Gradient flows in metric random walk spaces

Recently, motivated by problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes, there has been an increasing interest in the research of nonlocal partial differential equations. In the last years and with these problems in mind, we have studied some gradient flows in the general framework of a metric random walk space, that is, a Polish metric space (X, d) together with a probability measure assigned to each x∈X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in X$$\end{document}, which encode the jumps of a Markov process. In this way, we have unified into a broad framework the study of partial differential equations in weighted discrete graphs and in other nonlocal models of interest. Our aim here is to provide a summary of the results that we have obtained for the heat flow and the total variational flow in metric random walk spaces. Moreover, some of our results on other problems related to the diffusion operators involved in such processes are also included, like the ones for evolution problems of p-Laplacian type with nonhomogeneous Neumann boundary conditions.


Introduction
The digital world has brought with it many different kinds of data of increasing size and complexity. Indeed, modern devices allow us to easily obtain images of higher resolution, as well as to collect data on internet searches, healthcare analytics, social networks, geographic information systems or business informatics. The study and treatment of these big data is of great interest and value. To this aim, weighted discrete graphs provide the most natural and flexible workspace in which to represent the data. For this purpose, a vertex represents a data point and each edge is weighted according to an appropriately chosen measure of "similarity" between the corresponding vertices. Historically, the main tools for the study of graphs came from combinatorial graph theory. However, following the implementation of the graph Laplacian in the development of spectral clustering in the seventies, the theory of partial differential equations on graphs has obtained important results in this field. This has prompted a big surge in the research of nonlocal partial differential equations. Moreover, interest has been further bolstered by the study of problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes among other problems. Some references on these topics are given along the survey, see also [11,21,27,30,31,33,39,42,53,[57][58][59].
In the last years and with these problems in mind, we have studied some gradient flows in the general framework of a metric random walk space, that is, a Polish metric space (X , d) together with a probability measure m x assigned to each x ∈ X that encode the jumps of a Markov process. In particular, we have studied, in this framework, the heat flow, the total variational flow, and evolutions problems of Leray-Lions type with different types of nonhomogeneous boundary conditions. In doing so, we have been able to unify into a broad framework the study of these problems in weighted discrete graphs and other nonlocal models of interest. Specifically, together with the existence and uniqueness of solutions to the aforementioned problems, a wide variety of their properties have been studied (some of which are listed in the contents section), as well as the nonlocal diffusions operators involved in them. This survey is mainly based on the results that we have obtained in [46,47] (see also the more recent work [55]) and [48]. Related to the above problems, we have also studied (see [49]) the (BV , L p )-decomposition, p = 1 and p = 2, of functions in metric random walk spaces.
Let us shortly describe the contents of the paper. To start with, in Sect. 2 we introduce the general framework of a metric random walk space and we give important examples. Section 3 is devoted to the study of the heat flow. In our context, associated with the random walk m = (m x ) x∈X , the Laplace operator m is defined as Assuming that there exists a measure ν satisfying a reversibility condition with respect to the random walk, the operator − m generates in L 2 (X , ν) a Markovian semigroup (e t m ) t≥0 (Theorem 3.1) called the heat flow on the metric random walk space. Thanks to the generality of our framework the results that we obtain are applicable, for example, to the heat flow on graphs or on nonlocal models in R N associated to a nonsingular kernel. Moreover, we introduce a stability property for the random walk, called m-connectedness, which allows us, for example, to characterise the infinite speed of propagation of the heat flow (Theorem 3.8) and the ergodicity of the invariant measure associated with the random walk. Furthermore, the behaviour of the semigroup (e t m ) t≥0 as t → ∞ is of great importance in many applications; in this regard, and with the help of a Poincaré inequality, we obtain rates of convergence. Finally, we study the relation between this Poincaré inequality and the Bakry-Émery curvature condition.
In Sect. 4 we study the total variation flow. For this purpose, we introduce the 1-Laplacian operator associated with a metric random walk space, as well as the notions of perimeter and mean curvature for subsets of a metric random walk space. In doing so, we generalize results obtained in [44,45] for the particular case of R N with a nonsingular kernel as well as some results in graph theory. We then proceed to prove existence and uniqueness of solutions of the total variation flow in metric random walk spaces and to study its asymptotic behaviour with the help of some Poincaré type inequalities.
One motivation for the study of the 1-Laplacian operator comes from spectral clustering. Partitioning data into sensible groups is a fundamental problem in machine learning, computer science, statistics and science in general. In these fields, it is usual to face large amounts of empirical data, and getting a first impression of these data by identifying groups with similar properties is proved to be very useful. One of the most popular approaches to this problem is to find the best balanced cut of a graph representing the data, such as the Cheeger ratio cut [19]. Consider a finite weighted connected graph G = (V , E), where V = {x 1 , . . . , x n } is the set of vertices (or nodes) and E the set of edges, which are weighted by a function w ji = w i j ≥ 0, (i, j) ∈ E. The degree of the vertex x i is denoted by d i := n j=1 w i j , i = 1, . . . , n. In this context, the Cheeger cut value of a partition {S, S c } (S c := V \ S) of V is defined as and vol(S) is the volume of S, defined as vol(S) := i∈S d i . Furthermore, is called the Cheeger constant, and a partition {S, S c } of V is called a Cheeger cut of G if h(G) = C(S). Unfortunately, the Cheeger minimization problem of computing h(G) is NPhard [36,56]. However, it turns out that h(G) can be approximated by the second eigenvalue λ 2 of the graph Laplacian thanks to the following Cheeger inequality [20]: This motivates the spectral clustering method [43], which, in its simplest form, thresholds the second eigenvalue of the graph Laplacian to get an approximation to the Cheeger constant and, moreover, to a Cheeger cut. In order to achieve a better approximation than the one provided by the classical spectral clustering method, a spectral clustering based on the graph p-Laplacian was developed in [14], where it is showed that the second eigenvalue of the graph p-Laplacian tends to the Cheeger constant h(G) as p → 1 + . In [56] the idea was further developed by directly considering the variational characterization of the Cheeger constant h(G) The subdifferential of the energy functional | · | T V is the 1-Laplacian in graphs 1 . Using the nonlinear eigenvalue problem λ sign(u) ∈ 1 u, the theory of 1-Spectral Clustering is developed in [16][17][18]36]. In [46], we obtained a generalization, in the framework of metric random walk spaces, of the Cheeger inequality (1.1) and of the variational characterization of the Cheeger constant (1.2). Moreover, in Sect. 4 we introduce the concepts of Cheeger and calibrable sets in metric random walk spaces and characterise the calibrability of a set by using the 1-Laplacian operator. Furthermore, we study the eigenvalue problem of the 1-Laplacian and relate it to the optimal Cheeger cut problem. These results apply, in particular, to locally finite weighted connected discrete graphs, complementing the results given in [16][17][18]36].
Finally, in Sect. 5 we study p-Laplacian type evolution problems like the one given in the following reference model: where ⊂ X and ∂ m = {x ∈ X \ : m x ( ) > 0} is the m-boundary of . This reference model can be regarded as the nonlocal counterpart to the classical evolution problem where U is a bounded smooth domain in R n , and η is the outer normal vector to ∂U .
Nonlocal diffusion problems of p-Laplacian type with homogeneous Neumann boundary conditions have been studied in nonlocal models in R N associated to a non-singular kernel (see, for example, [4,5]) and also in weighted discrete graphs (see, for example, the work of Hafiene et al. [35]) with the following formulation: Another interesting approach is proposed by Dipierro et al. in [25] for the particular case of the fractional Laplacian diffusion (although the idea can be used for other kernels) with the following Neumann boundary condition, that we rewrite in the context of metric random walk spaces, (1. 6) or, alternatively, if one prefers a normalized boundary condition with respect to the underlying probability measure induced by the jump process under consideration, Therefore, in this latter case and as remarked in [25], when a particle of mass u(x) exits towards a point x ∈ ∂ m , a mass u(x) − ϕ(x) immediately comes back into according to the distribution 1 A similar probabilistic interpretation can be given for the Neumann boundary condition (1.5) but involving all of ∪ ∂ m . Anyhow, observe that the formulations (1.5) and (1.6) have an important difference in their definition regarding the domain of integration. In our work we study Problem (1.3) with the nonhomogeneous Neumann boundary conditions of Gunzburger-Lehoucq type and also of Dipierro-Ros-Oton-Valdinoci type.

Metric random walk spaces
Let (X , d) be a Polish metric space equipped with its Borel σ -algebra. Every measure considered in this survey is defined on this σ -algebra.
A random walk m on X is a family of probability measures (m x ) x∈X on X depending measurably on x, i.e., for any Borel set A of X and any Borel set B of R, the set {x ∈ X : m x (A) ∈ B} is Borel. When dealing with optimal transport problems we will further assume that each measure m x has finite first moment (see [52]).
Consequently, if ν is an invariant measure with respect to m and f ∈ L 1 (X , ν), it holds that The measure ν is said to be reversible for m if, moreover, the following detailed balance condition holds: i.e., for all bounded Borel functions f defined on X × X Note that the reversibility condition is stronger than the invariance condition. Of course, if ν(X ) < +∞ then ν can, and will, be normalized to a probability measure.
As mentioned by Ollivier in [52], a geometer may think of m x as a replacement for the notion of ball around x, while in probabilistic terms we can rather think of this data as defining a Markov chain whose transition probability from x to y in n steps is where m * 0 x = δ x . Of course, [X , d, m * n ] is also a metric random walk space. Moreover, if ν is invariant (reversible) for m, then ν is also invariant (reversible) for m * n .
We now give some well-known examples of metric random walk spaces which will aid in illustrating the generality of this abstract setting. In particular, Markov chains serve as paradigmatic examples that capture many of the properties of this general setting that we will encounter during this study. Applying Fubini's Theorem it is easy to see that the Lebesgue measure L N is a reversible (thus invariant) measure for this random walk.
Observe that, if we assume that in R N we have an homogeneous population and J (x − y) is thought of as the probability distribution of jumping from location x to location y, then, for a Borel set A in R N , m J x (A) is measuring how many individuals who started at x are arriving at A in one jump. The same ideas are applicable to the countable spaces given in the following two examples.
(2) Let K : X × X → R be a Markov kernel on a countable space X , i.e., Then, for is a metric random walk space for any metric d on X . Moreover, in Markov chain theory terminology, a measure π on X satisfying x∈X π(x) = 1 and is called a stationary probability measure (or steady state) on X . This is equivalent to the definition of invariant probability measure for the metric random walk space [X , d, m K ]. In general, the existence of such a stationary probability measure on X is not guaranteed (see, for instance, [51, Example 1.7.11]). However, for irreducible and positive recurrent Markov chains (see, for example, [37] or [51]) there exists a unique stationary probability measure. Furthermore, a stationary probability measure π is said to be reversible for K if the following detailed balance equation holds: By Tonelli's Theorem for series, this balance condition is equivalent to the one given in (2.1) for ν = π: (3) Consider a locally finite weighted discrete graph is defined as the number n of edges in the path. Then, G = (V (G), E(G)) is said to be connected if, for any two vertices x, y ∈ V , there is a path connecting x and y, that is, a path {x k } n k=0 such that x 0 = x and x n = y.
is connected, define the graph distance d G (x, y) between any two distinct vertices x, y as the minimum of the lengths of the paths connecting x and y. Note that this metric is independent of the weights. We will always assume that the graphs we work with are connected.
For x ∈ V (G) we define the weighted degree at the vertex x as Note that, by definition of locally finite graph, the sets N G (x) are finite. When w xy = 1 for every x ∼ y, d x coincides with the degree of the vertex x in a graph, that is, the number of edges containing vertex x.
For each x ∈ V (G) we define the following probability measure is a metric random walk space and it is not difficult to see that the measure ν G defined as is a reversible measure for this random walk. Given a locally finite weighted discrete graph G = (V (G), E(G)), there is a natural definition of a Markov chain on the vertices. We define the Markov kernel We have that m G and m K G define the same random walk. If ν G (V (G)) is finite, the unique reversible probability measure is given by (4) From a metric measure space (X , d, μ) we can obtain a metric random walk space, the so called -step random walk associated to μ, as follows. Assume that balls in X have finite measure and that Supp(μ) = X . Given > 0, the -step random walk on X starting at x ∈ X , consists in randomly jumping in the ball of radius centered at x with probability proportional to μ; namely ) .
If balls of the same radius have the same volume, then μ is a reversible measure for the metric random walk space [X , d, m μ, ].
(5) Given a metric random walk space [X , d, m] with reversible measure ν, and given a is a metric random walk space and it easy to see that ν is reversible for m . In particular, if is a closed and bounded subset of R N , we obtain the metric random walk space See Example 3.5 to understand how we may take advantage of this random walk.
From this point onwards, when dealing with a metric random walk space [X , d, m], we will assume that there exists an invariant and reversible measure for the random walk, which we will always denote by ν. Then, for simplicity, we will denote the metric random walk space by [X , d, m, ν]. Furthermore, we assume that the measure space (X , ν) is σ -finite.

The heat flow
Let [X , d, m, ν] be a metric random walk. For a function u : X → R we define its nonlocal gradient ∇u : and for a function z : The averaging operator on [X , d, m] (see, for example, [52]) is defined as when this expression has sense, and the Laplace operator Note that The invariance of ν is equivalent to the following property: In the case of the metric random walk space associated to a locally finite weighted discrete graph G (see Example 2.2), the above operator is the graph Laplacian studied by many authors (see e.g. [6,9,26,38]).
If the invariant measure ν is reversible, the following integration by parts formula is straightforward: In L 2 (X , ν) we consider the symmetric form given by Definition 3. 2 We denote e t m := T m t and say that {e t m : t ≥ 0} is the heat flow on the metric random walk space [X , d, m, ν].
By the Hille-Yosida exponential formula we have that As a consequence of (3.1), if ν(X ) < +∞, we have that the semigroup (e t m ) t≥0 conserves the mass. In fact, and, therefore, Associated with E m we define the energy functional We denote Note that for f ∈ D(H m ), we have

Remark 3.3
The functional H m is convex, closed and lower semi-continuous in L 2 (X , ν), and it is not difficult to see that ∂H m = − m . Consequently, − m is a maximal monotone operator in L 2 (X , ν) (see [13]). We can also consider the heat flow in then A is an m-completely accretive operator in L 1 (X , ν) [10]. Therefore, A generates a C 0 -semigroup (S(t)) t≥0 in L 1 (X , ν) (see [10,23] In the case that ν(X ) < ∞, we have that S(t) is an extension to L 1 (X , ν) of the heat flow e t m in L 2 (X , ν), that we will denote equally.
which is an homogeneous Neumann problem for the m-heat equation. See [5] for a comprehensive study of this problem in the case m = m J , . In Sect. 5 we will consider other types of Neumann problems.

Infinite speed of propagation and ergodicity
In this section we study the infinite speed of propagation of the heat flow (e t m ) t≥0 , that is, we study the conditions under which We will see that this property is equivalent to a connectedness property of the space, to the ergodicity of the m-Laplacian m , and to the ergodicity of the measure ν.
Let [X , d, m] be a metric random walk space with invariant measure ν. For a ν measurable set D, we set This is equivalent to requiring that, for every Borel set D ⊂ X with ν(D) > 0 and ν-a.e.
For locally finite weighted connected graphs we have the following result.
The next result establishes a relation between the m-connectedness of a metric random walk space and the infinite speed of propagation of the heat flow.
This concept is also equivalent to the m-connectedness of the metric random walk space: Consequently, m-connectedness is not a new concept; nevertheless, our aim with its introduction is to regard it as a kind of intrinsic geometric property of the metric random walk space. At the beginning of Sect. 4.1 we give another characterization of m-connectedness which justifies the choice of this terminology.

Functional inequalities
Suppose that ν is a probability measure. We denote the mean value of f ∈ L 1 (X , ν), or the expected value of f , by For f ∈ L 2 (X , ν), we denote the variance of f by

Definition 3.13
The spectral gap of − m is defined as  4) or, equivalently, where gap(− m ) is the best constant in the Poincaré inequality.
With such an inequality at hand and with a similar proof to the one done in the continuous setting (see, for instance, [8]), one can prove that if gap(− m ) > 0 then e t m u 0 converges to ν(u 0 ) with exponential rate λ = gap(− m ). In fact, we have: Theorem 3. 15 The following statements are equivalent: (i) There exists λ > 0 such that We finish this subsection by relating Poincaré inequalities with the Bakry-Émery curvature-dimension condition for the random walk. Observe that, since E m admits a Carré du champ (see [8]) defined by for all x ∈ X and f , g ∈ L 2 (X , ν), we can study the Bakry-Émery curvature-dimension condition in this context. Furthermore, we will address its relation with the spectral gap.
According to Bakry and Émery [7], we define the Ricci curvature operator 2 by iterating : This is well defined for f , g ∈ L 2 (X , ν). Moreover, we write, for f ∈ L 2 (X , ν), It is easy to see that Consequently, Furthermore, by (3.1) and (3.5), we get and, therefore, Definition 3. 16 The operator m satisfies the Bakry-Émery curvature-dimension condition The constant n is the dimension of the operator m , and K is said to be the lower bound of the Ricci curvature of the operator m . If there exists K ∈ R such that The use of the Bakry-Émery curvature-dimension condition as a possible definition of Ricci curvature in Markov chains was first considered in 1998 by Schmuckenschlager [54]. This concept of Ricci curvature in the discrete setting has been frequently used following the work by Lin and Yau [41] (see [40] and the references therein).
Integrating the Bakry-Émery curvature-dimension condition B E(K , n) we have Now, by (3.5) and (3.6), this inequality can be rewritten as or, equivalently, as Similarly, integrating the Bakry-Émery curvature-dimension condition B E(K , ∞) we have We call the inequalities (3.7) and (3.8) the integrated Bakry-Émery curvature-dimension conditions. Theorem 3.17 [46] Let [X , d, m, ν] be a metric random walk space. Assume that m is ergodic. Then, Consequently, on account of Theorem 3.17, we have the following result.
Theorem 3.18 [46] Let [X , d, m, ν] be a metric random walk with ν a probability measure. Assume that m is ergodic. Then,

Therefore, if m satisfies the Bakry-Émery curvature-dimension condition B E(K , n) with
In the case that m satisfies the Bakry-Émery curvature-dimension condition B E(K , ∞) with K > 0, we have In [46] there is an example that shows that, in general, the integrated Bakry-Émery curvature-dimension condition B E(K , n) with K > 0 does not imply the Bakry-Émery curvature-dimension condition B E(K , n) with K > 0.

Perimeter, curvature and total variation
Let [X , d, m, ν] be a metric random walk space. We define the m-interaction between two ν-measurable subsets A and B of X as Let us now introduce the concept of perimeter of a set in this general setting.

Definition 4.1
We define the m-perimeter of a ν-measurable subset E ⊂ X as It is easy to see that

Moreover, if ν(E) < +∞, we have
We may motivate this notion of perimeter as follows. Consequently, we have that In the case of Example 2.2(1), this concept is the same as the one studied in [45] and whose origin goes back to [12,15,24].
In the same spirit, we define a nonlocal notion of the mean curvature of the boundary of a set.

Definition 4.2
Let E ⊂ X be a ν-measurable set. For a point x ∈ X we define the m-mean curvature of ∂ E at x as Observe that Finally, associated to the random walk m = (m x ) and the invariant measure ν, we define the following space of bounded nonlocal variation functions and, as in the local case, we have that

Example 4.3 Let
the metric random walk space associated to a finite weighted discrete graph G. Then, which coincides with the anisotropic total variation defined in [32].
As in the local case, we have the following coarea formula relating the total variation of a function with the perimeter of its superlevel sets.

The 1-Laplacian and the TVF
Let [X , d, m, ν] be an m-connected metric random walk space.
As motivation, consider the formal nonlocal evolution equation In order to study the Cauchy problem associated to this equation, we will see in Theorem 4.8 that we can rewrite it as the gradient flow in L 2 (X , ν) of the functional F m : which is convex and lower semi-continuous. Following the method used in [3], the subdifferential of the functional F m is characterized as follows.
Theorem 4.5 [48] Let u ∈ L 2 (X , ν) and v ∈ L 2 (X , ν). The following assertions are equivalent: and The m-1-Laplacian is defined via this subdifferential in the following manner.
Definition 4. 6 We define in L 2 (X , ν) the multivalued operator m 1 by (u, v) ∈ m 1 if, and only if, −v ∈ ∂F m (u). As usual, we will write v ∈ m 1 u for (u, v) ∈ m 1 .
Chang in [16] and Hein and Bühler in [36] defined a similar operator in the particular case of finite graphs.

Example 4.7 Let [V (G), d G , (m G
x )] be the metric random walk space given in Example 2.2(3) with invariant measure ν G . By Theorem 4.5, we have that (u, v) ∈ m G 1 if, and only if, there As a consequence of Theorem 4.5, we can give the following existence and uniqueness result for the Cauchy problem which is a rewrite of the formal expression (4.1).

Asymptotic behaviour of the TVF and Poincaré type inequalities
Let [X , d, m, ν] be an m-connected metric random walk space. Definition 4. 10 We say that [X , d, m, ν] satisfies a ( p, q)-Poincaré inequality ( p, q ∈ [1, +∞[) if there exists a constant c > 0 such that, for any u ∈ L q (X , ν), or, equivalently, there exists a λ > 0 such that When [X , d, m, ν] satisfies a ( p, 1)-Poincaré inequality, we will say that [X , d, m, ν] satisfies a p-Poincaré inequality and write If a Poincaré inequality is satisfied then we obtain the following result on the asymptotic behaviour of the total variation flow.
The following result provides sufficient conditions for a Poincaré inequality to be satisfied by a metric random walk space. Two examples of metric random walk spaces in which a 1-Poincaré inequality is not satisfied are given in [48].
Theorem 4.12 [48] Suppose that ν is a probability measure and m x ν for all x ∈ X .
Let us see that, when [X , d, m, ν] satisfies a 2-Poincaré inequality, the total variation flow reaches the steady state in finite time.
Therefore, if we define the extinction time as then, under the conditions of Theorem 4.15, we have that, for u 0 ∈ L 2 (X , ν), .
To obtain a lower bound on the extinction time, we introduce the following norm which, in the continuous setting, was introduced in [50]. Given a function f ∈ L 2 (X , ν), we define Theorem 4.16 [48] Let u 0 ∈ L 2 (X , ν). If T * (u 0 ) < ∞ then

m-Cheeger and m-calibrable sets
Let [X , d, m, ν] be an m-connected metric random walk space.

Definition 4.17 Given a set
⊂ X with 0 < ν( ) < ν(X ), we define the m-Cheeger constant of as A ν-measurable set E ⊂ achieving the infimum in (4.6) is called an m-Cheeger set of . Furthermore, we say that is m-calibrable if it is an m-Cheeger set of itself, that is, if For ease of notation, we will denote for any ν-measurable set ⊂ X with 0 < ν( ) < ν(X ). It is well known (see [29]) that the classical Cheeger constant for a bounded smooth domain , is an optimal Poincaré constant, namely, it coincides with the first eigenvalue of the 1-Laplacian: In order to get a nonlocal version of this result, we introduce the following constant. For ⊂ X with 0 < ν( ) < ν(X ), we define Theorem 4.18 [48] Let ⊂ X with 0 < ν( ) < ν(X ). Then, Let us recall that, in the local case, a set ⊂ R N is called calibrable if The following characterization of convex calibrable sets is proved in [1].

Theorem 4.19 [1]
Given a bounded convex set ⊂ R N of class C 1,1 , the following assertions are equivalent: The next result is the nonlocal version of the fact that (a) is equivalent to (b) in Theorem 4.19.
We also have: Proposition 4.22 [48] Let ⊂ X be a ν-measurable set with 0 < ν( ) < ν(X ). Then, The above result relates the m-calibrability with the m-mean curvature, since it can be rewritten as Therefore, this is the nonlocal version of one of the implications in the equivalence between (a) and (c) in Theorem 4.19. However, the converse of Proposition 4.22 is not true in general, an example is given in [44] (see also [45]) for [R 3 , d, m J ], with d the Euclidean distance and J = 1 . An example of a graph for which the converse of Proposition 4.22 is not true is also given.

The eigenvalue problem for the 1-Laplacian
In this section we introduce the eigenvalue problem associated with the 1-Laplacian m 1 and its relation with the Cheeger minimization problem. For the particular case of finite weighted discrete graphs where the weights are either 0 or 1 this problem was first studied by Hein and Bühler ([36]); a more complete study was subsequently performed by Chang in [16].
Let [X , d, m, ν] be an m-connected metric random walk space.
We have the following relation between m-calibrable sets and m-eigenpairs of m 1 .
Theorem 4.24 [48] Let ⊂ X be a ν-measurable set with 0 < ν( ) < ν(X ). We have: In [48] we give an example showing that, in Theorem 4.24, the reverse implications of (i) and (ii) are false in general.

The m-Cheeger constant
Let [X , d, m] be an m-connected metric random walk space with ν a probability measure.
Assuming that ν(X ) = 1 is not a loss of generality since, for ν(X ) < +∞, we may work with 1 ν(X ) ν. Observe that the ratio λ m D = P m (D) ν(D) remains unchanged if we normalize the measure, and the same is true for the m-eigenvalues of the 1-Laplacian.
For a locally finite weighted discrete graph G = (V (G), E(G)) the Cheeger constant is defined as In [20] (see also [9]), the following relation between the Cheeger constant and the first positive eigenvalue λ 1 (G) of the graph Laplacian m G is proved: In this general context we define the following concept, which is consistent with the above definition on graphs. The above infimum is not attained in general, see, for instance, [48,Example 6.21].
We will now give a variational characterization of the m-Cheeger constant which generalizes the one obtained in [56] for the particular case of finite graphs. Recall that, given a function u : X → R, μ ∈ R is a median of u with respect to ν if We denote by med ν (u) the set of all medians of u.

Theorem 4.30
The following statements are equivalent: Recall that, for finite graphs, it is well known that the first non-zero eigenvalue coincides with the Cheeger constant (see [16]), that is, In our context we have: Theorem 4.31 [48] If λ = 0 is an m-eigenvalue of m 1 then h m (X ) ≤ λ.
In the next result we see that if the infimum in (4.8) is attained then h m (X ) is an meigenvalue of m 1 . Theorem 4.32 [48] Let be a ν-measurable subset of X such that 0 < ν( ) ≤ is an m-eigenpair of m 1 .
As a consequence of Proposition 4.25 and Theorem 4.32 we have the following result.
In [48] we give an example showing that (4.11) is not true in general.

Evolution problems of Leray-Lions type with nonhomogeneous Neumann boundary conditions
Let [X , d, m, ν] be an m-connected metric random walk space. We assume that m x ν for all x ∈ X . Definition 5.1 Given a ν-measurable set ⊂ X , we define its m-boundary as and its m-closure as We will assume that ν( m ) < +∞ in what follows.

Nonlocal Leray-Lions operators
For 1 < p < +∞, let us consider a function a p : e (x, y) ∈ X × X and for all r ; (a p (x, y, r ) − a p (x, y, s))(r − s) > 0 for ν ⊗ m x -a.e. (x, y) and for all r = s; there exist constants c, C > 0 such that |a p (x, y, r )| ≤ C 1 + |r | p−1 for ν ⊗ m x -a.e. (x, y) ∈ X × X and for all r , and a p (x, y, r )r ≥ c|r | p for ν ⊗ m x -a.e. (x, y) ∈ X × X and for all r .
An example of a function a p satisfying the above assumptions is being ϕ : X → R a ν-measurable function satisfying 0 < c ≤ ϕ ≤ C where c and C are constants. In particular, if ϕ = 1, we have that is the p-Laplacian operator on the metric random walk space.

Neumann boundary operators
We define the nonlocal Neumann boundary operator (of Gunzburger-Lehoucq type) by We also define the nonlocal Neumann boundary operator (of Dipierro-Ros-Oton-Valdinoci type) as These type of nonlocal boundary conditions where introduced, for the linear case, in [34] to develop a nonlocal vector calculus, and in [25] for diffusion with the fractional Laplacian.
For each of these Neumann boundary operators our main goal is to study the evolution problem 2) j = 1, 2, and the following associated Neumann problem Observe that div m a p is a kind of Leray-Lions operator for the random walk m. On account of (5.1), we have that div m a p u( Moreover, by the reversibility of ν with respect to m, we have that m x (X \ m ) = 0 for ν-a.e. x ∈ . Indeed, Consequently, div m a p u(x) = m a p (x, y, u(y) − u(x))dm x (y) for every x ∈ . Let The following integration by parts formula follows by the reversibility of ν with respect to m.

Neumann boundary conditions of Gunzburger-Lehoucq type
In this subsection we study the problem x ∈ .

(5.4)
We will assume that the following Poincaré type inequality holds: there exists a constant λ > 0 such that, for any u ∈ L p ( m , ν), It is shown in [48] (see also [4,5]) that, under rather general conditions, there are metric random walk spaces satisfying this kind of inequality.
The main tool to study Problem (5.4) is Nonlinear Semigroup Theory. In order to use this theory, we define the following operator on L 1 ( , ν) × L 1 ( , ν) associated to the problem. Observe that the space of definition is L 1 ( , ν) and not L 1 ( m , ν).
Consequently, B m a p ,ϕ is m-completely accretive in L p ( , ν).
If u 0 ∈ D(B m a p ,ϕ ) then the mild-solution is a strong solution.
If u 0 ∈ D(A m a p ,ϕ ), then the mild-solution is a strong solution. In particular, if u 0 ∈ L ∞ ( , ν), Problem (5.5) has a unique strong solution. For p ≥ 2 this is true for data in L p−1 ( , ν).
In [55] doubly nonlocal diffusion problems of Leray-Lions type with further nonlinearities on the boundary have also been studied.