Convergence of Brownian Motions on Metric Measure Spaces Under Riemannian Curvature-Dimension Conditions

We show that the pointed measured Gromov convergence of the underlying spaces implies (or under some condition, is equivalent to) the weak convergence of Brownian motions under Riemannian Curvature-Dimension (RCD) conditions.


Motivation
The aim of this paper is to characterize a probabilistic convergence of Brownian motions in terms of a geometric convergence of the underlying spaces. Our main results show that the pointed measured Gromov (pmG) convergence of the underlying spaces implies (or under some condition, is equivalent to) the weak convergence of Brownian motions under Riemannian Curvature-Dimension (RCD) conditions for the underlying spaces.
Let us consider the following motivating example: let a sequence of Riemannian manifolds {M n } n∈N converges to a (possibly non-smooth) metric measure space in the Gromov-Hausdorff (GH) sense. Let (B n , P n ) be a Brownian motions on each M n . Noting that (B n , P n ) can be determined only by the underlying geometric structure of the Riemannian manifolds M n , an important question is whether (Q) a sequence of Brownian motions on Riemannian manifolds also converges weakly to the Brownian motion on the GH-limit space.
This question does not make sense, however, without additional assumptions because there is a gap between the geometric and probabilistic convergences: the weak convergence of Brownian motions clearly involves the first-order differentiable structure of the underlying spaces although the GH convergence never sees any information of differentiable structures. Indeed, we have examples whereby the limit process is no more a diffusion process (see (ii) and (iii) in Remark 1.10).
In this paper, adopting as an additional assumption the uniform lower Ricci curvature bound of M n , we can answer (Q) affirmatively, which is an application of the main results in this paper. To be more precise, we obtain the equivalence between these geometric/probabilistic convergences in the framework of metric measure spaces under the synthetic lower Ricci curvature bound (called RCD in this paper), which is a substantially more general condition than the one assumed in (Q).
Let us explain the background issues in more detail. Generally, the GH-limit spaces of Riemannian manifolds with lower Ricci curvature bounds (called Ricci limit spaces) are so singular that they are not necessarily even topological manifolds and may have a dense singular set (see Example 4.3). However, they still have "Riemannian-like" structures and similar properties to smooth Riemannian manifolds with lower Ricci curvature bounds, which have been investigated initially by Cheeger-Colding [17,18,19].
The RCD condition, which was introduced by Ambrosio-Gigli-Savaré [5,2], Ambrosio-Mondino-Savaré [7] and Erbar-Kuwada-Sturm [22], is a proper generalization of the notion of lower Ricci curvature bounds to non-smooth spaces including Ricci limit spaces. It is known that RCD spaces include various finite-and infinite-dimensional non-smooth spaces, not only Ricci limit spaces, but also infinite-dimensional spaces such as Hilbert spaces with log-concave measures (related to various stochastic partial differential equations) (see further details in Section 4).
By recent developments of analysis on metric measure spaces, we can construct Brownian motions on RCD spaces by using a certain quadratic form, what is called Cheeger energy. This is a generalization of Dirichlet energy on smooth manifolds and induces a quasiregular strongly local conservative symmetric Dirichlet form (Ambrosio-Gigli-Savaré [4,5], Ambrosio-Gigli-Mondino-Rajala [2]), which is determined only by the underlying metric measure structure.
One of the important problems for Brownian motions on these non-smooth spaces is to characterize the weak convergence of Brownian motions in terms of some geometrical convergence of the underlying spaces, which we call the stability of Brownian motions. The significance of the stability can be explained from several different perspectives. From the standpoint of limit theorems of stochastic processes, the stability is interpreted as a geometric characterization of invariance principles for Brownian motions in the sense that Brownian motions on limit spaces are approximated by Brownian motions on converging spaces. From the viewpoint of "well-definedness", the stability also enables us to verfiy that Brownian motions in limit spaces are "well-defined" in the sense that Brownian motions intrinsically defined by Cheeger energies on limit spaces coincide with limit processes of Brownian motions on approximating spaces. From the perspective that Brownian motions are considered as "a map" assigning laws of diffusions (i.e., probability measures on path spaces) to each metric measure space, the stability reveals the interesting fact that this map is continuous with respect to the corresponding topologies (e.g., GH-topology of metric measure spaces/weak topology of probability measures on path spaces), which is one ideal aspect of Brownian motions but has not been focused on so much until now.
The main contribution of this paper is to prove the stability of Brownian motions in the general framework of RCD spaces, whereby various singular/infinite-dimensional spaces are included. Moreover, we show several equivalences of the weak convergence of Brownian motions and the pmG convergence of the underlying spaces. For references to other investigations regarding the stability problem, see the historical remarks (Section 1.3 below).

Main Results
In this paper, we always consider pointed metric measure (p.m.m.) spaces X = (X, d, m, x) whereby (X, d) is a complete separable geodesic metric space with nonnegative and nonzero Borel measure m which is finite on all bounded sets, and x is a fixed point in supp[m]. (1.1) For main theorems, we assume the following condition: The notion of CD(K, ∞) spaces was introduced by Sturm [62] and Lott-Villani [44], and the notion of RCD(K, ∞) spaces was introduced by Ambrosio-Gigli-Savaré [5] and Ambrosio-Gigli-Mondino-Rajala [2]. The CD(K, ∞) condition is a generalization of Ricci curvature bounded from below by K to metric measure spaces in terms of the K-convexity of the entropy on the Wasserstein spaces. Furthermore RCD(K, ∞) condition means the CD(K, ∞) and that the Cheeger energy is quadratic. We will explain the precise definition in Subsection 2.4. RCD spaces admit the GH limit spaces of Riemannian manifolds with lower Ricci curvature bounds, and also admit Alexandrov spaces (metric spaces satisfying a generalized notion of "sectional curvature≥ K") (Petrunin [52] and Zhang-Zhu [68]), cone spaces and warped product spaces (Ketterer [37,38]), and quotient spaces (Galaz-García-Kell-Mondino-Sosa [28]). Moreover, not only finite-dimensional spaces, but also several infinite-dimensional spaces related to stochastic partial differential equations are included such as Hilbert spaces with log-concave measures (Ambrosio-Savaré-Zambotti [8]). Under Assumption 1.1, we can always take constants c 1 , c 2 > 0 dependent only on K satisfying the following volume growth estimate (see [62,Theorem 4.24]) Here we mean B r (x n ) := {x ∈ X n : d(x, x n ) < r}. Taking C > c 2 , we set a weighted measure m n as follows: z n :=ˆX n e −Cd 2 n (x,xn) dm n (x), and m n := Under Assumption 1.1, the Cheeger energy Ch n on X n = (X n , d n , m n , x n ) (see Subsection 2.4.2) induces a quasi-regular conservative symmetric strongly local Dirichlet form, and there exists a conservative symmetric Markov process on X n , which is unique at quasi-every starting point in X n (see [5] for the case of m(X) = 1 and see [2] for the σ-finite case). Among equivalent Markov processes associated with Ch n , we choose a specific Markov process ({P x n } x∈Xn , {B n t } t≥0 ) corresponding to a modification {P n t } t≥0 of the heat semigroup {H n t } t≥0 associated with Ch n (See Section 3.1). The Markov process ({P x n } x∈Xn , {B n t } t≥0 ) is a diffusion at any starting point x ∈ X n \ N whereby N is a set of capacity zero. We call ({P x n } x∈Xn , {B n t } t≥0 ) Brownian motion on X n . The following main theorem states that the weak convergence of the Brownian motions can be characterized by the pmG convergence of the underlying spaces under Assumption 1.1 (we will give the definition of the pmG convergence in Subsection 2.3). Theorem 1.2 Suppose that Assumption 1.1 holds. Then the following (i) and (ii) are equivalent:

(ii) (Weak Convergence of the Laws of Brownian Motions)
There exist a complete separable metric space (X, d) and isometric embeddings ι n : Here (ι n (B n ), P mn n ) means the law of the embedded Brownian motion ι n (B n ) with the initial distribution m n and P(C([0, ∞); X)) denotes the set of all Borel probability measures on the continuous path space C([0, ∞); X). Remark 1.3 Several remarks for Theorem 1.2 are given below.
(i) The RCD(K, ∞) condition is stable under the pmG convergence (see [29,Theorem 7.2]), and therefore the limit space X ∞ also satisfies the RCD(K, ∞) condition so that the Brownian motion can be defined also on the limit space X ∞ .
In statement (ii) in Theorem 1.2, the initial distribution is absolutely continuous with respect to the reference measure m n . It is natural in the next step to ask how the case of the dirac measure δ xn is, which means the Brownian motions start at the point x n . We introduce several conditions below: (A) For any n ∈ N, m n (X n ) < ∞.
(B) For any r > 0 and any t > 0, sup n∈N p n (t, x n , ·) ∞,Br(xn) < ∞, whereby p n (t, x, y) is the density of the transition probability p n (t, x, dy) of {P n t } t≥0 with respect to the reference measure m n , and · ∞,Br(xn) means the essential supremum on the ball B r (x n ). Now we state the second main result. Theorem 1.4 Suppose that Assumption 1.1 holds. If, moreover, either (A), or (B) holds, then (i) (thus also (ii)) in Theorem 1.2 implies the following (iii) >0 : There exist a complete separable metric space (X, d) and isometric embeddings ι n : X n → X (n ∈ N) so that it holds that is a diffusion at every starting point x ∈ X n (without any exceptional set N ) for each n ∈ N, then (1.4) holds in P(C([0, ∞); X)). Remark 1.5 Several remarks for Theorem 1.4 are given below.
(i) Note that, in (1.4), the time interval of the path space is not [0, ∞) but (0, ∞). This is due to the ambiguity in the starting points of Markov processes associated with Dirichlet forms. However, since Brownian motions on RCD(K, ∞) spaces are conservative and the heat kernel p n (ε, x, dy) is absolutely continuous with respect to m n for every x and every ε > 0, each law (ι n (B n · ), P x n ) lives on C((0, ∞); X) for every x ∈ X n (not only quasi-every x)(see Section 3.1). Therefore statement (iii) >0 in Theorem 1.4 makes sense without quasi-every ambiguity of starting points x n if we restrict the time interval [0, ∞) to (0, ∞).
(ii) Concerning the last statement in Theorem 1.4, if X is an RCD * (K, N ) space for K ∈ R and 1 < N < ∞, then Brownian motions are Feller processes, under which Brownian motions can be constructed uniquely at every starting point (see Proposition 3.2). Thus ({P x n } x∈Xn , {B n t } t≥0 ) is a diffusion at every starting point x ∈ X.
(iii) Condition (B) is satisfied for any RCD * (K, N ) spaces according to the Gaussian heat kernel estimate by Jiang-Li-Zhang [34].
By Theorem 1.4, (ii) and (iii) in Remark 1.5, we have the following corollary. The notion of RCD * (K, N ) spaces was first introduced by Erbar-Kuwada-Sturm [22] and Ambrosio-Mondino-Savaré [7]. It is the class of metric measure spaces satisfying the reduced curvature-dimension condition CD * (K, N ) with the Cheeger energies being quadratic. Roughly speaking, RCD * (K, N ) condition is a generalization of Ricci curvature bounded from below by K and dimension bounded above by N to metric measure spaces, which is stronger than the RCD(K, ∞) condition. Next we consider the converse implication that the weak convergence of Brownian motions induces the pmG convergence of the underlying spaces. Define p n (t, x, x) = p n (t/2, x, ·)p n (t/2, x, ·) 2 2 , which can be defined for every x ∈ X. Let us consider the following condition: there exists t * > 0 and a constant M so that sup n∈N p n (t * , x n , x n ) < M < ∞. (1.5) Note that, since p n (t, x, x) is non-increasing function in t, if we find the time t * satisfying (1.5), then for any t > t * , the estimate (1.5) holds. We also note that if X n satisfies the RCD * (K, N ) with 1 < N < ∞ and {x n } n∈N is bounded, then (1.5) is satisfied for some constant M and t * because of the local Gaussian heat kernel estimate by Jiang-Li-Zhang [34]. Let diam(X n ) denote the diameter of X n : diam(X n ) := sup x,y∈Xn d n (x, y). We now state the following theorem: Theorem 1.7 Suppose that Assumption 1.1 and condition (1.5) hold. If, moreover, either K > 0, or sup n∈N diam(X n ) < D holds for some 0 < D < ∞, then (iii) >0 in Theorem 1.4 implies (i) and (ii) in Theorem 1.2 (therefore all statements (i), (ii) and (iii) >0 are equivalent).
As a corollary of Theorem 1.4, (ii) in Remark 1.5, Corollary 1.6 and Theorem 1.7, we give the following statement in which all statements (i), (ii), (iii) >0 , and (iii) ≥0 are equivalent under the RCD * (K, N ) condition and a uniform diameter bound. There exist a compact metric space (X, d) and isometric embeddings ι n : X n → X (n ∈ N) so that

Remark 1.9
We give several remarks for Corollary 1.8.
(i) The RCD * (K, N ) condition is stable under the pmG convergence (see [22]), and therefore the limit space X ∞ also satisfies the RCD * (K, N ) condition so that the Brownian motion can be defined at every starting point also on the limit space X ∞ .
(ii) The pmG convergence (see Definition 2.1) is equivalent to the pointed measured Gromov-Hausdorff convergence under the assumption in Corollary 1.8 (see [29,Theorem 3.33]).

Historical Remarks
Remark 1.10 Several historical remarks are given below.
(i) In Ambrosio-Savaré-Zambotti [8, Theorem 1.5], they investigated the weak convergence of Brownian motions on a fixed Hilbert space (as an ambient space) with varying logconcave measures and norms, which is a specific case of RCD(0, ∞) spaces. Their metrics d n are not necessarily isometric to the metric d in the ambient space, but each d n is equivalent to d. In the case that each d n is isometric to d, our results (Theorem 1.2 and 1.4) can be seen as a generalization of their result [8, Theorem 1.5] to general RCD(K, ∞) spaces.
(ii) In Ogura [48], under the condition of uniform upper bounds for heat kernels (not necessarily lower bound of Ricci curvatures) and the Kasue-Kumura (KK) spectral convergence, he studied the weak convergence of the laws of time-discretized Brownian motions on weighted compact Riemannian manifolds. The KK spectral convergence roughly means a uniform convergence of heat kernels and stronger than the mGH convergence. In his case, the Ricci curvature is not necessarily bounded from below and the limit process may be a jump process ( [48, 4.6]). The time-discretization is one possible approach for a convergence of stochastic processes on varying spaces, while we adopt in this paper a different approach, i.e., embedding into one common metric space X.
(iii) If we do not assume RCD conditions for a sequence of the underlying metric measure spaces, then limit processes are not necessarily diffusions. In Ogura-Tomisaki-Tuchiya [49], they considered a sequence of Euclidean spaces (R d , · 2 ) with certain underlying measures µ n whereby {(R d , · 2 , µ n )} n∈N do not necessarily satisfy RCD conditions.
They showed that diffusion processes on R d associated with the corresponding local Dirichlet forms converge to jump processes (or generally jump-diffusion processes) corresponding to certain non-local Dirichlet forms.
(iv) In Freidlin-Wentzell [25,26] and Albeverio-Kusuoka [1] (see also references therein), diffusion processes associated with SDEs on thin tubes in R d were studied. When thin tubes shrink to a spider graph, diffusion processes converge weakly to a one-dimensional diffusion on this spider graph. Their setting does not satisfy the RCD condition since spider graphs branch at points of conjunctions but RCD spaces are essentially nonbranching (see [55,Theorem 1.1]).
(v) In Athreya-Löhr-Winter [9], the weak convergence of certain Markov processes on treelike spaces was studied. When tree-like spaces converge in Gromov-vague sense, the corresponding processes also converge weakly. Their tree-like spaces admit 0-hyporbolic spaces, which are not necessarily included in RCD spaces.
(vi) In Suzuki [65], the author investigated the weak convergence of continuous stochastic processes on metric spaces converging in the Lipschitz distance. The Lipschitz convergence is stronger than the measured Gromov convergence (see [31, Section 3.C]).
Finally we list related studies not mentioned in Remark 1.10. In Suzuki [66], the author studied the weak convergence of non-symmetric diffusion processes on RCD spaces as a next step of the current paper. In Li [41,42], she studied a convergence of random ODE/SDE on manifolds. In Stroock-Varadhan [57], Stroock-Zheng [58] and Burdzy-Chen [16], approximations of diffusion processes on R d by discrete Markov chains on (1/n)Z d were investigated. In Bass-Kumagai-Uemura [12] and Chen-Kim-Kumagai [20], they studied approximations of jump processes on proper metric spaces by Markov chains on discrete graphs. Approximations of Markov processes on ultra-metric spaces were explored in Suzuki [64]. In Pinsky [53], he studied approximations of Brownian motions on Riemannian manifolds by random walks, while the case of sub-Riemannian manifolds was investigated by Gordina and Laetsch [30]. In Croydon-Hambly-Kumagai [21], in which it was assumed that a sequence of resistance forms converges with respect to the GH-vague topology and satisfies a uniform volume doubling condition, they showed the weak convergence of corresponding Brownian motions and local times. There are many studies about scaling limits of random processes on random environments (see, e.g., Kumagai [39] and references therein).

Organization of the Paper
The paper is structured as follows: First, the notation is fixed and preliminary facts are recalled in Section 2 (no new results are included), namely: basic notations and basic definitions (Subsection 2.1); L 2 -Wasserstein distance (Subsection 2.2); pmG convergence (Subsection 2.3); RCD(K, ∞) and RCD * (K, N ) spaces (Subsection 2.4); L 2 -convergence of the heat semigroup (Subsection 2.5). In Section 3, we state several properties about Brownian motions on RCD spaces. In Section 4 we present examples in which Assumptions 1.1 and the assumption in Corollary 1.8 are satisfied. These examples consist of weighted Riemannian manifolds and their pmG limit spaces, Alexandrov spaces, and Hlibert spaces with log-concave probability measures. In Section 5, we give the proof of Theorem 1.2 . In Section 6, we show the proof of Theorem 1.4. In Section 7, we prove Theorem 1.7. Finally, in Section 8, we prove Corollary 1.8.

Notation
Let N = {0, 1, 2, ...} and N := N ∪ {∞} denote the set of natural numbers and the set of natural numbers with {∞} respectively. For a complete separable metric space (X, d), we denote by B r (x) = {y ∈ X : d(x, y) < r} the open ball centered at x ∈ X with radius r > 0. By using B(X), we mean the family of all Borel sets in (X, d); and by B b (X), the set of real-valued bounded Borel-measurable functions on X. Let C(X) be the set of real-valued continuous functions on X, while C b (X), C ∞ (X), C 0 (X) and C bs (X) denote the subsets of C(X) consisting of bounded functions, functions vanishing at infinity, functions with compact support, and bounded functions with bounded support, respectively. Let Lip(X) and Lip b (X) denote the set of Lipschitz continuous functions, and the set of bounded Lipschitz continuous functions, respectively. For f ∈ Lip(X), we denote by Lip X (f ) the global Lipschitz constant of f . The set P(X) denotes all Borel probability measures on X. The set of continuous In particular, if d(γa,γ b ) |b−a| can be replaced by 1, we say that γ is unit-speed. A metric space X is called geodesic if for any two points x, y ∈ X, there exists a minimal geodesic {γ t } t∈[0,1] connecting x and y. Let
Let (X, d) be a complete separable metric space. Let P p (X) be the subset of P(X) consisting of all Borel probability measures µ on X with finite p-th moment: We equip P p (X) with the transportation distance W p , called L p -Wasserstein distance, defined as follows: is called an optimal coupling if q attains the infimum in the equality (2.1). It is known that, for any µ, ν, there always exists an optimal coupling q of µ and ν (e.g., [67, §4]). It is also known that (P p (X), W p ) is a complete separable geodesic metric space for 1 < p < ∞ if (X, d) is a complete separable geodesic metric space (e.g., [67, Theorem 6.18]).

Pointed Measured Gromov Convergence
In this subsection, we recall the definition of pmG convergence introduced in Gigli-Mondino-Savaré [29].
for any bounded continuous function f : X → R with bounded support.

Remark 2.2
We would like to remark on the pmG convergence in Definition 2.1.
(i) In general, the pmG convergence is strictly weaker than the pointed measured Gromov

RCD Spaces
In this subsection, we recall the definition of the RCD(K, ∞) condition, following [29]. We also recall several properties satisfied on RCD(K, ∞) spaces.

Relative Entropy
In this subsection, we recall the definition of the relative entropy functional Ent m : P 2 (X) → R := R ∪ {+∞}: Here dµ/dm denotes the Radon-Nikodym derivative. Let us write D(Ent m ) := {µ ∈ P 2 (X) : Ent m (µ) < ∞}. Although m might not be a probability measure, the entropy Ent m is well-defined and lower-semicontinuous thanks to condition (1.2). Indeed, by recalling (1.3): we can check that, for any ρm = µ ∈ D(Ent m ) with ρ = dµ dm , it holds that µ = zρe Cd 2 (·,x) m. Therefore we obtain which implies that Ent m is well-defined and lower-semicontinuous with respect to W 2 -topology. See [29, §4.1.1] for more details.

Cheeger Energy
In this subsection, we recall the Cheeger energy Ch on (X, d, m, x). For f ∈ Lip(X), the local Lipschitz constant |∇f | : X → R is defined as follows: Then we now recall the definition of Cheeger energy: (see [2,4]) If Ch(f ) < ∞, then the Cheeger energy can be written as an integral form by minimal weak upper gradient |∇f | w (see [4,2]):

Definition 2.3
The CD(K, ∞)/RCD(K, ∞) conditions are defined as follows: We say that (X, d, m) satisfies the curvature-dimension condition CD(K, connecting µ 0 and µ 1 so that We say that (X, d, m) satisfies the Riemannian curvature-dimension condition RCD(K, ∞) if the following two conditions hold: (ii-a) CD(K, ∞) (ii-b) the infinitesimal Hilbertianity, that is, the Cheeger energy Ch is a quadratic form: It is known that CD(K, ∞)/RCD(K, ∞) conditions are stable under the pmG convergence. Let {X n } n∈N be a sequence of RCD(K, ∞) spaces. If X n converges to X ∞ in the pmG sense, then the limit space X ∞ is also an RCD(K, ∞) space.

W 2 -gradient Flow of Relative Entropy
In this subsection, following [2,4], we recall the heat flow on the L 2 -Wasserstein space (P 2 (X), W 2 ), which is constructed by the gradient flow of the relative entropy functional. We also recall the stability of the heat flows under the pmG convergence.
Under the CD(K, ∞) condition, it is known that the gradient flow µ t = H t µ of the relative entropy exists uniquely for any initial measure µ ∈ D(Ent m ) and for any t ≥ 0 ( [2,4]). We call {H t } t≥0 heat flow on P 2 (X).
Theorem 2.5 (Theorem 7.7 in [29]) (Stability of heat flows) Let {X n = (X n , d n , m n , x n )} n∈N be a sequence of RCD(K, ∞) spaces converging to then the solution µ n t = H n t (µ n ) of the heat flow starting at µ n converges to the limit one Here ι n is an embedding X n → X corresponding to the pmG convergence (see Definition 2.1).

L 2 -gradient Flow of Cheeger Energy
We now recall the L 2 -gradient flow of Cheeger energy by Hilbertian theory of gradient flows (see e.g., [6]). We also recall the important fact that the heat flow in the previous section and the L 2 -gradient flow of Cheeger energy in this section coincide under the CD(K, ∞) condition.
Here the subdifferential ∂ − Ch of convex analysis is the multi-valued operator in L 2 (X; m) defined at all elements of the domain of the Cheeger energy f ∈ W 1,2 (X, d, m) by the family of inequalities The map H t : f 0 → f t is uniquely determined by (2.4) and define a contraction semigroup (not necessarily linear) on L 2 (X; m). The flow f 0 → f t = H t f is called L 2 -gradient flow of the Cheeger energy, and the semigroup {H t } t≥0 is called heat semigroup.
We recall that the L 2 -gradient flows of Cheeger energies and the W 2 -gradient flow of entropies are equivalent under the CD(K, ∞) condition.
For each θ ∈ [0, ∞), we define the following functions We define the following functions: for t ∈ [0, 1], Let P ∞ (X, d, m) be the subset of P 2 (X) consisting of µ which is absolutely continuous with respect to m and has bounded support.
(ii) A metric measure space (X, d, m) is said to satisfy the Riemannian curvature-dimension condition RCD * (K, N ) if the following two conditions hold: (ii-a) CD * (K, N ) (ii-b) the infinitesimal Hilbertianity, that is the Cheeger energy Ch is a quadratic form: Remark 2.8 The RCD * (K, N ) condition is stronger than the RCD(K, ∞) condition. If X is an RCD * (K, N ) space, then X is locally compact by the local volume doubling property according to Bishop-Gromov inequality [22, Proposition 3.6] (see also [63,Corollary 2.4]).
The RCD * (K, N ) condition is stable under the pmG convergence.  N ) spaces. If X n converges to X ∞ in the pmG sense, then X ∞ is also an RCD * (K, N ) space.

L 2 -convergence of Heat Semigroups Under the PmG Convergence
In Gigli-Mondino-Savaré [29], they introduced L 2 -convergences on varying metric measure spaces and showed a convergence of heat semigroups in this sense under the pmG convergence of the underlying spaces with the RCD(K, ∞) condition. We recall their results briefly. spaces. Assume that (X n , d n , m n , x n ) converges to (X ∞ , d ∞ , m ∞ , x ∞ ) in the pmG sense. Let (X, d) be a complete separable metric space and ι n : supp[m n ] → X be isometries as in Definition 2.1. We identify (X n , d n , m n ) with (ι n (X n ), d, ι n# m n ) and omit ι n .
(i) We say that u n ∈ L 2 (X, m n ) converges weakly to u ∞ ∈ L 2 (X, m ∞ ) if the following hold: sup n∈Nˆ| u n | 2 dm n < ∞ andˆφu n dm n →ˆφu ∞ dm ∞ ∀φ ∈ C bs (X), whereby recall that C bs (X) denotes the set of bounded continuous functions with bounded support.
(ii) We say that u n ∈ L 2 (X, m n ) converges strongly to u ∞ ∈ L 2 (X, m ∞ ) if u n converges weakly to u ∞ and the following holds: Let {H n t } t≥0 be the L 2 (X, m n )-semigroup corresponding to the Cheeger energy Ch n . Then the following theorem states that {H n t } t≥0 convergence strongly in L 2 under the pmG convergence of the underlying spaces. Then, for any u n ∈ L 2 (X, m n ) converging strongly to u ∞ ∈ L 2 (X, m ∞ ), we have, for any t > 0 H n t u n converges strongly to H ∞ t u ∞ in the sense of Definition 2.10.
Note that, in [29, Theorem 6.11], the above Theorem 2.11 was stated without the condition of the infinitesimal Hilbertian.

Brownian Motions on RCD(K, ∞) Spaces
Let (X, d, m) satisfy the RCD(K, ∞) condition. Let δ x denote the unit mass at x ∈ X, and define a kernel p(t, x, dy) by the action of the heat flow (see Subsection 2.4.4) Then we have that (see [5,2]) p(t, x, dy) is absolutely continuous with respect to m for any t > 0, and we denote the density by p(t, x, y). By [5, Theorem 6.1] and [2] (for the case of σfinite reference measures), the density p(t, x, y) is symmetric in x and y, and satisfies the Chapman-Kolmogorov formula. Moreover, the following action of semigroup {P t } t≥0 is a version of the linear heat semigroup {H t } t≥0 defined as the gradient flow of the Cheeger energy Ch (see Subsection 2.4.5) for any f ∈ L 2 (X; m). Furthermore P t is an extension of H t to a continuous contraction semigroup in Here E x Q denotes the expectation with respect to Q x and N is a set of zero-capacity with respect to the Cheeger energy (Ch, W 1,2 (X, d, m)). Such systems of Markov processes are unique up to zero-capacity sets.
Since {P t } t≥0 is a version of {H t } t≥0 , the systems of Markov process ({P x } x∈X , {B t } t≥0 ) and ({Q x } x∈X , {B t } t≥0 ) coincide except on zero-capacity sets. In this paper, we adopt ({P x } x∈X , {B t } t≥0 ) for representing a system of Brownian motions.
By the same argument of [8, Proof of (c) in Theorem 1.2], we have that P x (C((0, ∞))) = 1 for every x ∈ X (not only quasi-every x ∈ X).

(3.2)
Note that by the conservativeness and the strong locality of the Dirichlet form (E, F), considering that P x and Q x coincide for quasi-every x, we know that P x (C([0, ∞))) = 1 for quasi-every x ∈ X. However, the property (3.2) is not necessarily true in general, and this property is thanks to the absolute continuity of the heat kernel p(t, x, dy) with respect to m for any t > 0.
Remark 3.1 The diffusion process defined above is conventionally called Brownian motion ( [5]), but this may indicate other diffusion processes than the standard Brownian motion in some situations. For instance, when we take (X, d, m) = (R d , · 2 , 1 (2π) d/2 exp{− 1 2 x 2 2 }dx) whereby dx denotes the Lebesgue measure, and · 2 denotes the Euclidean distance. Then (X, d, m) satisfies RCD(0, ∞) and the diffusion induced by the Cheeger energy coincides with what is known as the Ornstein-Uhlenbeck process, which is different from the standard Brownian motion on R d .

Brownian Motions on RCD * (K, N ) Spaces
In this subsection, we show the Feller property of the heat semigroup on RCD * (K, N ) spaces. Therefore Brownian motions can be constructed uniquely at every starting point.

Proposition 3.2
Under the RCD * (K, N ) condition, the heat semigroup {H t } t≥0 has a Feller modification. That is, there exists a semigroup {P t } t≥0 so that P t f = H t f m-a.e. for any f ∈ L 2 (X, m) and any t > 0 and the following conditions hold:

Remark 3.3
The following proof is the result of a private communication with Prof. Kazuhiro Kuwae. Although the proof might be already known in some literature, we could not find good references and we give the proof for the sake of reader's convenience.
Proof. By [5, (iii) in Theorem 6.1], there exists a semigroup {P t } t≥0 which is a modification of {H t } t≥0 so that P t f ∈ Lip b (X) if f ∈ L ∞ (X, m). Before checking (F-1) and (F-2), we first give a heat kernel estimate. By [34, Theorem 1.2], we have the following Gaussian heat kernel estimate: there exist positive constants C i = C i (N, K) for i = 1, 2, 3 depending only on N, K so that for all x, y ∈ X and 0 < t. Here the heat kernel p(t, x, y) means the integral kernel of the heat semigroup P t f (x) =´X f p(t, x, y)m(dy) for t > 0. We now show condition (F-1). We already know P t f ∈ C b (X), so it suffices to show that P t f vanishes at infinity for f ∈ C ∞ (X) and t > 0, which is an easy consequence of (3.3) as follows: we may assume that f is compactly supported since every element in C ∞ (X) can be approximated by elements in C 0 (X) with respect to the uniform norm. Let K ⊃ supp[f ] be a compact set. By (3.3) and inf y∈K m(B √ t (y)) > 0 (by the lower semi-continuity of m(B r (x)) in x), we see that, for any ε > 0, there exists a compact set K ⊂ X so that |P t f (x)| ≤ˆK p(t, x, y)|f (y)|m(dy) Thus we have proved (F-1). Now we prove (F-2). We may assume f ∈ C 0 (X). Let K ⊃ supp[f ] be a compact set. For given ε > 0, take δ > 0 so that |f (x) − f (y)| < ε whenever d(x, y) < δ in x, y ∈ K. By the Gaussian estimate (3.3), we can choose a positive number T so that p(t, x, y) < ε for any 0 < t < T , and for any x ∈ X and y ∈ K satisfying d(x, y) ≥ δ. Then we have that, for any Thus we have shown that (F-2) holds.

Examples
In this section, several specific examples satisfying Assumption 1.1 or the assumption in Corollary 1.8 are given. In the first subsection, we explain weighted Riemannian manifolds whose weighted Ricci curvature is bounded below, and their pmG limit spaces. In the second subsection, we explain Alexandrov spaces, which are a generalization of the lower sectional curvature bound to metric spaces. In the third subsection, we give Hilbert spaces with logconcave probability measures.

Weighted Riemanniam Manifolds and pmG Limit Spaces
Let {(M n , g n , w n , x n )} n∈N be a sequence of pointed complete and connected weighted Ndimensional Riemannian manifolds whose weight satisfies w n = e −Vn for a twice continuously differentiable function V n ∈ C 2 (M n ). We write the corresponding pointed metric measure space M n = (M n , d gn , w n Vol n , x n ) whereby d gn denotes the distance function associated with the Riemannian metric g n ; Vol n denotes the Riemannian volume measure; and x n ∈ M n is a fixed point. Let the weighted Ricci curvature Ric Mn of M n be bounded from below by K: there exists K ∈ R so that whereby Ric gn means the Ricci curvature of (M n , g n ) and ∇ 2 means the Hessian. Then M n satisfies RCD(K, ∞) spaces ( [56,62]). Even when V n : M n → R is not in C 2 (M n ), if Ric gn ≥ K and V n : M n → R is K -convex (see [62]), then M n satisfies RCD(K + K , ∞). If, moreover, V n : M n → R is (K , N )-convex (see [22]), then M n satisfies RCD * (K + K , N + N ). The Brownian motion on M n is a Markov process whose infinitesimal generator A n is whereby ∆ Mn is the Laplace-Beltrami operator on M n . If M n satisfying RCD(K, ∞) (or, RCD * (K, N )) converges to a metric measure space M ∞ in pmG sense, then the limit space M ∞ satisfies RCD(K, ∞) (or, RCD * (K, N )), respectively (see [29,22]). Thus we can apply our main results and obtain the weak convergence of the Brownian motions.
We have various singular examples appearing as the limit space. See e.g., [17,Example 8]. We give one of the simplest examples included in this framework.

Example 4.1 (Collapsing: Torus → Circle)
Let S 1 ⊂ R 2 be the unit circle. Let d S 1 be the shortest path distance on S 1 , that is, the distance between x and y is defined by the infimum over lengths of geodesics on S 1 connecting x and y. Let be the normalized Hausdorff measure on (S 1 , d S 1 ). Let T n = S 1 × S 1 be a two-dimensional flat torus with a metric d n = d S 1 ⊗ 1 n d S 1 and the normalized Hausdorff measure H n on (T n , d n ), whereby Then (T n , d n , H n ) satisfies the RCD * (0, 2) for any n ∈ N and converges to (S 1 , d S 1 , H S 1 ) in the measured Gromov sense. Thus we can apply our result (Corollary 1.8) and the weak convergence of the Brownian motions is equivalent to the pmG convergence of the underlying spaces.

Alexandrov Spaces
We explain Alexandrov spaces, which are a generalization of lower bounds of sectional curvatures to metric spaces. We refer the reader to [15] for basic theory of Alexandrov spaces. Let (X, d) be a locally compact length space. For a triple of points p, q, r ∈ X, a geodesic triangle pqr is a triplet of geodesics joining each two points. Let M N (K) be the N -dimensional complete simply connected space of constant sectional curvature K. For a geodesic triangle pqr, we denote by p q r a geodesic triangle in M 2 (K) whose corresponding edges have the same lengths as pqr.
A locally compact length space (X, d) is said to be an Alexandrov space with Curv ≥ K if for every point x ∈ X, there exists an open set U x including x so that for every geodesic triangle pqr whose edges are totally included in U x , the corresponding geodesic triangle p q r satisfies the following condition: for every point z ∈ qr and z ∈ q r with d(q, z) = d( q, z), we have d(p, z) ≥ d( p, z). Let X = (X, d, H) be an N -dimensional Alexandrov space with Curv ≥ K and H be the Hausdorff measure (see e.g., [15] for details). According to [52,68], X satisfies CD * ((N − 1)K, N ). Moreover, as was shown in [40], X satisfies the infinitesimal Hilbertian condition, and as a result, X satisfies RCD * ((N − 1)K, N ). Thus we can apply our results (Theorem 1.2, 1.4) and if a sequence of pointed Alexandrov spaces X n with Curv ≥ K converges to the limit space X ∞ in the pmG sense, then the Brownian motions on X n converge weakly to the limit Brownian motion on X ∞ . We give several examples.

Example 4.2 (Cone → Interval)
Let X n ⊂ R 3 be a cone defined by X n = {(x, y, z) ∈ R 3 : y 2 +z 2 = 1 n x, 0 ≤ x < 1}∪{(x, y, z) ∈ R 3 : y 2 + z 2 = 1 n , x = 1}. Let d n be the shortest path distance on X n and H n be the normalized Hausdorff measure on X n . Then (X n , d n , H n ) satisfies RCD * (0, 2) and converges to ([0, 1], | · |, m) in the measured Gromov sense, whereby m is a measure on [0, 1]. Thus we can apply our result (Corollary 1.8) and the weak convergence of the Brownian motions is equivalent to the pmG convergence of the underlying spaces. As a second example, we give a sequence of polygons made by the barycentric subdivision. The limit space has dense singularities. Let X = (X, d) be a polyhedron in R 3 with the shortest path metric d on X. Then we can check that X is an Alexandrov space with Curv≥ 0, which is also an RCD * (0, 2) space. For any vertices p ∈ X, let ∠(X, p) denote the sum of all inner angles of at p of faces T 's such that p is a vertex of T . Now we construct a sequence of polyhedra {M n } n∈N inductively. Let M 1 be a tetrahedron in R 3 with the barycenter o. Let M n be defined. Then we define M n+1 as follows: Take a monotone decreasing sequence {ε n } n∈N so that ε n → 0 as n → ∞ with 0 < ε n < 1 and ε := Π ∞ n=1 (1 − ε n ) > 0. We take the barycentric subdivision of M n . Keep the original vertices in M n in the same positions and move the new vertices generated by the barycentric subdivision outward along rays emanating from o so small that, for the new polyhedra M n+1 generated by the new and original vertices, we have for any vertex p ∈ M n . See [50,p. 632,Examples. (2)] for more details.
Let d n and H n be the shortest path distance and the Hausdorf measure on M n . Then there exists the Hausdorff-limit of M n = (M n , d n ), denoted by M ∞ . The limit space M ∞ is a two-dimensional Alexandorv space with nonnegative curvature. In particular, (M n , d n , H n ) converges to (M ∞ , d ∞ , H ∞ ) in the measured Gromov sense. The limit space M ∞ also satisfies the RCD * (0, 2) by the stability of RCD * (K, N ) spaces under the measured Gromov convergence (see [22]). The set of singular points in M ∞ is dense (see [50]). Since each diameter of M n is obviously uniformly bounded by the construction, we can apply our result (Corollary 1.8) and the weak convergence of the Brownian motions is equivalent to the pmG convergence of the underlying spaces.

Hilbert Space with Log-concave Measures
In this subsection, we give a specific class of RCD(0, ∞) spaces, which is a Hilbert space with log-concave measures. This subsection follows [8].
Let H be a separable Hilbert space, which would be a finite-or infinite-dimensional space, with an inner product ·, · and the corresponding norm · . A Borel probability measure γ Let K = supp[γ] and A = A(γ) be the smallest closed linear subspace containing K. We write canonically so that h 0 is the element of the minimal norm in K and H 0 is a closed linear subspace in H. Let C 1 b (A) be the set of all Φ : A → R which are bounded, continuous and Fréchet differentiable with a bounded continuous gradient ∇Φ : A → H 0 . Then, according to [8,Theorem 1.2], the following bilinear form becomes closable and the closed form becomes a symmetric quasi-regular Dirichlet form E = E · ,γ : In [8], the corresponding semigroup {P t } t≥0 associated with (E, F) satisfies EVI 0 property, which is equivalent to the RCD(0, ∞) condition of (H, · , γ) according to [5]. Let {H n = (H, · n , γ n , x n )} n∈N be a sequence of pointed Hilbert spaces with log-concave probability measures satisfying the above conditions. Then we can apply our results (Theorem 1.  Then γ n becomes a log-concave probability measure. Therefore the diffusion process associated with the Dirichlet form E n in (4.1) is a solution of the following SDE: If, V n converges to V ∞ uniformly and x n → x ∞ , then it is easy to check that γ n converges to γ ∞ weakly and (R N , · 2 , γ n , x n ) converges to (R N , · 2 , γ ∞ , x ∞ ) in the pmG sense. Thus we can apply our results (Theorem 1.2, 1.4) and the solution to SDE (4.2) on H n = (H, · n , γ n , x n ) converges weakly to the limit one on H ∞ .
(b) (SDE on Variable Convex Subsets with Variable Convex Potentials) Let H = R N with 1 < N < ∞ and U n ⊂ R N be a convex open set. We consider a convex functional V n ∈ C 1,1 (U n ) and V n ≡ +∞ on R N \ U n with´U n e −Vn dx < ∞. Take Then γ n becomes a log-concave probability measure. Therefore the diffusion process associated with the Dirichlet form E n in (4.1) is a solution of the following SDE: Here n is an inner normal vector to ∂U n and L n is a continuous monotone non-decreasing process which increases only when X t ∈ ∂U n .
If the closure U n converges to a closed convex subset U ∞ ⊂ R N in the Hausdorff sense (see e.g., [31]), x n → x ∞ and V n converges to V ∞ uniformly, then it is easy to check that γ n converges to γ ∞ weakly and (R N , · 2 , γ n , x n ) converges to (R N , · 2 , γ ∞ , x ∞ ) in the pmG sense. Thus we can apply our results (Theorem 1.2, 1.4) and the solution to (4.3) on U n converges weakly to the solution to the limit SDE on U ∞ .

Proof of Theorem 1.2
We first show the implication of (ii) =⇒ (i) in Theorem 1.2.
Proof of (ii) =⇒ (i) in Theorem 1.2. If we assume (ii), then it is obvious that the initial distributions m n converge weakly to m ∞ . Since the weak convergence of m n to m ∞ is equivalent to the convergence of m n to m ∞ in the sense of (2.2) (easy to check), we finish the proof of the implication (ii) =⇒ (i) in Theorem 1.2.
Proof of (i) =⇒ (ii) in Theorem 1.2. By Definition 2.1, there exist a complete separable metric space (X, d) and a family of isometric embeddings ι n : X n → X such that, for any bounded continuous function f : X → R with bounded support, we havê Set the notation for the laws of Brownian motions as follows: B mn n := (ι n (B n ), P mn n ), B xn n := (ι n (B n ), P xn n ).
Hereafter we identify ι n (X n ) with X n and we omit ι n for simplifying the notation.
To show the weak convergence of the Brownian motions, we have two steps. The first step is to show the weak convergence of finite-dimensional distributions, and the second is to show tightness. We first show the weak convergence of finite-dimensional distributions in the case that the initial distribution is the Dirac measure δ xn .
Proof. Since the limit Brownian motion B x∞ ∞ is conservative, it suffices to show the statement only for f 1 , f 2 , ..., f k ∈ C b (X) ∩ L 2 (X; m ∞ ). In fact, for any ε > 0 and T > 0, there exists R = R(ε, T ) so that the open ball B R (x ∞ ) satisfies If we know that E xn (f (B n t )) converges to E x∞ (f (B ∞ t )) for any f ∈ C b (X) ∩ L 2 (X; m ∞ ), then we know that Therefore, for any f 1 , ..., f k ∈ C b (X), and any small δ > 0, we can choose R > 0 large enough so that Thus we may show the proof only for Recall that we have the following equality (see Subsection 3.1): for every x ∈ X n . Here recall that {P n t } t≥0 is the semigroup defined in (3.1) by the action of the heat flow whereby P t is a modification of the heat semigroup H t and P n t f (x) can be defined for every point x ∈ X n if f ∈ C b (X) ∩ L 2 (X; m ∞ ). Since the Brownian motion ({P x n } x∈Xn , {B n t } t≥0 ) is constructed by the Kolmogorov extension theorem with the integral kernel p n (t, x, dy) of {P n t } t≥0 as in Section 3.1, the equality (5.1) holds for every point x ∈ X n By using the Markov property, for all n ∈ N, we have By [2,Theorem 7.3], P n k is bounded Lipschitz on X n whose Lipschitz constant depends only on the curvature lower-bound K.
For later arguments, we extend P n k to the whole space X (note that P n k is defined only on each X n ). The key point is to extend P n k to the whole space X preserving its Lipschitz regularity and bounds. Here H denotes the same Lipschitz constant of the original function P n k . Then P n k is a bounded Lipschitz continuous function on the whole space X with the same Lipschitz constant H and the same bound. Moreover P n k = P n k on the original domain X n . The function P n k is called McShane extension of P n k . We now return to the proof of Lemma 5.1. We have that Therefore it suffices to show (I) n → 0 and (II) n → 0 as n → ∞.
We first discuss to show (I) n → 0. Since P n Thus we have We next show that (II) n → 0. By (5.4)  On the other hand, we have that P n k converges to P ∞ k L 2 -strongly in the sense of Definition 2.10. We give a proof below. Lemma 5.3 P n k converges to P ∞ k in the L 2 -strong sense in Definition 2.10. Proof. By Theorem 2.11, the statement is true for k = 1. Assume that the statement is true when k = l. Since we have P n l+1 = P n t l+1 −t l (f (n) l+1 P n l ), by Theorem 2.11, it is sufficient to show f l+1 P n l → f l+1 P ∞ l strongly in L 2 . This is obvious to be true because P n l → P ∞ l strongly (the assumption of the induction), f l+1 ∈ C b (X) and P n l is bounded uniformly in n thanks to (5.4). Thus the statement is true for any k ∈ N.
We return to the proof of Lemma 5.1. Proof of Lemma 5.1. By using Lemma 5.3 and (5.6), it is obvious to check that whereby F | X∞ means the restriction of F into X ∞ . The R.H.S. P ∞ k of the above equality is clearly independent of choices of subsequences and thus the limit F | X∞ is independent of choices of subsequences. Thus we conclude that Now we return to show (II) n goes to zero. By (5.7), we have that Here · ∞,X∞ means the uniform norm on X ∞ . Thus we finish the proof of Lemma 5.1.
We next show the weak convergence of finite-dimensional distributions for the case that initial distributions are W 1 -convergent, which includes m n for the case of m n (X n ) = ∞.
Lemma 5.4 Let {ν n } n∈N ⊂ P(X n ) be a sequence of probability measures on X n ⊂ X converging to ν ∞ ∈ P(X ∞ ) in W 1 -distance. Then, for any k ∈ N, 0 = t 0 < t 1 < t 2 < · · · < t k < ∞ and f 1 , f 2 , ..., f k ∈ C b (X) ∩ L 2 (X; m ∞ ), the following holds: By the Kantorovich-Rubinstein duality (see e.g., [67,Theorem 5.10]), we have According to (5.4) and (5.5), we have that P n k is bounded and sup n∈N Lip( P n k ) < L < ∞ for some constant L. Thus we have that ˆX P n k dν n −ˆX P n k dν ∞ ≤ LW 1 (ν n , ν ∞ ). (5.8) Since P n k converges to P ∞ k uniformly in C b (X ∞ ) by (5.7), and ν n converges to ν ∞ in the W 1 -distance, by using (5.8), we have that Thus we have completed the proof.
We now show the weak convergence of finite-dimensional distributions for the case that initial distributions are 1 mn(Xn) m n , which corresponds to the case of m n (X n ) < ∞. Lemma 5.5 Let m n (X n ) < ∞ for any n ∈ N. Then, for any k ∈ N, 0 = t 0 < t 1 < t 2 < · · · < t k < ∞ and f 1 , f 2 , ..., f k ∈ C b (X), the following holds: Proof. Because of m n (X n ) < ∞, we have f ∈ L 2 (X, m n ) for all f ∈ C b (X) for any n ∈ N.
Since m n converges weakly to m ∞ in P(X), for any ε > 0, there exists a compact set K ⊂ X so that sup n∈N m n (K c ) < ε.
Thus, by (5.3), for any δ > 0, there exists a compact set K ⊂ X so that Take r > 0 so that K ⊂ B r (x n ) := {x ∈ X : d(x n , x) < r}. Let 1 R r denote the following function: (r < R) Then 1 R r ∈ C bs (X). Thus, by Theorem 2.11 and (5.9), for any δ > 0, there exists r > 0 so that In the fourth line, the first δ comes from using (5.9) and the second δ comes from using the tightness of the single measure m ∞ . The the middle term in the fourth line converges to zero thanks to the L 2 -strong convergence of the heat semigroup P t in the sense of Definition 2.10. Note that the total mass m n (X n ) → m ∞ (X ∞ )(≤ ∞) because of the pmG convergence. Thus we have completed the proof.
Now we show the tightness of {B mn }. For later arguments, we show the tightness for more general initial distributions ν n than m n . Lemma 5.6 Let ν n ∈ P(X n ) satisfy the following conditions: (i) ν n → ν ∞ weakly in P(X); (ii) ν n is absolutely continuous with respect to m n with dν n = φ n dm n and there exists a positive constant M so that, for any r > 0, sup n∈N φ n ∞,Br(xn) < M < ∞.
Proof. Let us denote the law of h(B n ) for h ∈ Lip b (X) as follows: It is easy to show that Lip b (X) strongly separates points in C b (X), that is, for every x and ε > 0, there exists a finite set Therefore, by [23, Corollary 3.9.2] with Lemma 5.4, the following two statements are equivalent: (i) {B νn } n∈N is tight in P(C([0, ∞), X)); (ii) {B νn,h } n∈N is tight in P(C([0, ∞), R)).
We note that, although [23,Corollary 3.9.2] gives sufficient conditions for tightness only in the càdlàg space D([0, ∞); X), since the law of each Brownian motion B mn n for n ∈ N has its support on the space of continuous paths C([0, ∞); X), the tightness in D([0, ∞); X) implies the tightness in C([0, ∞), X). See, e.g., [24,Lemma 5 in Appendix] for this point.
Since ν n converges weakly to ν ∞ in P(X), the set of the laws of the initial distributions {(h(B n 0 ), P νn n )} n∈N = {h # m n } n∈N is clearly tight in P(R). For δ > 0, let us define The desired result we would like to show is the following: lim η→0 sup n∈NˆX n L n,h η,T dν n = 0, (5.10) for any T > 0. By conditions (i) and (ii) in this lemma, for any ε > 0, there exists R > 0 so thatˆX Here P x n,r is a conservative diffusion process associated with (Ch r n , F r n ) Ch r n (f ) = 1 2ˆY r n |∇f | 2 w,Y r n dm n,r , F r n := {f ∈ L 2 (Y r n ; m n,r ) : Ch r n (f ) < ∞}.
Recall that |∇f | 2 w,Y r n means the minimal weak upper gradient on Y r n (see Subsection 2.4.2). We note that the Cheeger energy Ch r n on the closed ball Y r n is also quadratic because of [5,Theorem 4.19]. Since closed balls are not necessarily convex subset in X n , the closed ball Y r n is not necessarily an RCD(K, ∞) space. However, we can still construct the Brownian motion on Y r n since we have that (Ch r n , F r n ) is quadratic ( [5,Theorem 4.19]) and [d(x, ·)] ≤ m n,r ([5, (iv) Theorem 4.18]) for any fixed x ∈ Y r n , which imply that (Ch r n , F r n ) becomes a quasi-regular Dirichlet form by the same manner of [5, Lemma 6.7] and [8, Theorem 1.2] (see also [2, §7.2]). Here [f ] means the energy measure of the Cheeger energy (see [5, (4.21)]) and [d(x n , ·)] ≤ m n,r means d[d(x n , ·)] dm n,r (y) ≤ 1 m n,r -a.e. y ∈ Y r n .
Note that although [5, Lemma 6.7] assumed the RCD(K, ∞) condition, only the quadraticity of the Cheeger energy and [d(x n , ·)] ≤ m n,r are used to construct the Brownian motions, and the CD(K, ∞) condition is not necessary (see also [43, §4] for more detailed studies of the Cheeger energies and Brownian motions on subsets in RCD(K, ∞) spaces). We first estimate (I) n,η . By Lyons-Zheng decomposition ( [45], and see also [ T −s (r T )), P We now estimate (II) n,η . We have the following estimate: (II) n,η = P m n,R sup 0≤s,t≤T |t−s|≤h Here c > 0 is a constant independent of n. We resume to prove Theorem 1.2.
Proof of Theorem 1.2. It is easy to check that conditions (i) and (ii) in Lemma 5.6 are satisfied with ν n = m n in the both cases of m n (X n ) = ∞ and m n (X n ) < ∞. Thus we have shown the tightness. By using Lemma 5.5, we have completed the proof of (i) =⇒ (ii) in Theorem 1.2 in the case of m n (X n ) < ∞. Moreover, we can check easily that the conditions in Lemma 5.4 are satisfied with ν n = m n in the case of m n (X n ) = ∞ (see [29,Remark 4.6]). Therefore, we have completed the proof of (i) =⇒ (ii) in Theorem 1.2 in the case of m n (X n ) = ∞. We finish the proof of (i) =⇒ (ii) in Theorem 1.2.
6 Proof of Theorem 1.4 To show statement (iii) >0 , it suffices to show the following statement: for any ε > 0, (iii) ≥ε There exist a complete separable metric space (X, d) and isometric embeddings ι n : We first show the case of condition (A), that is, m n (X n ) < ∞.
Since we have already shown the weak convergence of the finite-dimensional distributions under the general RCD(K, ∞) condition for starting points x n in Lemma 5.1, it suffices to prove the tightness: In fact, we have that, for any Borel measurable functions F : C([ε, ∞), X) :→ R, E xn (F (B n ε+· )) = E xn (E B n ε (F )) =ˆX n E y F (B n · ) p n (ε, x n , dy) =ˆX n E y (F (B n · ))p n (ε, x n , y) m n (dy) = E mn (p n (ε, x n , B n 0 )F (B n · )) = E mn (p n (ε, x n , B n ε )F (B n ε+· )).
In the third line above, we used the invariance property of m n with respect to the heat semigroup {P n t } t≥0 whereby 1 m n (X n )ˆX n P n t f dm n = 1 m n (X n )ˆX n f dm n .
Note that inf n∈N m n (X n ) > 0 because m ∞ (X ∞ ) > 0 by assumption that m ∞ is non-zero, and m n (X n ) → m ∞ (X ∞ ). Since {B mn n } n∈N is tight by Lemma 5.6, by using the tightness criterion with respect to the entropy [29, Proposition 4.1], we have the tightness of {B xn n } n∈N .
Proof of Theorem 1.4 in the case of (A). By the weak convergence of the finite-dimensional distributions in Lemma 5.1, and the tightness in Lemma 6.1, we have finished the proof of Theorem 1.4 for the case (A).
7 Proof of Theorem 1.7 Proof of Theorem 1.7: The goal of the proof is to show the pmG convergence of X n to X ∞ , that is, for any f ∈ C bs (X) (recall C bs (X) means the set of bounded continuous functions with bounded supports), we havê X f dm n →ˆX f dm ∞ as n → ∞.
We first consider the case of K > 0. The case of K > 0: Let λ 1 n be the spectral gap of Ch n : Ch n (f ) f 2 L 2 (mn) : f ∈ Lip(X n ) \ {0},ˆX n f dm n = 0}.

(7.2)
Since the global Poincaré inequality holds under the CD(K, ∞) condition with a positive In the following argument, however, we give another proof for tightness of {B n } n∈N in P(C([0, ∞), X)) by using the heat kernel estimate.
Proof of (i) =⇒ (iii) ≥0 in Corollary 1.8. Since we have already shown the weak convergence of the laws of finite-dimensional distributions in Lemma 5.1 for the general RCD(K, ∞) case, what we should prove is only the tightness of the Brownian motions on C([0, ∞]; X). Lemma 8.1 {B n } n∈N is tight in P(C([0, ∞), X)).
Proof. Since x n converges to x ∞ in (X, d), the set of the laws of the initial distributions {B n 0 } n∈N = {δ xn } n∈N is clearly tight in P(X). Thus it suffices to show the following (see [13,Theorem 12.3]): for each T > 0, there exist β > 0, C > 0 and θ > 1 such that, for all n ∈ N for all x, y ∈ X n and 0 < t ≤ D 2 . Here constants C 1 , C 2 , c, ν only depend on the given constants N, K, D. Note that the constant C 3 in (3.3) can be taken as zero under sup n∈N diam(X n ) < D according to [60,61] (note that the MCP condition is satisfied under the assumption of Corollary 1.8).
Thus we have completed the proof of Corollary 1.8.