Asymptotic evolution of acyclic random mappings

An acyclic mapping from an n element set into itself is a mapping ϕ such that if ϕ k ( x ) = x for some k and x , then ϕ ( x ) = x . Equivalently, ϕ (cid:96) = ϕ (cid:96) +1 = . . . for (cid:96) suﬃciently large. We investigate the behavior as n → ∞ of a sequence of a Markov chain on the collection of such mappings. At each step of the chain, a point in the n element set is chosen uniformly at random and the current mapping is modiﬁed by replacing the current image of that point by a new one chosen independently and uniformly at random, conditional on the resulting mapping being again acyclic. We can represent an acyclic mapping as a directed graph (such a graph will be a collection of rooted trees) and think of these directed graphs as metric spaces with some extra structure. Informal calculations indicate that the metric space valued process associated with the Markov chain should, after an appropriate time and “space” rescaling, converge as n → ∞ to a real tree ( R -tree) valued Markov process that is reversible with respect to a measure induced naturally by the standard reﬂected Brownian bridge. Although we don’t prove such a limit theorem, we use Dirichlet form methods to construct a Markov process that is Hunt with respect to a suitable Gromov-Hausdorﬀ-like metric and evolves according to the dynamics suggested by the heuristic arguments. This process is similar to one that appears in earlier work by Evans and Winter as


Introduction
A mapping ϕ from the set [n] := {1, 2, . . . , n} into itself may be represented as a directed graph with vertex set [n] and directed edges of the form (i, ϕ(i)), i ∈ [n]. The resulting directed graph has the feature that every vertex has out-degree 1 (with self-loops -corresponding to fixed points -allowed), and any such graph corresponds to a unique mapping. For example, the mapping ϕ : [18] → [18] in Table 1 corresponds to the directed graph in Figure 1.
The directed graph may be decomposed into a number of connected components. Each of these components consists of a single directed cycle (possibly a self-loop) plus trees rooted at each vertex on the directed cycle (such a tree may be a trivial tree consisting of only the root, meaning that the only pre-image of that point is its predecessor on the directed cycle). We call such rooted trees the tree components of the graph.
Aldous and Pitman (AP94) describe a procedure for associating a mapping of [n] into itself with a lattice reflected bridge path of length 2n, that is, with a function b : {0, 1, . . . , 2n} → {0, 1, 2, . . .} such that b(0) = b(2n) = 0 and |b(k + 1) − b(k)| = 1 for 0 ≤ k < 2n. The exact details of the procedure aren't important for us. However, we note that a tree component with vertices corresponds to a lattice positive excursion path from 0 with 2 steps. Such a segment of path records the distance from the root plus 1 in a depth-first-search of the tree component. Of course, the mapping cannot be recovered from just the lattice reflected bridge path. For one thing, some extra marking of distinguished points of the zero set of the lattice path is required to split the lattice path up into sub-paths corresponding to components of the directed graph.
Once this is done, the mapping is uniquely specified by the lattice path up to a relabeling of the vertices: that is, if two mappings ϕ and ψ correspond to the same lattice reflected bridge path, then ψ = π • ϕ • π −1 for some permutation π of [n]. Conversely, if ψ = π • ϕ • π −1 for some permutation π of [n], then the lattice path corresponding to ψ may be obtained from the lattice path corresponding to ϕ by composition with a bijective map of {0, 1, . . . , 2n} that preserves lengths of excursions above all levels. That is, if the lattice path corresponding to ϕ has k excursions above some level h, then the same is true of the lattice path corresponding to ψ.
Suppose now that a mapping of [n] into itself is chosen uniformly at random from the n n possibilities. This is equivalent to choosing the image of each point of [n] independently and uniformly at random from [n]. The corresponding lattice reflected bridge path is not uniformly distributed. However, it is shown in (AP94) that if the lattice reflected bridge path is turned into a continuous time process by holding it constant between integer time points, time is rescaled by 2n, and space is rescaled by n 1 2 to produce a function from [0, 1] into R + , then this stochastic process with càdlàg sample paths converges in distribution to twice a standard reflected Brownian bridge (that is, twice the Brownian bridge reflected at 0 that goes from position 0 at time 0 to position 0 again at time 1). In particular, the proportion of vertices that lie on directed cycles converges to the proportion of time the standard reflected Brownian bridge spends at 0, which is, of course, 0, so that asymptotically almost all vertices are not roots of tree components. The asymptotics of the cyclic vertices jointly with the tree vertices are described in (AP94) using the local time at 0 of the reflected bridge and that paper also describes an auxiliary "marking" procedure for describing the joint asymptotics of the the component sizes. Some later results in this same vein may be found in (AMP05; AP02; DS97; DG99; GL00; DG04; Pit02).
A mapping ϕ from [n] into itself is acyclic if the only directed cycles in the corresponding directed graph are self-loops. That is, each x ∈ [n] is either a fixed point of ϕ (so that x is a vertex on a self-loop) or ϕ k (x) = x for any k. Equivalently, ϕ = ϕ +1 = . . . for sufficiently large. For such a mapping, each graph component consists of single tree component with a self-loop attached to the root, and no auxiliary marking procedure is necessary to recover the mapping up to a permutation from the corresponding lattice reflected bridge path. It is not hard to show that if we turn the lattice reflected bridge path for a uniformly chosen acyclic random mapping into a continuous time process indexed by [0, 1] as above, then the resulting process also converges to twice a standard reflected Brownian bridge -as one would expect from the observation that the cyclic vertices are asymptotically negligible for a uniformly chosen random mapping, In this paper we are interested in the asymptotic behavior as n → ∞ of a simple Markov chain that randomly evolves an acyclic mapping from [n] into itself. At each step of the chain, a point of [n] is chosen uniformly at random and the image of this point is re-set to a new image chosen independently and uniformly at random from [n], conditional on the resulting mapping being acyclic. It is clear for each n that this chain is reversible with respect to the uniform distribution on the set of acyclic mappings from [n] into itself and that the chain converges to this distribution at large times.
In terms of the corresponding directed graphs, the chain evolves as follows. A directed edge is first chosen uniformly at random and deleted. The deleted edge is then replaced by another directed edge with the same initial vertex but a uniformly chosen final vertex, conditional on the resulting graph having no cycles other than self-loops. Note that the effect of such a step is the following.
• If the deleted edge is a self-loop, its deletion turns the graph component that contained the edge into a rooted subtree. Otherwise, the deletion of the directed edge splits the graph component that contained it into two pieces, one of which contains a self-loop and the other of which is a rooted subtree.
• In either case, the addition of the new directed edge either attaches the root of the subtree to itself by a self-loop, producing an extra graph component, or the new directed edge attaches the root to a vertex chosen uniformly outside the subtree (possibly to a vertex outside the subtree but within the same former graph component). All such possibilities are equally likely.
The effect on the corresponding lattice bridge path is to remove an excursion above some level, insert a suitable time-space translation of it at some time point in the lattice bridge path outside the excursion, and then close up the gap left by the removal (more precisely, this transformation may need to be followed by a bijective map of {0, 1, . . . , 2n} that preserves lengths of excursions above all levels because of the way that the labeling of vertices in the directed graph is used to construct the corresponding lattice bridge path).
In order to understand the asymptotic behavior of this sequence of chains as n → ∞, we need to embed the state space of each chain into a common state space that will also be the state space of the limit process.
To begin with, we erase all of the self-loops in the directed graph corresponding to an acyclic mapping. This produces a forest of subtrees rooted at vertices that were formerly on self-loops. We then connect the roots of these subtrees by directed edges to a single adjoined point to produce a tree rooted at the adjoined point. Keeping in mind the rescaling identified by Aldous and Pitman, we think of this tree as a one-dimensional cell complex by regarding each edge as a segment of length n − 1 2 . We thus have a metric space with a distinguished base point (the root). This pointed metric space is an instance of a rooted compact real tree (R-tree): see Section 2 for the precise definition of a R-tree -for the moment, all that is important for explaining our results is that a R-tree is a metric space that is, in some sense, "tree-like". We regard two rooted compact R-trees as being equal if one can be mapped into the other by an isometry that preserves the root. If two mappings ϕ and ψ are related by a relabeling ψ = π • ϕ • π −1 for some permutation π of [n], then they correspond to the same rooted compact R-tree.
Before we continue with the motivation of our results, we need to indicate how the rooted compact R-tree associated with a mapping from [n] into itself may be constructed directly from the corresponding lattice reflected bridge path. We begin by introducing some general notation that is useful later.
We associate each f ∈ Ω 1 + with a compact metric space as follows. Define an equivalence relation ∼ f on [0, 1] by letting Consider the pseudo-metric d T f on [0, 1] defined by This pseudo-metric becomes a true metric on the quotient space T f := [0, 1]/ ∼ f . The resulting metric space is compact and is an instance of a rooted compact R-tree if we define the root to be the image of 0 under the quotient map.
Suppose that the function f ∈ Ω 1 + is obtained by first linearly interpolating the lattice reflected bridge path associated with an acyclic mapping ϕ of [n] into itself to produce a function in Ω 2n + and then rescaling time by 2n and space by n 1 2 . The corresponding pointed metric space T f is the rooted compact R-tree associated with ϕ that we described above.
Any metric space of the form T f for f ∈ Ω 1 + has two natural Borel measure on it. Firstly, there is the "uniform" probability measure ν T f given by the push-forward of Lebesgue measure on [0, 1] by the quotient map. We call this measure the weight on T f . Secondly, there is the natural length measure µ T f , which is the one-dimensional Hausdorff measure associated with the metric d T f restricted to points of T f that are not "leaves" (see Section 2 for a more precise definition). When f is associated with a map of [n] into itself as above, then µ T f is just the "Lebesgue measure" on the cell complex T f that assigns mass n − 1 2 to each edge of T f (recall that we have rescaled so that each edge has length n − 1 2 ). Now, if we speed up time by a factor of n 1 2 in our Markov chain for evolving mappings of [n] into itself and look at the corresponding rooted compact R-tree-valued process, then it is reasonable at the heuristic level that we should obtain in the limit as n → ∞ a continuous time Markov process with the following informal description. The state space of the limit process is the space consisting of rooted compact R-trees T equipped with a probability measure ν T : we call such objects weighted rooted compact R-trees. We note that, as in the special case of R-trees of the form T f for f ∈ Ω 1 + , an arbitrary compact R-tree has a canonical length measure µ T given by the restriction of the one-dimensional Hausdorff measure associated with the metric to the set of points that aren't leaves. The process evolves away from its state at time 0 by choosing a point (t, v) at rate dt ⊗ µ T (dv) in time and on the current tree T , and at time t the subtree above v (that is, the subtree of points on the other side of v from the root) is re-attached at a point w chosen according to ν T (conditional on w being outside the subtree).
In general, the measure µ T may have infinite total mass. For example, if f ∈ Ω 1 + is chosen according to the distribution of standard reflected Brownian bridge, so that T 2f is the rooted compact R-tree that arises from a limit as n → ∞ of uniform acyclic random mappings of [n] into itself, then µ T 2f almost surely has infinite total mass. Consequently, the above specification of the dynamics of the limit process does not make rigorous sense for general weighted rooted compact R-trees.
The aim of this paper is to use Dirichlet form methods to construct a suitably well-behaved Markov process with evolution dynamics that conform to the heuristic description. Our main result in this direction is stated precisely as Theorem 7.5 at the end of the paper after we have developed the necessary background and notation to describe the result and the requisite technical machinery to prove it.
We stress that we do not obtain a convergence result. The process we construct has no obvious Feller-like properties and it is not clear how to define its dynamics for all starting points (as opposed to almost all starting points with respect to the symmetrizing measure, which is all the Dirichlet form approach provides). Consequently, it is not clear how standard techniques such as martingale problem methods might be used to establish convergence.
The process we construct is somewhat similar to the process constructed in (EW06) as the heuristic limit of a sequence of natural chains based on the subtree prune and regraft (SPR) tree rearrangement transformations from phylogenetics. Both processes involve the relocation of a subtree whose root is chosen according to the length measure on the current tree. However, the state space of the process in (EW06) consists of weighted unrooted compact R-trees, whereas we work with weighted rooted compact R-trees and the root plays a crucial role in defining the dynamics. The symmetrizing measures are, as a consequence, rather different: the measure in (EW06) is the distribution of the Brownian continuum random tree, which is the R-tree "inside" twice a standard Brownian excursion, whereas our symmetrizing measure is the distribution of the R-tree "inside" twice a standard reflected Brownian bridge. However, many of the steps in the construction are quite similar so we omit several arguments and simply refer to the analogous ones in (EW06).
We note that Markov processes with reflected bridge paths as their state space and continuous sample paths have been studied in (Zam03; Zam02; Zam01). These processes are reversible with respect to the distribution of a Bessel bridge of some index.
2 Weighted R-trees Definition 2.1. A metric space (T, d) is a real tree (R-tree) if it satisfies the following axioms.
Axiom 0: The space (T, d) is complete.
Axiom 2: For every injective continuous map ψ : Axiom 1 says simply that there is a unique "unit speed" path between any two points x and y. We write [x, y] for the image of this path and call it the segment with endpoints x and y. Axiom 2 implies that the image of any injective path connecting two points x and y coincides with the segment [x, y], and so such a path may be re-parameterized to become the unit speed path. Thus, while Axiom 1 is satisfied by many other spaces such as R d with the usual metric, Axiom 2 captures the essence of "treeness" and is only satisfied by R d when d = 1. See (Dre84; DT96; DMT95; DMT96; Ter97; Chi01) for background on R-trees. In particular, (Chi01) shows that a number of other definitions are equivalent to the one above. Much of this content is synthesized and combined with other material on probability on R-trees in (Eva07). Also, some probabilistic aspects of R-trees are reviewed in (LG06).
We define the η-trimming, R η (T ) of a compact R-tree (T, d) for η > 0 to be the set of points x ∈ T such that x belongs to a segment [y, z] with d(x, y) = d(x, z) = η -see Figure 3. The skeleton of (T, d) is the set for all x, y ∈ T (see Section 2.4 of (EPW06) or Section 2 of (EW06)). h T Figure 3: A R-tree T and its η-trimming R η (T ). The R-tree T consists of both the solid and dashed segments, whereas the R-tree R η (T ) consists of just the solid segments.
In the following, we are interested in compact R-trees (T, d) equipped with a distinguished base point ρ ∈ T (called the root) and a probability measure ν on the Borel σ-field B(T ) (called the weight). We call such objects weighted rooted compact R-trees. We say that two weighted rooted compact R-trees (X, d X , ρ X , ν X ) and (Y, d Y , ρ Y , ν Y ) are weighted rooted isometric if there exists a bijective isometry Φ between the metric spaces (X, d X ) and (Y, d Y ) such that Φ(ρ X ) = ρ Y and the push-forward of ν X by Φ is ν Y , that is, The property of being weighted rooted isometric is an equivalence relation. We write T wr for the collection of equivalence classes of weighted rooted compact R-trees.
In order to define a metric on T wr , we first recall the definition of the Prohorov distance between two probability measures (see, for example, (EK86)). Given two probability measures α and β on a metric space (X, d) with the corresponding collection of closed sets denoted by C, the Prohorov distance between them is where C ε := {x ∈ X : inf y∈C d(x, y) < ε}. The Prohorov distance is a metric on the collection of probability measures on X.
We are now in a position to define the weighted rooted Gromov-Hausdorff distance between the two weighted rooted compact R-trees (X, d X , ρ X , ν X ) and (Y, Note that the set on the right hand side is non-empty because X and Y are compact, and hence bounded in their respective metrics. Note also that ∆ GH wr (X, Y ) only depends on the weighted rooted isometry classes of X and Y .
It turns out that the function ∆ GH wr satisfies all the properties of a metric except the triangle inequality. To rectify this, put where the infimum is taken over all finite sequences of weighted rooted compact R-trees Z 1 , . . . Z n with Z 1 = X and Z n = Y (the exponent 1 4 is not particularly important, any sufficiently small number would suffice). Note again that d GH wr (X, Y ) only depends on the weighted rooted isometry classes of X and Y .
From now on, we think of ∆ GH wr and d GH wr as being defined on T wr × T wr . Parts (i) and (ii) of the following result are analogous to Lemma 2.3 of (EW06), part (iv) is analogous to Proposition 2.4 of (EW06), part (v) is a re-statement of Lemma 2.6 of (EW06), and part (vi) is analogous to Theorem 2.5 of (EW06). The results in (EW06) are for R-trees with weights but without roots, but the addition of roots does not present any new difficulties (cf. the passage from R-trees without weights or roots to R-trees without weights but with roots in Section 2.3 of (EPW06)). The space T in part (iv) is the collection of isometry classes of compact R-trees (without weights or roots) and we refer the reader to Section 2.1 of (EPW06) for the definition of the associated Gromov-Hausdorff distance d GH .
Proposition 2.2. (i) The map ∆ GH wr has the properties: (ii) The map d GH wr is a metric on T wr .
(iii) For all X, Y ∈ T wr , (vi) The metric space (T wr , d GH wr ) is complete and separable.
We note that an extensive study of spaces of metric spaces equipped with measures is given in (Stu06a; Stu06b), and the theory of weak convergence for random variables taking values in such spaces is developed in (GPW06).

Trees and continuous paths
Definition 3.1. The space of positive excursion paths is the set Ω ++ ⊂ Ω + ⊂ C(R + , R + ) given by The following result is a slight generalization of Lemma 3.1 in (EW06). The latter result was for the special case of R-trees constructed from positive excursion paths rather than general positive bridge paths. The proof goes through unchanged.
We root a R-tree (T f , d T f ) coming from a positive bridge path in f ∈ Ω 1 + by taking the root to be the point corresponding to 0 ∈ [0, 1] under the quotient map. We equip (T f , d T f ) with the weight ν T f given by the push-forward of Lebesgue measure on [0, 1] by the quotient map.
For a positive bridge path f ∈ Ω 1 + , we identify the length measure µ T f on the associated compact R-tree (T f , d T f ) as follows (the discussion is essentially the same as that in Section 3 of (EW06) which considered R-trees coming from positive excursion paths).

A path transformation connecting reflected Brownian bridge and Brownian excursion
Write P + for the law of the standard Brownian bridge reflected at 0 that goes from 0 at time 0 to 0 at time 1. Write P ++ for the law of standard Brownian excursion. Thus, P + is a probability measure on Ω 1 + and P ++ is a probability measure on Ω 1 ++ . We show in this section how various computations for P + can be reduced to computations for P ++ using a result of Bertoin and Pitman.
Denote byΩ + the subset of Ω + consisting of functions f with the properties: • the closed set {t ∈ [0, ] : f (t) = 0} is perfect (that is, has no isolated points) and has Lebesgue measure zero; • for 0 ≤ t ≤ , • the function t → L(t; f ) is continuous; • the set of points of increase of the function t → L(t; f ) coincides with {t ∈ [0, ] : f (t) = 0}.
Note that if f ∈Ω + , then L(·; f ) is not identically 0 (indeed, L(·; f ) has 0 as a point of increase). Of course, P + (Ω 1 + ) = 1. For f ∈ Ω + , set For f ∈ Ω + and u ∈ [0, ], set The following result is elementary and we leave the proof to the reader. The construction it describes is illustrated in Figure 5. Then, e ∈ Ω ++ and f = K ← (·; e, U (f )).
The next result, which is Lemma 3.3 of (BP94), says that under P + the path-valued random variable f → K → (·; f ) + f has law P ++ , the random variable f → U (f ) is uniformly distributed on [0, 1], and these two random variables are independent. In order to apply Proposition 4.2, we need to understand for a fixed positive bridge path f ∈Ω 1 + how the measure m f of (2) or (5) is related to the analogous measure for the associated positive excursion path K → (·; f ) + f ∈ Ω 1 ++ .    s(e, s, a) .
The function t → L(t; f ) is constant on each of the sets A k , and so the same is true of the function t → K → (t; f ). Write c k for this constant. The function ξ maps B k bijectively into C k and the restriction of ξ to B k is the translation (s, a) → (s, a + c k ).
Therefore, ξ is a measure-preserving bijection between the set Γ(f ) equipped with the measure ds ⊗ da and the set Γ * (e, U (f )) equipped with the measure ds ⊗ da.  As explained in the Introduction, the dynamics of the process we wish to construct involves "picking" a point v in a rooted compact R-tree (T, d T , ρ T ) according to the length measure µ T and then re-rooting the subtree above v (that is, the subtree consisting of points of x ∈ T such that v ∈ [ρ T , x[) at a new location w. When T = T f for some f ∈ Ω 1 + , this re-rooting of a subtree corresponds to a rearrangement of f by relocating an excursion of f above some level. We introduce the following notation to describe such rearrangements -see also Figure 7.  (t +s(f, s, a) − s(f, s, a)), t > s(f, s, a).
That is,f s,a is the sub-excursion of f that straddles (s, a) shifted to start at position 0 at time 0, andf s,a is f with the sub-excursion that straddles (s, a) excised and the resulting gap closed up.

Standard Brownian excursion and length measure
We first recall a result (Proposition 5.2 below) that appears as Corollary 5.2 in (EW06). It says that if we pick an excursion e according to the standard excursion distribution P ++ and then pick a point (s, a) ∈ Γ(e) according to the σ-finite measure ds ⊗ da 1 s(e, s, a) − s(e, s, a) so that the time point s(e, s, a) is picked according to the σ-finite measure m e , then the following objects are independent: (a) the length of the excursion above level a that straddles time s; (b) the excursion obtained by taking the excursion above level a that straddles time s, turning it (by a shift of axes) into an excursionê s,a above level zero starting at time zero, and then Brownian re-scalingê s,a to produce an excursion of unit length; (c) the excursion obtained by taking the excursioně s,a that comes from excisingê s,a and closing up the gap, and then Brownian re-scalingě s,a to produce an excursion of unit length; (d) the starting time s(e, s, a) of the excursion above level a that straddles time s rescaled by the length ofě s,a to give a time in the interval [0, 1].

Moreover,
• the length in (a) is "distributed" according to the σ-finite measure • the unit length excursions in (b) and (c) are both distributed as standard Brownian excursions (that is, according to P ++ ); The change of variable w =Ȗ (u, v, r) gives That is,ẽ f,v is the result of taking the excursion starting and ending at times v and δ(f, v), respectively, and shifting the time and space axes to obtain an excursion that starts at position 0 at time 0. Given w , denote by f v,w ∈ Ω 1 + the path defined as follows. If w > v (so that w > δ(f, v)), then In other words, the excursion of f starting at time v is first moved so that it starts at w and then the resulting gap left between times v and δ(f, v) is closed up -see Figure 8.
That is, a starting point v of an excursion is chosen according to the measure m f corresponding to length measure µ T f on the R-tree associated with f , this excursion is then relocated so that it starts at a uniformly chosen point w ∈ [0, 1] \ [v, δ(f, v)], and finally the resulting gap is closed up. Define a measure J + on Ω 1 + × Ω 1 + by Proposition 6.3. The measure J + is symmetric.
Proof. Given e , e ∈ Ω 1 ++ , v ∈ [0, 1], and r ∈]0, 1], define e • (·; e , e , v, r) ∈ Ω 1 ++ by e • (t; e , e , v, r) That is, e • (·; e , e , v, r) is the excursion that arises from Brownian re-scaling e and e to have lengths r and 1 − r, respectively, and then inserting the re-scaled version of e into the re-scaled version of e at a position that is a fraction v of the total length of the re-scaled version of e .
By construction, the measure J + is concentrated on pairs (f , f ) ∈ Ω 1 + × Ω 1 + such that f is obtained from f by the re-location of an excursion. If we shift the starting point of this excursion in space and time to the origin to obtain an element of Ω ++ , then the σ-finite law of this shifted excursion is Informally, this is the law of the excursion under J + , but we note while J + is concentrated on pairs (f , f ) of the form (f, f v,w ) for some v, w ∈ [0, 1], the value of v and the corresponding excursioñ e f,v cannot be uniquely reconstructed from (f , f ). Arguing as in the proof of Proposition 6.3, this law is given by We need the following properties of this law.
Proposition 6.4. (i) For 0 < t ≤ 1, and hence (ii) For x > 0, and hence Proof. (i) By the remarks prior the statement of the proposition, the quantity in the first claim is and a straightforward integration shows that this has the stated value. The second claim follows by an equally straightforward integration by parts.
An integration by parts shows that the quantity in the second claim is π 2 6 , as required.

A Dirichlet form
Recall that any f ∈ Ω 1 + is associated with a R-tree (T f , d T f ) that arises as a quotient of [0, 1] under an equivalence relation defined by f . Moreover, we may equip this R-tree with the root ρ T f that is the image of 0 under the quotient and the weight ν T f that is the push-forward of Lebesgue measure on [0, 1] by the quotient map.
Definition 7.1. Define the probability measure P on T wr to be the push-forward of the probability measure P + on Ω 1 + by the map Define the measure J on T wr × T wr to be twice the push-forward of J + by the map Proposition 7.2. (i) The measure J is symmetric.
(ii) For each compact subset K ⊂ T wr and open subset U such that K ⊂ U ⊆ T wr , (iii) The function ∆ GH wr is square-integrable with respect to J, that is, Proof. (i) This is immediate from Proposition 6.3.
(ii) By construction, the measure J has the following description. Firstly, a weighted rooted compact R-tree T ∈ T wr is chosen according to P. A point v ∈ T is chosen according to the length measure µ T and another point w ∈ T is chosen according to the renormalization of the weight ν T outside of the subtree S T ,v of points "above" v (that is, of points x such that v belongs to the segment [ρ T , x[). The subtree S T ,v is then pruned off and re-attached at w to form a new R-tree T . More formally, the R-tree T can be identified as the set T equipped with new metric d T given by x, y ∈ S T ,v , d(x, y), x, y ∈ T \ S T ,v , d(x, v) + d(w, y), x ∈ S T ,v , y ∈ T \ S T ,v , d(y, v) + d(w, x), y ∈ S T ,v , x ∈ T \ S T ,v .
With this identification, ρ T = ρ T and ν T = ν T .
Moreover, it is immediate that d P (f * ν T , ν T ) ≤ ε, and so f ∈ F ε T ,T . Note also that S T ,w = S T ,v as sets, and ν T (S T ,w ) = ν T (S T ,v ), and so a similar argument shows that the map g : T → T defined by g(x) := x, x ∈ T \ S T ,w , w, x ∈ S T ,w belongs to F ε T ,T . Thus, ∆ GH wr (T , T ) ≤ ε as required.
Now let K and U be as in the statement of part (ii). The result is trivial if K = ∅, so we assume that K = ∅. Since T wr \ U and K are disjoint closed sets and K is compact, we have that From what we have just observed, if 0 < ε < c, then, by Proposition 6.4, 1 max(ẽ f,v ) > ε/2 1 ζ ẽ f,v > ε < ∞, as required.
(iii) By an argument similar to that in part (ii) and Proposition 6.4, we have Here, as usual, L 2 (T wr , P) is equipped with the inner product (f, g) P := P(dx) f (x)g(x). Part (i) (respectively, parts (ii) and (iii)) of the following result may be proved in the same manner as Lemma 7.1 (respectively, Lemma 7.2 and Theorem 7.3) of (EW06), with Proposition 7.2 above playing the role played in (EW06) by Lemma 6.2 of that paper. Our setting is slightly different, in that we are working with weighted rooted compact R-trees rather than just weighted compact R-trees, but this doesn't require any significant changes in the arguments. In particular, the analogues of Lemmas 7.5, 7.6 and 7.7 of (EW06) go through quite straightforwardly to establish the tightness property required for part (iii) to hold.
(ii) The set L is a vector lattice and an algebra, and L ⊆ D * (E). Hence, if D(E) denotes the closure of L in D * (E)), then (E, D(E)) is also a Dirichlet form.
(iii) There is a recurrent P-symmetric Hunt process X = (X t , P T ) on T wr with Dirichlet form (E, D(E)).