Discrete Dirichlet forms as traces of Feller’s one-dimensional diffusions

: Our primary purpose is to compute explicitly traces of the Dirichlet forms related to Feller’s one-dimensional diffusions on countable sets via Fukushima’s method. For discrete measures, the obtained trace form can be described as a Dirichlet form on the graph.


Introduction
I n this paper, we are interested in diffusion processes which have important applications in several domains such as biology, economy and finance. Thus from this research we draw our attention to discretize a Dirichlet form by computing its trace form through the scaling function θ as well as the speed measure µ associated to Feller 's one-dimensional diffusions operators d dµ d dθ on an open interval Ω. In order to compute traces of Feller's Dirichlet forms on a countable set concerning a functional Tr, we shall apply the methods developed in [1,2] for the transient case. Hence in this situation, we have to determine the extended Dirichlet spaces D ext via Feller's classification of the boundaries. At to end of this paper, we will apply the obtained theoretical results to the Feller's Dirichlet form associated with the 1-order Bessel operator.
We arranged this paper as follows. In §2, we briefly review Feller's one-dimensional diffusions and basic notations to construct the Dirichlet form associated with Feller's operators. Then, in §3, we compute traces of Dirichlet form explicitly concerning discrete measures supported by countable sets F. Finally, examples in §4 such as 1-order Bessel processes on a weighted graph (N, ω) illustrate the detailed results. We emphasis that a Bessel process on the half-line is also the radial component of the standard Brownian motion in the Euclidean space R n (we refer to [3] for more details).

Preliminaries
Let Ω be an open interval of the real line R. We set E (θ) a Dirichlet form with domain D (θ) in H = L 2 (Ω, µ) defined by Let a positive Radon measure ν with support F ⊂ Ω that charges no sets having zero capacity. We consider H aux = L 2 (F, ν) and assume that trace operator is closed. We shall designate byĚ the trace form of E (θ) w.r.t Tr (or w.r.t the measure ν) and byĎ its domain.

Feller's Dirichlet forms
In this section we will introduce a Dirichlet form associated with the Feller's operator on some subsets of

R.
Let Ω = (0, ∞) and we consider a continuous strictly increasing function θ : Ω → R. Assume that θ ∈ AC loc (Ω) such that where γ > 0 and γ −1 = 1 γ ∈ L 1 loc (Ω). Obviously θ(dx) = γ −1 (x) dx. Moreover, the scaling function θ can be also regarded as a scaling measure on Borel subsets J of R given by Giving a speed measure µ with full support Ω as with γ > 0 and γ ∈ L 1 loc (Ω). We denote by , We define a Dirichlet form E with a densely defined domain D in L 2 (Ω, µ) by (1) Lemma 1. Each function from D is continuous and a.e. differentiable on Ω.

Proof.
As each function f ∈ D is absolutely continuous w.r.t dθ, it is a composition of continuous functions, hence continuous. Since θ is strictly increasing then by Lebesgue's theorem it is a.e. differentiable. Besides every locally absolutely continuous function is a.e. differentiable. Hence f is a.e. differentiable.
Using Lemma 1 together with the fact that 1 , and We claim that E is a regular strongly local Dirichlet form in L 2 (Ω, µ) [4]. According to the first Kato's representation Theorem [5] we can associate to E a positive self-adjoint operator L which is defined by (we refer to [6, §2] and [1, Correspondingly the Feller's canonical operator ∆ µ,θ is defined as (see [4, p. 63-64])

Feller's classification of the boundaries
We let for some c > 0. The left boundary 0 can be classified as follows: ( see [7, p. 151-152]) Analogously to the right boundary ∞.

Definition 1 ([4]). (a)
We say that the boundary 0 (resp. ∞) is approachable whenever We designate by D ext the extended Dirichlet space of E which is introduced in [4].
Henceforth, making use the theorem below to induce the extended Dirichlet space D ext .
We remark that endpoints 0 and ∞ are non-exit.

Traces of one-dimensional diffusions on countable sets
Let F = {p k ∈ Ω, k ∈ N} be a countable set. Let (α k ) k∈N be a sequence of in the right half axis (0, ∞). We define a discrete measure ν on F by where δ p k denote the Dirac measure centred on some fixed point p k in F.
Let H aux = ℓ 2 (F, µ) be a real Hilbert space equipped with a scalar product given by On the other hand,Ě is the trace of E on F w.r.t ν (see [1,9,10]). As by assumptions E is transient, hence we can adopt the technique developed in [2] to evaluateĚ .
We shall now discuss the following case: 0 is approachable and non-regular.
According to Theorem 1 the extended domain D ext is given by
We are now in position to compute E -orthogonal projection Π.
Theorem 2. Let f ∈ D ext . Then Π f a solution from D ext of the Sturm-Liouville problem Π f = f on F.
Proof. Assume that t = θ −1 is absolutely continuous and t ′ ∈ L 2 loc (Ω). Hence, making use [11, Let f ∈ D ext . Given that Π the E -orthogonal projection from D ext onto (ker(Tr)) ⊥ , then which is equivalent to 1 is the dual bracket between C ∞ c (Ω \ F) and its dual C ∞ c (Ω \ F) ′ . Accordingly, we obtain − 1 On the other hand, the closedness of the operator Tr yields the closedness of Ker(Tr) as well. Hence f − Π f ∈ Ker(Tr) and Tr f = Tr Π f . Thus f = Π f on F. It is easy to proof the converse so we omit.
We are in position now to compute explicitly the E -orthogonal projection Π f solution of the boundary value problem (9).
Proof. On the light of the Theorem 2, it suffices to show that the function given by (11) solves the following ODE Indeed the solution of the homogeneous differential equation (12) is given by where M k ,N k ∈ R. Hence, we obtain This leads to achieve

Theorem 3.
For each function f ∈ dom(Tr). It holds dom(Ě ) = ran(Tr) anď where dΠ f dx (p + k ) and dΠ f dx (p − k+1 ) are the right derivative at p k and the left derivative at p k+1 respectively.

Traces of the Feller's Dirichlet forms related to 1-order Bessel's process on N
Let us consider a speed measure µ defined on Ω = (0, ∞) by µ(dx) = 2x 3 dx.