we get Further, owing to Feller's canonical form, we can define the differential operator L on I by

Feller's boundary classification
Let us introduce the following quantities for some r 1 < c < r 2 .
It is well known ( [JYC09]) that the boundaries r 1 and r 2 of I can be classified w.r.t the Feller operator d dm d ds into four classes as follows (we refer the reader to [Ito06, or [Man69, Definition 2.3. We say that the boundary r 1 (resp. r 2 ) is approachable whenever According to the inequality (3.2), if r 1 , (resp. r 2 ) is approachable, then for any element from D (s) we have u(r 1 ) = lim x↓r 1 ,x∈I u(x) < ∞, (resp. u(r 2 ) = lim x↑r 2 ,x∈I u(x) < ∞) and u ∈ C([r 1 , r 2 )), (resp. u ∈ C((r 1 , r 2 ])). One has in particular that space D (s) is a uniformly dense sub-algebra of C([r 1 , r 2 ]) if both r 1 and r 2 are approachable (for more details we refer to [CF12, Chap. II]).
Remark 2.5. Regularity condition of boundaries r i , i = 1, 2 (see [RW00, chpa.5] for more details) is very like the concept of irreducibility which has the following probabilistic terminology P x (H y < ∞) > 0, ∀x, y ∈ I, where H y is the hitting time of {y} relative to a diffusion process. Likewise, ([RW00, chap.5]) it allows us to characterise one-dimensional diffusion essentially by a scaling function s and a speed measure m.

Extended Dirichlet space
Let us now introduce the extended Dirichlet space of E ([CF12, chap.1]), which we denote by D e .
Definition 2.6. Let (E, D) be a closed symmetric form on L 2 (I, m). Denote by D e the totality of m-equivalence classes of all m-measurable functions f on I such that |f | < ∞ [m] and there exists an E-Cauchy sequence {f n , n ≥ 1} ⊂ D such that lim n→∞ f n = f, m − a.e. on I. {f n } ⊂ D in the above is called an approximating sequence of f ∈ D e . We call the space D e the extended space attached to (E, D). When the latter is a Dirichlet form on L 2 (I, m), the space D e will be called its extended Dirichlet space.
To determine the extended Dirichlet form in our case we shall using the following proposition which is mentioned by [CF12,p.66].
Proposition 2.7. Assume that both r 1 and r 2 are approachable but non-regular. If we let In this case, the boundaries r 1 and r 2 are non-exit points.
Remark 2.8. We have the following discussion about the boundary conditions at r 1 and r 2 on D(L): (i) If r 1 (resp. r 2 ) is an exit endpoint then we have the boundary condition at r 1 (resp. r 2 ) lim (ii) If r 1 (resp. r 2 ) is an entrance endpoint then we have the boundary condition at r 1 (resp. r 2 ) (iii) If r 1 , (resp. r 2 ) is a natural endpoint then there is no boundary condition needed.

Computation of the trace of transient formĚ
In addition, let x ∞ = lim k→∞ x k which can be finite or not. Next, we will investigate the following two cases for a transient Dirichlet form E: (a) V has no accumulation point.
(b) V has x ∞ as an accumulation point.
We have the following definition of capacity which is a set function associated to Dirichlet form and it plays an important role to measure the size of sets adapted to the form.
Definition 3.1. We define the 1-capacity Cap associated with the Dirichlet form (E, D) by for an open set U ⊂ I, and The following lemma show that a diffusion process associated with a Dirichlet form E enjoys a strong irreducibility property which means that any two point of I are connected for a diffusion process (we refer to [Fuk14]).
Proof. An elementary identity leads to By Hölder inequality, we get Then we get by integrating the both hand side on each compact set K ⊂ I that there is a positive constant C K such that i.e. each point of I has a positive capacity relative to the Dirichlet form (E, D).
We shall start by the first case (a) which said that V has no accumulation point, i.e.

V has no accumulation point
Let (a k ) k∈N be a sequence of real numbers such that a k > 0 for all k ∈ N. Let us consider the atomic measure defined as follows We define now the Hilbert space 2 (V, µ) equipped with the product given by LetĚ be the trace of E on the discrete set V (see [FOT11,BBST19]). We shall adopt the method elaborated in [BBST19] to compute explicitlyĚ.
We quote that since functions with finite support are dense in 2 (V, µ), the operator J has dense range. Obviously We shall first determine the extended domain D e of the trace formĚ according to Proposition 2.7.
Accordingly we can computeĚ following [BBST19, Prop.3.1]. To that end we designate by P the orthogonal projection in the Dirichlet space (E, D e ) onto the E-orthogonal complement of ker J. Clearly We can define the quadratic fromĚ as followš Since J is closed in (D e , E), then, from [BBST19, Prop.3.1], the formĚ is closed in 2 (V, µ).
Lemma 3.3. Let u ∈ D. Then P u is the unique solution in D e of the following Sturm-Liouville problem Proof. Let u ∈ D(J). Since P is the E-orthogonal projection from D e onto (KerJ) ⊥ , then we obtain Multiplying the latter equation by a positive term 1 ρ . P u be a solution with smooths coefficients on (x 1 , ∞) \ V . Moreover since J is a closed operator then KerJ is also closed, hence u−P u ∈ KerJ and P u ∈ D(J), then P u ∈ D e and hence P u is s-absolutely The converse is obvious.
Let us now compute explicitly the E-orthogonal projection P u the solution of the boundary value problem (3.4).
Lemma 3.4. Let u ∈ D. P u can be expressed in the following way for all x ∈ [x k , x k+1 ] and k ∈ N, where c is a constant in (x 1 , ∞) such that s(c) = 0.
Proof. In fact the differential equation (3.4) has the solution for all k ∈ N as follows where C 1 and C 2 are two real constants to be determined according to the boundary conditions. Then, we have which leads to get the following expression for all x ∈ [x k , x k+1 ], and c ∈ (x 1 , ∞).
Lemma 3.5. For every u ∈ D e . It holdš where (P u) (x + k ) and (P u) (x − k+1 ) are the right derivative at x k and the left derivative at x k+1 respectively.
Proof. Let u ∈ D e . A straightforward computation leads tǒ . (3.9) From the expression of P u we can compute its derivative (P u) for all k ∈ N, Finally we obtain the trace formĚ of pure jump typě (3.10) In the sequel we shall recall some definition of weighted graphs to construct the discrete Dirichlet forms. We define a function c : V −→ (0, ∞) which can be interpreted as a killing term or as a potential. We say that two vertices x, y ∈ V are neighbors or connected by an edge if b(x, y) > 0 and we write x ∼ y.
In order to describe the trace formĚ we have to introduce the next form For every u ∈ 2 (V, µ). Set (3.11) Then,Ě = Q | ranJ . Proof. Let us rewrite the trace formĚ as followš ∃r > 0 : |x k+1 − x k | = r) and b(x k+1 , x k ) = 0 otherwise. Moreover, if µ(V ) = k∈N a k = ∞ and b(x k , x k+1 ) > 0 for all k ∈ N, the condition (A) from [KL12] is fulfilled which yields the assertion.
Then, the associated self-adjoint discrete operatorĽ is given by where for all k ∈ N. We get .

V has an accumulation point
Now we consider the second case (b) where the sequence (x k ) k∈N of the set V is convergent and it has x ∞ as an accumulation point . We keep the same definitions as in the third section. We consider again the case where E is transient Dirichlet form. For u ∈ D P u is the unique solution in D e of the differential equation with boundary condition which can be expressed as follows for all x ∈ (x ∞ , ∞) and for all fixed arbitrary c ∈ (x 1 , ∞).
We can compute now the traceĚ which is decomposed into the sum of a non-local and a killing Dirichlet form.
Lemma 3.9. For every u ∈ D e , it holdš . (3.16) Proof. Since the end-point ∞ is an approachable boundary, i.e., s(∞) < ∞, then we have lim x→∞ P u(x) = lim x→∞ u(x) = 0 and s([x ∞ , ∞)) < ∞. Owing to this argument we can obtain the following explicit computation ofĚ. Let u ∈ D e . We geť where (P u) (x + k ) and (P u) (x − k+1 ) are the right derivative at x k and the left derivative at x k+1 respectively. Since Therefore we obtaiň

Trace of the Dirichlet form related to one-dimensional diffusion process w.r.t mixed type measure
Let (x k ) k be a sequence of negative numbers which converges to 0 and so the set V ⊂ (r 1 , 0) has 0 as an accumulation point. We consider a new measure on (x 1 , ∞) of mixed type, i.e. measure which has an absolutely continuous part and a discrete part as follows a k δ x k , ∀a k > 0, k ∈ N and µ abs = 1 (0,∞) ρ(x)dx.
is the support of the measure µ. In order to compute the trace of E w.r.t measure µ we shall define the trace operator J by

Obviously, we have
Then the Sturm-Liouville problem has P u a unique solution of We can express the general solution of P u in the same way as the third section for all x ∈ [x k , x k+1 ] and k ∈ N, and c is an arbitrary fixed point in (r 1 , ∞).
We introduceĎ max the space of the trace formĚ by We denote byĚ (J) andĚ (c) the quadratic forms such that whereĚ (c) andĚ (J) are the strongly local type and non-local type Dirichlet forms respectively. We quote that the trace of the Dirichlet formĚ decomposed intoĚ (c) andĚ (J) . In fact, let us stress that this decomposition is mentioned by [BM20] for dimension n = 3 and V = (0, 1) ∪ N.

Conservativeness of traces of one-dimensional diffusions
In this section we further assume that E is conservative. Our aim is to establish necessary and sufficient conditions ensuring the trace formĚ to inherit conservativeness property.
The form E is said to be conservative if T t 1 = 1 for some and for every t > 0, where T t stands for L ∞ -semi-group induced by the Dirichlet form E . Let us now start with the case where the set V has no accumulation point.
Theorem 4.1. Assume that µ is infinite. Then the discrete Dirichlet form on the graph Proof. The conservativeness of the Dirichlet form E is equivalent to the fact that the equationL has no nontrivial bounded solution (we refer the reader to [KL12]).
Thus by induction and the recursive formula we get The latter formula gives rise to two observations (which can be proved by induction): 1. u(x k ) has the sign of u(x 1 ) for all k ∈ N. This is if u(x 1 ) > 0, then u(x k ) > 0, for all k ∈ N and if u(x 1 ) < 0, hence u(x k ) < 0, for all k ∈ N.
2. u(x k ) is monotone, depending on the sign of u(x 1 ).
Hence without loss of generality we may and shall assume that u(x 1 ) > 0. In this case (u(x k )) k∈N is positive and strictly monotone increasing sequence. Accordingly, making use of formula (4.6) we derive and Finally we achieve Obviously the latter product is finite provided ∞ k=1 s(x k+1 ) − s(x k ) k j=1 a j < ∞ and then we get a bounded non-zero solution.
In the other sense, we suppose that ∞ k=1 s(x k+1 ) − s(x k ) k j=1 a j = ∞. Then summing over k in formula (4.6) and keeping in mind that the sequence (u(x k )) k∈N is increasing. We obtain (4.10) which finishes the proof.
Theorem 4.2. If µ is a finite measure. ThenĚ [u] is not conservative.
Proof. Since µ is finite, i.e., µ({x k , k ∈ N}) = ∞ k=1 a k < ∞, then conservativeness and recurrence are equivalent. We have already compute the trace of a transient Dirichlet form E. Therefore, according to [FOT11, Lemma 6.2.2., p.317] transience property is inherited by the trace form which yields thatĚ is transient and hence it is not conservative.
The case where V is a finite set can be resolved easily. For the case where V has x ∞ as an accumulation point, we remark that the trace formĚ has a killing part. Hence owing to theoretical results (see [BM20]) it can't be conservative.
is not conservative.
Proof. Non-conservativeness of the trace formĚ follows from the fact that 1 ∈Ď anď Theorem 4.4. Assume that set V is infinite and accumulates at x = x ∞ . Theň for each u ∈Ď. MoreoverĚ is not conservative.
Proof. For every u ∈ 2 (V, µ). We set which is acting on It is easy to check that Q is closed, then the trace formĚ is the restriction of Q to ranJ. For any sequence (u n ) n∈N ⊂Ď with 0 ≤ u n ≤ 1 and u n ↑ 1 µ − a.e., we have for every v ∈Ď By dominated convergence theorem we get Since v = 0 on I, we get the non-conservativeness ofĚ.
5 Application : traces of the one-dimensional diffusion related to Bessel's process For each n ∈ N, n ≥ 2. We consider the speed measure m defined on I = (0, ∞) by We define the scaling function s as follows We shall be concerned with the Dirichlet form E with domain D ⊂ L 2 (I, 2x 2ν+1 dx) defined by Since n ≥ 2 the Bessel process is transient [CF12,p.126] which yields that associated Dirichlet form is transient too. We can easily check that for n ≥ 3, (i.e. ν ≥ 1 2 ) we obtain r 1 = 0 is a non-approachable boundary, i.e., s(0) = ∞. Whereas the boundary point r 2 = ∞ is an approachable boundary, i.e., s(∞) < ∞.
In these circumstances, the selfadjoint adjoint operator related toĚ, which we denote by L is the generator of the Bessel process of index ν on the half-line. Moreover we have the following description of L. Set Then We start with the case that the sequence (x k ) k∈N diverges and so it has no accumulation point. Accordingly we consider the discrete measure defined as the first section by µ = k∈N a k δ x k , which is supported by an infinite countable set V = {x k , k ∈ N } ⊂ (0, ∞).
Remark 5.1. In our case for n ≥ 2, r 1 = {0} is an entrance boundary and it is well known that Cap({0}) = 0 (we refer to [JYC09] for more details) and for each element To compute the trace of the general Bessel's Dirichlet formĚ with domainĎ ⊂ 2 (V, µ) we have to apply Theorem 3.7. with scaling function ds(x) = 1 x 2ν+1 dx to obtain the following expressionĚ Let u ∈ 2 (V, µ). Set We haveĚ = q | ranJ . Indeed, q is a closed quadratic form together with the fact thatĚ is the closure of q restricted to ranJ. Hence, µ(V ) = ∞ leads tǒ D := D(Ě) = {u ∈ 2 (V, µ) : for all u ∈ D(Ě). (5.1) In this case, we can determine the discrete Bessel operator associated with trace formĚ as follows for each k ∈ Ň .
(5.2) For the conservativeness property of the general Bessel's Dirichlet forms we have following result as an application of the Theorem 3.1. If µ is infinite, thenĚ is conservative if and only if Finally we consider the case where the sequence (x k ) k∈N converges to x ∞ . According to Lemma 3.9. we obtaiň For every u ∈ 2 (V, µ). Put We can easily check that trace formĚ is the closure of Q | ranJ . We assume that µ(V ) = ∞. Regarding conservativeness property, according to Theorem 4.4., the trace formĚ is not conservative.