Averaging Principle and Shape Theorem for a Growth Model with Memory

TL;DR

This paper introduces a unified approach to analyze a class of self-interacting random growth models in Euclidean space, establishing an averaging principle and shape theorem that describe their large-scale behavior.

Contribution

It develops a general averaging principle and shape theorem for growth models with memory, linking the limiting shape to the invariant measure of an associated Markov chain.

Findings

Proves an averaging principle for self-interacting growth models.

Establishes a shape theorem describing the asymptotic growth shape.

Shows the limiting shape can be computed via an invariant measure.

Abstract

We present a general approach to study a class of random growth models in $n$-dimensional Euclidean space. These models are designed to capture basic growth features which are expected to manifest at the mesoscopic level for several classical self-interacting processes originally defined at the microscopic scale. It includes once-reinforced random walk with strong reinforcement, origin-excited random walk, and few others, for which the set of visited vertices is expected to form a "limiting shape". We prove an averaging principle that leads to such shape theorem. The limiting shape can be computed in terms of the invariant measure of an associated Markov chain.

Figures

Equations

$$\begin{array}[]{ccc}\includegraphics[width=85.35826pt]{once_reinforced_a=2_grid2000}&\includegraphics[width=85.35826pt]{once_reinforced_a=3_grid2000}&\includegraphics[width=99.58464pt]{once_reinforced_a=100_grid2000}\end{array}$$

$$\begin{array}[]{ccc}\includegraphics[width=85.35826pt]{cookies_propor_white_bk.pdf}&\includegraphics[width=85.35826pt]{cookies_max_white_bk.pdf}&\includegraphics[width=85.35826pt]{cookies_11_spectral_white_bk.pdf}\end{array}$$

$$\displaystyle D=\big{\{}x\in\mathbb{R}^{n}:x=\rho\theta,\,\theta\in\mathbb{S}^{n-1},\,0\leq\rho\leq r(\theta)\big{\}}\,.$$

$$Leb(r)=n^{-1}{\int }_{{\mathbb{S}}^{n-1}}r(\theta {)}^{n}d\sigma (\theta )=n^{-1}\Vert r{\Vert }_n^n.$$

$$F:\mathcal{D}(F)\subset C_{+}({\mathbb{S}}^{n-1})\times {\mathbb{R}}^n\to C_{+}({\mathbb{S}}^{n-1}),$$

$$H(r,\xi ):C_{+}({\mathbb{S}}^{n-1})\times {\mathbb{S}}^{n-1}\to {\mathbb{R}}^n$$

$$(f\star g_{\eta })(z):=\frac{1}{{\omega }_n}{\int }_{{\mathbb{S}}^{n-1}}f(\theta )g_{\eta }(⟨z,\theta ⟩)d\sigma (\theta ),z\in {\mathbb{S}}^{n-1},$$

$$\begin{array}[]{cc}\includegraphics[width=113.81102pt]{ap_id_line}&\includegraphics[width=128.0374pt]{ap_id_sphere_jet}\end{array}$$

$$y_{r,x}:={\omega }_n{\int }_{{\mathbb{S}}^{n-1}}r(\theta {)}^{n-1}F(r,x,\theta )d\sigma (\theta ).$$

$$\lim_{\epsilon,\eta\to 0}\epsilon^{-1}\mathbb{E}\big{[}Leb(r+\epsilon g_{\eta}(\langle\xi,\cdot\rangle))-Leb(r))\big{]}=y_{r,x}$$

$$\eta (ϵ,r,x):={ϵ}^{1/n}{y}_{r,x}^{-1/(n-1)},$$

$${\mathcal{F}}_{t^{-}}:=\sigma \{R_s^{ϵ},x_s^{ϵ},{\xi }_s:s\le t^{-}\},$$

$${\xi }_t\stackrel{d}{\sim }F(R_{t^{-}}^{ϵ},x_{t^{-}}^{ϵ},\cdot ).$$

$$R_t^{ϵ}(\theta )$$

$$x_t^{ϵ}$$

$$\begin{array}[]{ccc}\includegraphics[width=56.9055pt]{SphereTrueEps=1e-4N=100T=16}&\includegraphics[width=56.9055pt]{SquareTrueEps=1e-4N=100T=16}&\includegraphics[width=56.9055pt]{DiamondTrueEps=1e-4N=100T=16}\\ \includegraphics[width=85.35826pt]{SphereEps=1e-6N=100T=16drift-1}&\includegraphics[width=85.35826pt]{SquareEps=1e-6N=100T=16drift-1}&\includegraphics[width=85.35826pt]{DiamondEps=1e-6N=1000T=16drift-1}\\ \includegraphics[width=85.35826pt]{SphereEps=1e-6N=1000T=16alpha09}&\includegraphics[width=85.35826pt]{SquareEps=1e-6N=1000T=16alpha09}&\includegraphics[width=85.35826pt]{DiamondEps=1e-6N=1000T=16alpha09}\\ \end{array}$$

$$\displaystyle\big{(}\mathcal{L}^{\epsilon}\mathsf{f}\big{)}(r,x)$$

$$\displaystyle:=\frac{1}{\epsilon}\Big{[}\int_{\mathbb{S}^{n-1}}\mathsf{f}\Big{(}r+\epsilon y^{-1}_{r,x}g_{\eta(\epsilon,r,x)}(\langle\xi,\cdot\rangle),H(r,\xi)\Big{)}F(r,x,\xi)d\sigma(\xi)-\mathsf{f}(r,x)\Big{]},$$

$$b(r,x)(\theta ):=\frac{{\omega }_n}{y_{r,x}}F(r,x,\theta ),b^{ϵ}(r,x):=b(r,x)\star g_{\eta (ϵ,r,x)},$$

$$h(r,x):={\int }_{{\mathbb{S}}^{n-1}}H(r,\xi )F(r,x,\xi )d\sigma (\xi )-x.$$

$$R_t^{ϵ}(\theta )$$

$$x_t^{ϵ}$$

$$({\mathcal{L}}^{ϵ,r}\mathsf{f})(x)$$

$$r_t(\theta )$$

$$\overline{b}(r)(\theta )$$

$$\displaystyle\mathcal{A}_{p}(a):=\big{\{}r\in C_{+}(\mathbb{S}^{n-1})\,:\;$$

$$\displaystyle\mathcal{A}^{\sf e}_{p}(a):=\big{\{}(r,x)\in\mathcal{D}(F):$$

$$\displaystyle x\in\text{Image}(H(r^{\prime},\cdot))\text{ for some }r^{\prime}\in\mathcal{A}_{p}(a),r^{\prime}\leq r\big{\}}$$

$$\Vert F(r,x,\cdot )-F(r^{\prime },x^{\prime },\cdot ){\Vert }_p$$

$$|H(r,z)-H(r^{\prime },z^{\prime })|$$

$$\Vert \overline{b}(r)-\overline{b}(r^{\prime }){\Vert }_p$$

$$\overline{F}(a):=\sup \{F(r,x,\theta ):(r,x)\in {\mathcal{A}}_2^{\mathsf{e}}(a),\theta \in {\mathbb{S}}^{n-1}\}<\infty .$$

$$\displaystyle\sup_{(r,x)\in\mathcal{A}^{\sf e}_{\infty}(a)}\sup_{t_{0}\geq 0}\mathbb{E}_{x}\left[\Big{\|}\frac{1}{t}\int_{t_{0}}^{t_{0}+t}[b(r,x^{1,r}_{s})-\overline{b}(r)]ds\Big{\|}_{2}^{2}\right]\leq\lambda(t,a)\,,$$

$$\displaystyle\zeta^{\epsilon}(\delta):=\inf\big{\{}t>0:\,\|R^{\epsilon}_{t}\|_{\infty}>\delta^{-1}\big{\}}\,,$$

Full text

This research was supported in part by NSF grant DMS-1613091, UBACYT grant 20020160100147BA and PICT 2015-3154.

1. Introduction

Random growth processes arise in great variety in a large class of physical and biological phenomena, network dynamics, etc. Starting from seminal works of Eden [14] and Hammersley and Welsh [18], a series of mathematical models have been developed to capture and understand the evolution and pattern formation of growth processes. Our motivation stems from Laplacian growth models, which are characterized by the fact that the rate at which each portion of the boundary of the domain grows is determined by the harmonic measure of the domain from some given point, which we call source. The list includes Diffusion Limited Aggregation (dla) [40], its generalization – Dielectric Breakdown Model (dbm) [33], Hastings-Levitov process [21]; Internal dla (idla) [13, 29], abelian sandpiles and rotor aggregation [31]. It also includes once-reinforced random walk with strong reinforcement (orrw) [11], and origin-excited random walk (oerw) [28], for which the set of visited vertices is expected to form a limiting shape. For models such as dla, dbm or Hastings-Levitov, the source is at infinity, while in models such as idla, the source is at the origin. Whenever the source is fixed, the process of growing in time domains is Markovian. In contrast, the latter process is non-Markovian in orrw or excited random walks, where the source is moving and depends strongly on the last hitting point of the boundary and current shape of the domain.

In general, lattice growth models of this type are elusive, specially when the source is at infinity or when it is not fixed. A notable exception is idla for which Lawler, Bramson and Griffeath obtained a shape theorem (see [29]). Specifically, here particles are emitted in steps, one by one, from the source which is always located at the origin, and perform simple random walk until they visit an unvisited vertex. Each particle waits at the source until the previous one hits the external boundary, before being emitted. Gravner-Quastel [16] and Levine-Peres [30] generalize [29] and relate idla under more general, albeit still fixed, source locations to pde free boundary problems (a Stefan problem in [16], and an obstacle problem in [30] who also obtain analogous shape theorems for rotor-router and divisible sandpile models). An interesting variant is the Uniform idla, where upon hitting the boundary, the particle (source) is moved at a point chosen at random uniformly in the domain, and it is shown in [2] that the limiting shape of Uniform idla is the Euclidean ball.

Beyond these two examples, there is little understanding of such growth processes, despite substantial recent advances for first passage percolation. In particular, it is conjectured that for both orrw and oerw the evolution leads to the formation of an asymptotic shape as time goes to infinity (see [27, 28]), but there is no clear vision on how to attack the problem. Recall that in orrw the particle performs random walk on $\mathbb{Z}^{n}$, but each edge (or vertex) increases its conductance by a fixed strength $a>0$ after the first time it is traversed. A phase transition is expected in terms of $a$, with a limiting shape conjectured for all $a$ large enough. In the oerw model, the particle receives a (one-time) small drift towards the origin whenever it reaches an unvisited vertex (instead of the conductance change of the orrw), and a shape theorem is conjectured to hold, no matter how small this positive drift is. We refer the reader to [3, 26] for background on excited random walks, and to [2, 28] for discussions on various idla type processes and reinforced walks, all of whom share certain similar features. In particular, heuristically, whenever the self-interaction tends to attract the walker towards the bulk of its existing range, the boundary of the latter should change at a much slower rate than that of the walker, providing a natural setting to witness averaging.

While non-lattice isotropic models are more amenable to rigorous analysis (see [23, 34]), this typically requires having random conformal maps, hence restricted to dimension $n=2$. By focusing instead on the evolutions of star-shaped domains in $\mathbb{R}^{n}$, we are able to handle any $n\geq 2$, and mention in passing that, on the deterministic side, the works [5, 6] are close in spirit to our averaged equation (1.12).

We consider here a general random growth model in $\mathbb{R}^{n}$ which is specified by two rules $F$, $H$ and a scaling parameter $\epsilon>0$. The rule $F$, which is allowed to depend on the whole geometry of the domain and the position of the source, determines the (random) point at the boundary where the particle, upon starting at the prescribed position, called source, is going to hit the boundary of the domain. For example, $F$ may be the Harmonic measure at the boundary of the domain from the source. After the particle hits the boundary, the domain grows around the hitting point with a volume increase of $\epsilon$, followed by the particle jumping to the next source position, according to the rule $H$.

More precisely, fixing a small parameter $\epsilon>0$, we consider evolving domains $(D_{t}^{\epsilon})_{t\geq 0}$ in $\mathbb{R}^{n},n\geq 2$, which form simply-connected star-shaped compact sets (i.e. they can be parametrized by a function $R_{t}^{\epsilon}$ defined on the sphere $\mathbb{S}^{n-1}$). It is a pure jump Markov process that starts with an initial domain $D^{\epsilon}_{0}\ni 0$ and particle position $x^{\epsilon}_{0}$ and evolves at a Poisson rate of $\epsilon^{-1}$ by increasing the domain around randomly chosen boundary points (or equivalently, spherical angles $\xi_{t}\in\mathbb{S}^{n-1}$). The probability density for choosing boundary points to evolve is given by the hitting kernel $F(R^{\epsilon}_{t^{-}},x^{\epsilon}_{t^{-}},\cdot)$, which is a probability density on the sphere $\mathbb{S}^{n-1}$. After each hitting at the boundary at a point $\xi_{t}$, the particle is instantaneously transported according to the specified rule $H(R^{\epsilon}_{t^{-}},\xi_{t})$ to a point in $\mathbb{R}^{n}$ that can depend on both the domain and the last hitting position. Contrary to the rule $F$, we note that $H$ is deterministic in this article. In principle, one can consider a more comprehensive model where $H$ is a probability density function on $\mathbb{R}^{n}$, representing random transportation of the particle after each hitting of the current domain boundary.

The process $(R^{\epsilon}_{t},x^{\epsilon}_{t})_{t\geq 0}$ of evolving domains in $\mathbb{R}^{n}$ together with the position of the driving particle coupled to the former is, by construction, Markov (though each marginal is in general non-Markovian). The aim is to construct a continuum simplified model of “random walk interacting with its range”, allowing for general hitting kernel and non-trivial redistribution after each interaction, while inferring whether the evolving domain has an asymptotic shape.

The averaging principle has been extensively studied in the theory of dynamical systems, see e.g. [4, 7, 15, 20, 24, 35, 39] and references therein. Usually one identifies a slow variable and a fast variable. Under suitable conditions the fast variable achieves equilibrium in a time scale for which the slow variable does not evolve macroscopically. Hence, as the scale parameter $\epsilon\to 0$ one expects the slow variable to move according to a system where the fast variable is integrated with respect to its invariant measure, which may depend on the slow variable as well. In our model, the averaging property that one expects in models such as orrw and oerw is explicitly shown in terms of the process $(R_{t}^{\epsilon},x_{t}^{\epsilon})$, where as $\epsilon\to 0$, the variable $R_{t}^{\epsilon}$ serves as the slow variable, while $x_{t}^{\epsilon}$ acts as the fast one (and though the literature on averaging is large, we found no averaging principle that fits our case, involving a Markov jump process in infinite dimensions). The averaging principle is close in spirit to hydrodynamic limits, a standard tool in the study of interacting particle systems (see [12, 25, 38] and references therein). A hydrodynamic limit is proved for a continuous version of idla in [16], yielding in turn a shape theorem, thanks to the scale invariance of this model (as in Lemma 3.1 below). As mentioned before, in this process particles are emitted from fixed sources. One of our goals here is to derive similar results for self-interacting random walks, where the source is clearly moving.

Under certain mild conditions on our model features (namely, the rules $F$ and $H$), we prove in Theorem 1.4 an averaging principle. It allows us to identify the limiting infinite-dimensional ode governing the evolving domain as the slower dynamics of the pair, yielding in Theorem 1.9 the limiting shape result as a stationary solution of the limiting ode. Then, in Theorem 1.10 we verify our assumptions for a certain class of models, and in some instances compute explicitly their limiting shape.

Let $\mathbb{S}^{n-1}$ be the unit sphere in $\mathbb{R}^{n},n\geq 2$ equipped with its Euclidean surface area measure $\sigma(\cdot)$ and for any $1\leq p\leq\infty$ let $\|f\|_{p}$ denote the $L^{p}(\mathbb{S}^{n-1})$ norm of $f$ with respect to $\sigma(\cdot)$. We denote by $C_{+}(\mathbb{S}^{n-1})$ the space of strictly positive continuous functions on $\mathbb{S}^{n-1}$.

Definition 1.1**.**

A simply-connected compact set $D\subseteq\mathbb{R}^{n}$ is called star-shaped with respect to $0\in D$, if the line segment connecting [math] and any $x\in\partial D$ is entirely contained in $D$.

Any star-shaped $D$ is uniquely represented by a non-negative function $r:\mathbb{S}^{n-1}\to\mathbb{R}_{+}$ as

[TABLE]

Hereafter, by a slight abuse of notation, we identify any $r\in C_{+}(\mathbb{S}^{n-1})$ with its graph, which encloses a star-shaped domain $D$ and denote by $Leb(r)$ the Lebesgue measure (or volume) of that domain $D$. Namely,

[TABLE]

Let $\mathcal{D}(F)$ be an open subset of $C_{+}(\mathbb{S}^{n-1})\times\mathbb{R}^{n}$ such that $\{x:(r,x)\in\mathcal{D}(F)\}$ is non-empty for any $r\in C_{+}(\mathbb{S}^{n-1})$. The measurable map

[TABLE]

assigns to each $(r,x)\in\mathcal{D}(F)$ a strictly positive, continuous probability density function $F(r,x,\xi)$ with respect to $\sigma(\cdot)$. This function represents the rule whereby a particle starting from $x\in\mathbb{R}^{n}$ chooses a point $r(\xi)\xi$, $\xi\in\mathbb{S}^{n-1}$ at the boundary of the domain enclosed by $r$, to be the center of the (small) bump we add on the domain boundary $r$. The measurable map

[TABLE]

assigns for each $r\in C_{+}(\mathbb{S}^{n-1})$ and $\xi\in\mathbb{S}^{n-1}$ the transported (source) location $x=H(r,\xi)$ of a particle that hits the domain boundary $r$ at angle $\xi$. Assuming that $(r^{\prime},H(r,\xi))\in\mathcal{D}(F)$ for any $r^{\prime}\geq r$ and $F(r,x,\cdot)d\sigma$-a.e. $\xi$, guarantees that a.s. the iterative composition of the rules $H$ and $F$ is well defined (per our dynamics (1.7)). The small bump we add is in the form of a suitable spherical approximate identity $g_{\eta}(\cdot)$, as defined next.

Definition 1.2**.**

A collection of continuous functions $g_{\eta}:[-1,1]\to\mathbb{R}_{+}$ is called a spherical approximate identity if $1\star g_{\eta}=1$, $\|f\star g_{\eta}\|_{2}\leq\|f\|_{2}$ and $\|f\star g_{\eta}-f\|_{2}\to 0$ as $\eta\to 0$, for every $f\in L^{2}(\mathbb{S}^{n-1})$, where (see [9, (2.1.1)]),

[TABLE]

$\omega_{n}=\sigma(\mathbb{S}^{n-1})=\frac{(2\pi)^{n/2}}{\Gamma(n/2)}$ is the surface area of $\mathbb{S}^{n-1}$, and $\langle z,\theta\rangle$ denotes the scalar product associated with the Euclidean norm $|\cdot|$ in $\mathbb{R}^{n}$. We call such collection local if in addition $g_{\eta}(\langle z,\cdot\rangle)$ is supported on the spherical cap of (Euclidean) radius $2\eta$ centered at $z$ and $\eta^{n-1}\|g_{\eta}\|_{\infty}$ are uniformly bounded.

Utilizing [9, Section 2.1] we characterize in Lemma A.1 the collections $g_{\eta}$ that form a local spherical approximate identity (see also Figure 3).

Throughout we set the positive function on $\mathcal{D}(F)$,

[TABLE]

Noting that for $\xi$ of density $F(r,x,\cdot)d\sigma$

[TABLE]

(c.f. proof of Proposition 1.7), we add at each update a bump $(\epsilon/y_{r,x})\,g_{\eta}(\langle\xi,\cdot\rangle)$ on the current boundary $r$, where $\{g_{\eta}\}$ is a local spherical approximate identity, so that for $\epsilon\ll 1$, the volume of $D_{t}^{\epsilon}$ should grow at a nearly constant, unit rate. Using the $\epsilon$-dependent

[TABLE]

as our spherical-scale parameter yields a bump $\epsilon y^{-1}_{r,x}g_{\eta}$ on the boundary $r$ of about $\epsilon^{1/n}$ height (in the radial direction), uniformly in $(r,x)$, which is supported in case of a Euclidean ball of unit surface area (namely, $r\equiv\omega_{n}^{-1/(n-1)}$), on spherical caps of radius $2\epsilon^{1/n}$. Clearly, when adding such $\epsilon$-dependent bumps to our boundary function, the star-shaped domain evolves by a localized bump and the new domain remains star-shaped. Specifically, fixing $\epsilon\in(0,1]$ and starting at some $(R_{0}^{\epsilon},x_{0}^{\epsilon})$ we construct the Markov jump process $(R^{\epsilon}_{t},x^{\epsilon}_{t})_{t\geq 0}$ of jump rate $\epsilon^{-1}$ and state space $C_{+}(\mathbb{S}^{n-1})\times\mathbb{R}^{n}$, as follows. For a sequence $\{T^{\epsilon}_{i}\}_{i\in\mathbb{N}}$ of auxiliary Poisson arrival times of rate $\epsilon^{-1}$, starting at $T_{0}^{\epsilon}=0$, we freeze $(R^{\epsilon}_{t},x^{\epsilon}_{t})$ during each of the intervals $[T^{\epsilon}_{i},T^{\epsilon}_{i+1})$, while as each $t=T^{\epsilon}_{i}$, $i\geq 1$, conditional on the canonical filtration

[TABLE]

let

[TABLE]

That is, given $\mathcal{F}_{t^{-}}$ the random $\xi_{t}\in\mathbb{S}^{n-1}$ has the density $F(R_{t^{-}}^{\epsilon},x^{\epsilon}_{t^{-}},\cdot)$ with respect to $\sigma(\cdot)$. Then, update $(R_{t^{-}}^{\epsilon},x^{\epsilon}_{t^{-}})$ according to

[TABLE]

(recall the definitions (1.2) of $y_{r,x}$ and (1.4) of $\eta(\epsilon,r,x)$).

The generator $\mathcal{L}^{\epsilon}$ of the Markov process $(R^{\epsilon}_{t},x^{\epsilon}_{t})_{t\geq 0}$ is

[TABLE]

for any $\mathsf{f}:C_{+}(\mathbb{S}^{n-1})\times\mathbb{R}^{n}\to\mathbb{R}$ in the domain of $\mathcal{L}^{\epsilon}$. For $(r,x)\in\mathcal{D}(F)$ and $\theta\in\mathbb{S}^{n-1}$ let

[TABLE]

Considering (1.8) for $\mathsf{f}(r,x)=r(\theta)$ the evaluation map at fixed $\theta\in\mathbb{S}^{n-1}$ and using (1.1), we get for $(R^{\epsilon}_{t}(\theta))_{t\geq 0}$ the decomposition

[TABLE]

where $\Sigma_{t}^{\epsilon}(\theta)$ is an $\mathcal{F}_{t}$-martingale. Similarly, taking $\mathsf{f}(r,x)=x\cdot\vec{e}_{i}$, $i=1,\ldots,n$, in (1.8) yields

[TABLE]

for some $\mathbb{R}^{n}$-valued, $\mathcal{F}_{t}$-martingale $M_{t}^{\epsilon}$. For $r\in C_{+}(\mathbb{S}^{n-1})$ let $(x^{\epsilon,r}_{t})_{t\geq 0}$ denote the $\mathbb{R}^{n}$-valued Markov jump process evolving by (1.10) in the frozen domain $R^{\epsilon}_{s}\equiv r$. Its generator is thus

[TABLE]

for a suitable collection of functions $\mathsf{f}:\mathbb{R}^{n}\to\mathbb{R}$. Consider also the deterministic dynamics $t\mapsto r_{t}\in C_{+}(\mathbb{S}^{n-1})$ given by

[TABLE]

The probability measures $\nu_{r}$ on $\mathbb{R}^{n}$ for $r\in C_{+}(\mathbb{S}^{n-1})$ will be specified in Assumption (E), with Proposition 1.6 establishing the existence and uniqueness of the solution for the infinite-dimensional ode (1.12). For every $a\in(0,1)$ and $p\geq 1$, we define the collections

[TABLE]

and assume the following Lipschitz properties of $F$, $H$ and $\overline{b}$ throughout $\mathcal{A}^{\sf e}_{\infty}(a)$.

Assumption** (L).**

For any $a\in(0,1)$, there exists $K=K(a)$ finite such that uniformly for $(r,x),(r^{\prime},x^{\prime})\in\mathcal{A}^{\sf e}_{\infty}(a)$, $z,z^{\prime}\in\mathbb{S}^{n-1}$, we have that

[TABLE]

Moreover, for any $a\in(0,1)$,

[TABLE]

Our other two assumptions concern the ergodicity of the particle process in a frozen domain and the convergence to $b(\cdot,\cdot)$ of the drift of $R^{\epsilon}_{t}$ when $\epsilon\to 0$.

Assumption** (E).**

For any $r\in C_{+}(\mathbb{S}^{n-1})$ the process $(x_{t}^{1,r})_{t\geq 0}$ of generator (1.11) has a unique invariant probability measure $\nu_{r}$, such that

[TABLE]

where $\lambda(t,a)\to 0$ as $t\to\infty$, for any fixed $a\in(0,1)$.

Assumption** (C).**

For the $\mathcal{F}_{t}$-stopping times

[TABLE]

any fixed $t\geq 0$ and $\delta>0$,

[TABLE]

Remark 1.3**.**

From Definition 1.2 we know that $\|b^{\epsilon}(r,x)-b(r,x)\|_{2}\to 0$ as $\epsilon\to 0$, for any fixed $(r,x)\in\mathcal{D}(F)$. For Assumption (C) we need this to hold at the $\epsilon$-dependent $(R^{\epsilon}_{t\wedge\zeta^{\epsilon}},x^{\epsilon}_{t\wedge\zeta^{\epsilon}})$, but see parts (c) and (d) of Proposition 1.6 for simple sufficient conditions for Assumptions (E) and (C), respectively.

Equipped with these assumptions, we next state our main result. For technical reasons, we need to introduce a stopping time $\sigma^{\epsilon}(\delta)$ giving a lower bound to $F$, as needed to apply the crucial Lemma 2.1.

Theorem 1.4** (Averaging principle).**

Under Assumptions (L), (E) and (C), starting at $R^{\epsilon}_{0}=r_{0}\in C_{+}(\mathbb{S}^{n-1})$, for the $\mathcal{F}_{t}$-stopping time

[TABLE]

and any $T<\infty$, $\delta>0$, we have that for any $\iota>0$

[TABLE]

where $\{r_{t}\}_{t\geq 0}$ denotes the unique $C_{+}(\mathbb{S}^{n-1})$-solution of the ode (1.12) starting at $r_{0}$ (see Proposition 1.6(b)).

Remark 1.5**.**

With minor modifications of the proof, we can accommodate in Theorem 1.4 any random initial data such that $R_{0}^{\epsilon}\to r_{0}$ in probability. It is crucial to have $r_{0}$ strictly positive, since the function $b(r,x)$ blows up when $y_{r,x}\to 0$, hence (1.15) fails near $r\equiv 0$. Of course, if for any $a\in(0,1)$,

[TABLE]

then we can dispense with the stopping time $\sigma^{\epsilon}(\delta)$ in (1.21).

The next proposition, whose proof is deferred to the appendix, clarifies the implications of our assumptions.

Proposition 1.6**.**

(a) If condition (1.13) of Assumption (L) holds, then for every $a\in(0,1)$ there exists $C=C(a,K)=C(a)<\infty$ such that for all $(r,x),(r^{\prime},x^{\prime})\in\mathcal{A}^{\sf e}_{\infty}(a)$,

[TABLE]

Similarly, if (1.13) holds with $p=\infty$, then the same applies for (1.23).

Further, conditions (1.13) and (1.14) imply that for all $(r,x),(r^{\prime},x^{\prime})\in\mathcal{A}^{\sf e}_{\infty}(a)$,

[TABLE]

*while if (1.13) holds for both $p=2$ and $p=\infty$, with (1.14), (1.16) and (1.22) holding as well, then (1.15) must also hold.

(b) Condition (1.15) at $p=\infty$, together with Condition (1.16) imply that starting at any $r_{0}\in C_{+}(\mathbb{S}^{n-1})$ the ode (1.12) admits a unique $C_{+}(\mathbb{S}^{n-1})$-solution on $[0,\infty)$. Further, up to time $T$ the solution of (1.12) is in $\mathcal{A}_{\infty}(a_{1})$ for some $a_{1}(r_{0},T)>0$.

(c) To verify Assumption (E), it suffices to show that for any $a\in(0,1)$ there exist $n_{0}(a)\in\mathbb{N}$ and $\delta=\delta(a)>0$, such that the jump transition probability measure $\mathsf{P}_{r}$ of the embedded Markov chain $\{x_{T_{i}}^{1,r}\}$ satisfies the uniform minorisation condition*

[TABLE]

*for any $(r,x)\in\mathcal{A}^{\sf e}_{\infty}(a)$ and some probability measure $m_{r}(\cdot)$ on $\mathbb{R}^{n}$.

(d) Assumption (C) holds if for any $a\in(0,1)$,*

[TABLE]

(e) Condition (1.16) implies that for some positive $\delta_{n}(r_{0},T)$ and any $0<\delta<\delta_{n}$

[TABLE]

with the $\mathcal{F}_{t}$-stopping times $\zeta^{\epsilon}(\cdot)$ of (1.18) and

[TABLE]

Recall (1.3) that the random dynamics (1.6) has expected volume increase of $\epsilon(1+o(1))$ at each Poisson jump, (irrespective of the precise choice of $\eta(\epsilon,r,x)\to 0$ as $\epsilon\to 0$). We thus expect the following result (whose proof is also deferred to the appendix), about the linear growth of the volume of the deterministic dynamics (1.12).

Proposition 1.7**.**

If the solution $(r_{t})_{t\geq 0}$ to the ode (1.12) belongs to $C_{+}(\mathbb{S}^{n-1})$ for all $t\geq 0$, then $Leb(r_{t})=Leb(r_{0})+t$. Further, (1.3) holds throughout $\mathcal{D}(F)$.

Under the following scaling invariance of $F$ and $H$, we will deduce a shape theorem for the process $(R_{t}^{1})_{t\geq 0}$, from the averaging principle of Theorem 1.4.

Assumption** (I).**

For any scalar $c>0$, if $(r,x)\in\mathcal{D}(F)$ then $(cr,cx)\in\mathcal{D}(F)$ and

[TABLE]

Definition 1.8**.**

(a) A function $\psi\in C_{+}(\mathbb{S}^{n-1})$ is called invariant (shape) for the ode (1.12), if starting at $r_{0}=\psi$ yields

[TABLE]

(b) A function $\psi\in C_{+}(\mathbb{S}^{n-1})$ is called attractive (shape) for the ode (1.12) and a collection $\mathcal{C}$ of initial data, if starting at any $r_{0}\in\mathcal{C}$, the solution $t\mapsto r_{t}\in C_{+}(\mathbb{S}^{n-1})$ exists, with

[TABLE]

In general, invariant shapes may not be unique, nor are they necessarily attractive. See Example 3.4.

Theorem 1.9** (Shape theorem).**

*Suppose Assumption (I) holds and (1.21) applies without the stopping time $\sigma^{\epsilon}(\delta)$ (see Remark 1.5).

(a) If a function $\psi$ with $Leb(\psi)=1$ is invariant for the ode (1.12), then for any $c>0$, $1\leq T<\infty$ and $\iota>0$,*

[TABLE]

*(b) If a function $\psi$ with $Leb(\psi)=1$ is attractive for the ode (1.12) and a collection $\mathcal{C}$ of initial data, then for any $\iota>0$ and $r_{0}\in\mathcal{C}$, *

[TABLE]

Our main application is a model of random growth on $\mathbb{R}^{n}$ motivated by the expected mesoscopic behavior of orrw and oerw on $\mathbb{Z}^{n}$, where to gain regularity we consider $F$ and $H$ defined via a smoothed version of the evolving domain. Specifically, fix $\eta>0$ and $g=g_{\eta}$ as in (A.14) for some $\phi_{\eta}\in C^{3}([-1,1])$. Then, $\widetilde{r}:=r\star g\in C^{3}(\mathbb{S}^{n-1})$ for every $r\in L^{2}(\mathbb{S}^{n-1})$ (see (1.1)). We set

[TABLE]

where $G_{\widetilde{r}}(x,y)$ denotes the Green’s function of the Laplacian $-\Delta$ on star-shaped domain $D\subseteq\mathbb{R}^{n}$ with Dirichlet boundary conditions at $\widetilde{r}=\partial D$ and $\frac{\partial}{\partial\mathbf{n}}$ is the inward normal derivative on $\partial D$. Similarly, fix a locally Lipschitz function $\alpha\colon\mathbb{R}_{>0}\times\mathbb{S}^{n-1}\to\mathbb{R}_{\geq 0}$ such that

[TABLE]

and set (see Section 4 for the probabilistic interpretation),

[TABLE]

Another natural hitting rule $F$ chooses a boundary point with probability “proportional to a function of the distance to the particle”. Specifically, fixing a locally Lipschitz $\varphi:(0,\infty)\to(0,\infty)$ and with $\widetilde{r}$ as above, replace (1.32) by

[TABLE]

while keeping the rule $H$ of (1.34). We have the following results for these rules.

Theorem 1.10**.**

(a) The Averaging Principle of Theorem 1.4 holds under either (1.32)-(1.34) or (1.33)–(1.35), without the stopping times $\sigma^{\epsilon}(\delta)$ of (1.20).

(b) In case $\alpha(\ell,z)=\alpha(z)\ell$ and $\varphi(t)=t^{\beta}$, the Shape Theorem 1.9 also holds. In particular, for $\alpha(\ell,z)=\gamma\ell$ with $\gamma\in[0,1)$ fixed, the centered Euclidean ball is an invariant shape.

(c) For $\gamma=0$, $\beta<0$ and setting $\widetilde{r}=r$ in (1.32) and (1.35), the centered Euclidean ball is uniquely attractive among $C^{1}_{+}(\mathbb{S}^{n-1})$ initial data.*

Remark 1.11**.**

Similar conclusions apply for other transportation rules, such as

[TABLE]

which sends the particle to a statistical center of the domain. Note that $H(r)$ may possibly be outside the domain, and if start-shaped $D$ is invariant for the ode corresponding to (1.36) and (1.32), then so are all translations of $D$ which are star-shaped. Also, for such $H(\cdot)$ which depends on the domain only, trivially the invariant measure $\nu_{r}$ is the Dirac mass at $H(r)$ and Assumption (E) holds.

The rest of the article is organized as follows. In Section 2 we prove Theorem 1.4 and in Section 3 we deduce the shape result, Theorem 1.9. In Section 4 we apply these theorems to concrete growth models and prove Theorem 1.10.

2. Proof of Theorem 1.4

We start with bounding the Wasserstein 2-distance between any two measures on a compact, connected Riemannian manifold, by the $L^{2}$-distance between their densities with respect to the Riemannian measure.

Lemma 2.1**.**

Let $M\subseteq\mathbb{R}^{n}$ be a connected Riemannian manifold without boundary compactly embedded in $\mathbb{R}^{n}$, equipped with its Riemannian distance $d(\cdot,\cdot)$ and measure $\sigma(\cdot)$. Let $\mu,\nu$ be probability distributions on $M$ having densities $p,q$ respectively with respect to $\sigma(\cdot)$, where in addition $p(x)\geq c>0$ for all $x\in M$. Then, there exists $C=C(M,c)<\infty$ such that

[TABLE]

where

[TABLE]

is the Wasserstein $2$-distance between $\mu$ and $\nu$, and $\|\cdot\|_{2}:=\|\cdot\|_{L^{2}(\sigma)}$.

Proof of Lemma 2.1. .

By [36, Theorem 1], we have the variational representation

[TABLE]

In the last step, we have used the Poincaré inequality $\|f-\overline{f}_{M}\|_{2}\leq c(M)\|\nabla f\|_{2}$, where $\overline{f}_{M}$ denotes the $\sigma$-weighted average of $f$ in $M$ and $c(M)$ is the Poincaré constant. ∎

The proof of Theorem 1.4 is based on considering an auxiliary process in which the slow variable is frozen (this is a standard tool for proving averaging principles, see [20, 39]). Set

[TABLE]

Given the main process $(R^{\epsilon}_{t},x_{t}^{\epsilon})_{t\geq 0}$, we consider a family (indexed by $\epsilon>0$) of auxiliary dynamics $(\widehat{R}_{t}^{\epsilon},\widehat{x}_{t}^{\epsilon})_{t\geq 0}$ defined piecewise on each time interval $[k\Delta,(k+1)\Delta)$ with $k\in\mathbb{N}$, on the same probability space $(\Omega,\mathcal{F},\mathbb{P})$ as the main process, as follows. Inductively for every $k\in\mathbb{N}$, take the same Poisson clock $\{T^{\epsilon}_{i}\}_{i\in\mathbb{N}}$ used in constructing the main process, and starting at $\widehat{x}_{k\Delta}^{\epsilon}=x^{\epsilon}_{k\Delta}$, let $(\widehat{x}^{\epsilon}_{t})_{t\in[k\Delta,(k+1)\Delta)}$ have the marginal distribution of the Markov jump process in the frozen domain $R^{\epsilon}_{k\Delta}$ defined as in (1.11). That is, $(\widehat{x}^{\epsilon}_{t})_{t\in[k\Delta,(k+1)\Delta)}$ jumps at each $T^{\epsilon}_{i}\in[k\Delta,(k+1)\Delta]$, $i\in\mathbb{N}$ in the frozen domain $R^{\epsilon}_{k\Delta}$, by first using probability density $F(R^{\epsilon}_{k\Delta},\widehat{x}^{\epsilon}_{T_{i}^{\epsilon-}},\cdot)$ to choose a spherical angle $\widehat{\xi}_{i}$, then applying the rule $H(R^{\epsilon}_{k\Delta},\widehat{\xi}_{i})$. We further put requirement on the joint law such that at each jump, $\widehat{\xi}_{i}$ and $\xi_{i}$ of (1.5) achieve $\mathbb{E}[\,d(\xi_{i},\widehat{\xi}_{i})^{2}]^{1/2}$ within twice the Wasserstein $2$-distance $W_{2}(\mu,\nu)$ on $\mathbb{S}^{n-1}$, where $\mu=F(R^{\epsilon}_{T_{i}^{\epsilon-}},x^{\epsilon}_{T_{i}^{\epsilon-}},\cdot)d\sigma$, $\nu=F(R^{\epsilon}_{k\Delta},\widehat{x}^{\epsilon}_{T_{i}^{\epsilon-}},\cdot)d\sigma$. Inductively the above procedure defines $(\widehat{x}_{t})_{t\geq 0}$ on $(\Omega,\mathcal{F},\mathbb{P})$.

We then define $(\widehat{R}^{\epsilon}_{t})_{t\geq 0}$ on $(\Omega,\mathcal{F},\mathbb{P})$ as the dynamics driven by the ode, with $\widehat{R}^{\epsilon}_{0}=R^{\epsilon}_{0}$,

[TABLE]

With the auxiliary processes in place, we proceed to the proof of the theorem. By Proposition 1.6(b), starting at $r_{0}\in C_{+}(\mathbb{S}^{n-1})$, the solution $(r_{t})_{t\geq 0}$ to the ode (1.12) exists and is unique in $C_{+}(\mathbb{S}^{n-1})$. Fixing $\delta>0$, since $R^{\epsilon}_{t}\geq R^{\epsilon}_{0}=r_{0}$, for any $\epsilon>0$ the stopped at $\zeta^{\epsilon}(\delta)$ of (1.18), sample path $R^{\epsilon}_{t\wedge\zeta^{\epsilon}(\delta)}$ remains within $\mathcal{A}_{\infty}(\mathsf{a})$, provided $\mathsf{a}\in(0,\delta\wedge\inf_{\theta}r_{0}(\theta))$. Hereafter, we only apply Assumptions (L) and (E) with Lipschitz constant $K(\mathsf{a})$, resp. convergence rate $\lambda$ in (1.17), depending on such fixed $\mathsf{a}$, for the stopped processes.

Next, by (1.9), for any $u\leq t$ we have per $\theta$,

[TABLE]

By [10, Proposition 8.7], for the stopped martingale $\Sigma^{\epsilon}_{s\wedge\zeta^{\epsilon}(\delta)}(\theta)$,

[TABLE]

which together with Fubini, implies that

[TABLE]

Further, since spherical convolution is a contraction in $L^{2}(\mathbb{S}^{n-1})$ (per Definition 1.2), and the Lipschitz bound (1.23) holds throughout $\mathcal{A}^{\sf e}_{\infty}(\mathsf{a})$, hence the norms $\|b(R^{\epsilon}_{s\wedge\zeta^{\epsilon}(\delta)},x^{\epsilon}_{s\wedge\zeta^{\epsilon}(\delta)})\|_{2}$ are uniformly bounded,

[TABLE]

for some finite $C=C(K,\delta)$ and all $0\leq u\leq t$. Consequently, by (2.3)-(2.5) and Cauchy-Schwarz, for any $0\leq u\leq t$,

[TABLE]

We let hereafter $\tau=\zeta^{\epsilon}(\delta)\wedge\sigma^{\epsilon}(\delta)$ (for $\zeta^{\epsilon}$ and $\sigma^{\epsilon}$ of (1.18) and (1.20), respectively), and rely on (2) to establish the following two lemmas.

Lemma 2.2**.**

In the setting of Theorem 1.4, we have that

[TABLE]

Lemma 2.3**.**

In the setting of Theorem 1.4, we have that

[TABLE]

While deferring the proofs of Lemmas 2.2 and 2.3 to the end of the section, we note that these lemmas together with the definition (1.27) of $\kappa^{\epsilon}$ yield that

[TABLE]

In view of Proposition 1.6(d), this in turn results with

[TABLE]

(provided $\delta<\delta_{n}$). The conclusions of Lemma 2.2 and Lemma 2.3 are thus strengthened to apply with $\sigma^{\epsilon}(\delta)$ instead of $\tau$, so combining these lemmas with Markov’s inequality completes the proof of the theorem.

Turning to establish Lemmas 2.2 and 2.3, we start with the following key bound.

Lemma 2.4**.**

In the setting of Theorem 1.4, for some finite $C=C(K(\mathsf{a}),\delta)$ we have

[TABLE]

Proof of Lemma 2.4. .

Per (1.11), for each $k\in\mathbb{N}$ the auxiliary process $(\widehat{x}^{\epsilon}_{t})_{t\in[k\Delta,(k+1)\Delta)}$ admits the decomposition

[TABLE]

for some $\mathbb{R}^{n}$-valued, $\mathcal{F}_{t}$-martingale $\widehat{M}^{\epsilon}_{k\Delta,t}$ (setting $\widehat{M}^{\epsilon}_{k\Delta,t}\equiv 0$ when $t\leq k\Delta$). Taking the difference of (2.10) with (1.10), we have that for $t\in[k\Delta,(k+1)\Delta)$,

[TABLE]

is an $\mathbb{R}^{n}$-valued martingale.

Considering the generator of $(x_{t}^{\epsilon}-\widehat{x}^{\epsilon}_{t})_{t\in[k\Delta,(k+1)\Delta)}$, we have by [10, Proposition 8.7] and (1.14) that for any $t\in[k\Delta,(k+1)\Delta)$,

[TABLE]

where the inner conditional expectation $\mathsf{E}$ is only over $(\xi_{s},\widehat{\xi}_{s})$, having marginal densities $F(R^{\epsilon}_{s},x^{\epsilon}_{s},\cdot)$ and $F(R^{\epsilon}_{k\Delta},\widehat{x}^{\epsilon}_{s},\cdot)$, with respect to $\sigma(\cdot)$ on $\mathbb{S}^{n-1}$. By the coupling we chose, and Lemma 2.1 with $M=\mathbb{S}^{n-1}$, $p=F(R^{\epsilon}_{s},x^{\epsilon}_{s},\cdot)$ bounded below by $\delta$ when $s<\tau$, we have in (2) for $s<\tau$,

[TABLE]

using (1.13) in the last line. Consequently, we obtain from (2) that

[TABLE]

using in the last line (2) and that $\Delta\geq\epsilon$. From (2.11) at $t\wedge\tau$, Cauchy-Schwarz inequality, (2), (2) and $\Delta\geq\epsilon$, we have that for some finite $C=C(K,\delta)$ and any $t\in[k\Delta,(k+1)\Delta)$, $k=0,1,\ldots,\lfloor T/\Delta\rfloor$,

[TABLE]

utilizing also that $h(\cdot,\cdot)$ is Lipschitz, as in (1.24) of Proposition 1.6(a). By Gronwall’s inequality for $e(s)=\mathbb{E}[{\bf 1}_{\{s\leq\tau\}}|x_{s}^{\epsilon}-\widehat{x}_{s}^{\epsilon}|^{2}]$, starting at $e(k\Delta)=0$, we get that

[TABLE]

For our choice (2.1) of $\Delta=\Delta(\epsilon)$, the rhs of (2.14) is bounded by $C\epsilon$ for some (other) finite $C=C(K,\delta)$, as claimed. ∎

Proof of Lemma 2.2. .

Per (1.9) and (2.2), for any $t\geq 0$ we have that

[TABLE]

Consequently, by Cauchy-Schwartz,

[TABLE]

where the first term tends to zero as $\epsilon\to 0$ by Assumption (C) and the uniform boundedness of the integrand. The remaining two terms are bounded via (1.23), (2)-(2) and Lemma 2.4 by

[TABLE]

for some generic constants $C=C(K,\delta)$, hence tending to zero as well. ∎

Proof of Lemma 2.3. .

Per (1.17) and the fact that the event $\{k\Delta\leq\zeta^{\epsilon}(\delta)\}$ is measurable on $\sigma\left(R^{\epsilon}_{s},s\leq k\Delta\right)$, we have that uniformly for $k=0,1,\ldots,\lfloor T/\Delta\rfloor-1$,

[TABLE]

It then follows from (2), (1.15) and (2) that for some finite $C=C(K,\delta)$ and any $t\in[0,T]$,

[TABLE]

where $\lambda(\Delta/\epsilon,\mathsf{a})\to 0$ as $\epsilon\to 0$ by (1.17) since $\Delta/\epsilon\to\infty$. We proceed to bound

[TABLE]

via Gronwall’s inequality. Per (2.2) and (1.12), for any $t\geq 0$ we have that

[TABLE]

By (1.15) and (2) we have that for any $t\leq T$,

[TABLE]

Gronwall’s inequality and Lemma 2.2 yield that

[TABLE]

converge to zero when $\epsilon\to 0$, as required. ∎

3. Proof of Theorem 1.9.

The following intuitive coupling enables to transfer the Averaging Principle for the family of processes $(R^{\epsilon}_{t})$ as the scale parameter $\epsilon\to 0$ on finite time horizons, into a shape result for $(R^{1}_{t})$ of scale $1$ as time $t\to\infty$.

Lemma 3.1** (coupling).**

Fix $\epsilon>0$. Under Assumption (I), with

[TABLE]

there exists a coupling such that

[TABLE]

Proof.

Let $\{T^{\epsilon}_{i}\}_{i\in\mathbb{N}}$ with $T_{0}^{\epsilon}=0$ denote the sequence of Poisson jump times of rate $\epsilon^{-1}$ used in constructing $(R_{t}^{\epsilon},x^{\epsilon}_{t})_{t\geq 0}$, for some fixed $\epsilon>0$. Set $T^{1}_{i}:=\epsilon^{-1}T^{\epsilon}_{i}$, $i\in\mathbb{N}$. By scaling properties of exponential distribution, $\{T^{1}_{i}\}_{i\in\mathbb{N}}$ has the law of a sequence of Poisson arrival times of rate $1$, as such we construct $(R^{1}_{t},x^{1}_{t})_{t\geq 0}$ using $\{T^{1}_{i}\}_{i\in\mathbb{N}}$, on the same probability space $(\Omega,\mathcal{F},\mathbb{P})$ as $(R^{\epsilon}_{t},x^{\epsilon}_{t})_{t\geq 0}$.

Starting with $(R_{0}^{\epsilon},x^{\epsilon}_{0})=(\epsilon^{1/n}R^{1}_{0},\epsilon^{1/n}x^{1}_{0})$ on $(\Omega,\mathcal{F},\mathbb{P})$, suppose we have succeeded in coupling $(R^{\epsilon}_{s},x_{s}^{\epsilon})$ with $(\epsilon^{1/n}R^{1}_{s/\epsilon},\epsilon^{1/n}x^{1}_{s/\epsilon})$ as in (3.1) up to time $(T_{i}^{\epsilon})^{-}$ for some $i\in\mathbb{N}$. Then by (1.28), for any $s\leq(T^{\epsilon}_{i})^{-}$ we have

[TABLE]

The induction hypotheses and the construction (1.5), (1.6) yield at $t=T_{i}^{\epsilon}$, per $\theta$

[TABLE]

By coupling the jumps of $x^{\epsilon}_{t^{-}}$ and $x^{1}_{t^{-}/\epsilon}$ at $t=T^{\epsilon}_{i}$ such that $\xi^{\epsilon}_{t}=\xi^{1}_{t/\epsilon}$, we deduce $R^{\epsilon}_{t}=\epsilon^{1/n}R^{1}_{t/\epsilon}$, $\xi^{\epsilon}_{t}=\xi^{1}_{t/\epsilon}$ for $t=T^{\epsilon}_{i}$, and by (1.7), (1.29) also $x^{\epsilon}_{t}=\epsilon^{1/n}x^{1}_{t/\epsilon}$. During $t\in[T^{\epsilon}_{i},T^{\epsilon}_{i+1})$, all processes stay put, hence continuing extends the coupling to all $t\geq 0$. ∎

Proof of Theorem 1.9..

We only prove part (b), whereas the proof of part (a) is similar. By Theorem 1.4 and Lemma 3.1, we firstly have for any $t<\infty$ and $\iota>0$,

[TABLE]

where $(r_{t})_{t\geq 0}$ is the continuous solution of (1.12) with initial data $r_{0}\in\mathcal{C}\cap C_{+}(\mathbb{S}^{n-1})$, and $Leb(r_{t})=Leb(r_{0})+t$. By the triangle inequality, we have that

[TABLE]

By (3.2), the first term vanishes for any $t\geq 1$, and upon taking another limit as $t\to\infty$, the second term vanishes as well by (1.30). We obtain the claims upon setting $N=\epsilon^{-1}$. ∎

Problem 3.2**.**

It remains open to remove the strict positivity of initial condition in Theorem 1.4, hence to be able to take $c=0$ in (1.31), which would correspond to a genuine shape theorem.

We have the following general characterization of invariant shapes.

Proposition 3.3**.**

Under Assumption (I), $\psi\in C_{+}(\mathbb{S}^{n-1})$ is invariant for the ode (1.12) if and only if $\overline{b}(\psi)=n^{-1}\psi/Leb(\psi)$.

Proof.

We prove the “only if” part, while the converse “if” direction can be checked directly. Assumption (I) implies that for any $c>0$, $y_{cr,cx}=c^{n-1}y_{r,x}$ and hence $b(cr,cx)=c^{-(n-1)}b(r,x)$. From (1.11) the value of $(\mathcal{L}^{1,cr}(\mathsf{f}\circ c^{-1}))(cx)$ is independent of $c$. The natural coupling $x_{t}^{1,cr}=cx_{t}^{1,r}$ implies that $\nu_{cr}(\cdot)$ of Assumption (E) are merely the push-back of $\nu_{r}(\cdot)$ under scaling $x\mapsto cx$, resulting with $\overline{b}(cr)=c^{-(n-1)}\overline{b}(r)$. Per Definition 1.8(a), an invariant solution $(r_{t})_{t\geq 0}$ starting at $r_{0}=\psi$ is such that $r_{t}=c_{t}\psi$ with $(c_{t}^{n}-1)Leb(\psi)=t$. From the ode (1.12) it is not hard to infer that $\overline{b}(\psi)\propto\psi$. Further, by taking derivative of (1.12) in $t$ we identify the proportionality constant to be $n^{-1}/Leb(\psi)$. ∎

However, invariant shapes may not be unique.

Example 3.4**.**

Consider $H\equiv 0$ (the origin) and $F(r,0,\cdot)=r(\cdot)/\int rd\sigma\in C_{+}(\mathbb{S}^{n-1})$. Then it is easy to check that

[TABLE]

Since this choice of $F$ and $H$ satisfies Assumption (I), by Proposition 3.3, any $r\in C_{+}(\mathbb{S}^{n-1})$ is invariant for (1.12), and not attractive except when starting from itself.

We provide sufficient condition for the centered Euclidean ball $\mathbb{B}$ to be attractive for (1.12), where we denote henceforth by $\mathbb{B}$ the constant $1$ function on $\mathbb{S}^{n-1}$. Unfortunately, the condition (3.3) is rather hard to check.

Proposition 3.5**.**

Suppose the ode (1.12) has $C^{1}(\mathbb{S}^{n-1})$-solution $(r_{t})_{t\geq 0}$ for any $r_{0}\in\mathcal{C}\subset C^{1}(\mathbb{S}^{n-1})$, and that for any $r\in C^{1}(\mathbb{S}^{n-1})$, it holds

[TABLE]

Then $\mathbb{B}$ is attractive for (1.12) for the collection $\mathcal{C}$ of initial data.

Proof.

Set $\mathsf{osc}(r)=\max_{\theta}r(\theta)-\min_{\theta}r(\theta)$. Since $r_{t}$ is $C^{1}$ for all $t\geq 0$, we have

[TABLE]

and similarly for $\operatorname{arg\,min}_{\theta}r_{t}$. Therefore, combined with (3.3) we have that

[TABLE]

Set $\mathbf{r}_{t}:=r_{t}/(Leb(r_{0})+t)^{1/n}$. Then we have that $\mathsf{osc}(\mathbf{r}_{t})=\mathsf{osc}(r_{t})/(Leb(r_{0})+t)^{1/n}$ and

[TABLE]

This yields

[TABLE]

as $t\to\infty$, for any $r_{0}\in\mathcal{C}$. Equivalently, for some constant $c_{n}$ such that $Leb(c_{n}\mathbb{B})=1$,

[TABLE]

This is exactly the definition (1.30) of attractive shapes with $\psi=\mathbb{B}$. ∎

4. Applications: Proof of Theorem 1.10

We consider the two applications of Theorems 1.4 and 1.9, introduced previously in Theorem 1.10, with the main one being a simplified model for the growth of the range of oerw (with $F(r,x,\cdot)$ the density of the harmonic measure). Indeed, our choice of (1.34) is motivated by basic features of orrw and oerw in the mesoscopic scale. The ideal choice of $H$ to be closer to these models would be $H(r,z,\epsilon)=(r(z)-\epsilon^{1/n})z$, with (1.34) an independent of $\epsilon$, rule of the same type. Similarly, our choice of $F$ for (1.32) corresponds to taking a simplified continuous model, where the random walk is replaced by a Brownian motion (see the probabilistic interpretation provided in Section 4.1). An advantage of using such continuous model is that our proofs work verbatim when instead of Brownian motion, the particle follows an elliptic diffusion whose generator is a uniformly elliptic second-order divergence form operator $\mathcal{L}=-\text{div}A\nabla$ (so the Green’s function used in the definition (1.32) be the one for $\mathcal{L}$). Indeed, recall Dahlberg’s theorem [8, Theorem 3 and remark], that for a Lipschitz domain $D\subset\mathbb{R}^{n}$, harmonic measures from any point $x\in D$ are mutually absolutely continuous with respect to the $(n-1)$-dimensional Hausdorff measure on $\partial D$, hence their Radon-Nikodym derivative which is the Poisson kernel $P(D,x,\cdot)$ exists and belongs to $L^{2}_{\text{loc}}(\partial D)$. If the domain is more regular, so is the Poisson kernel. Per [22, page 547], if $\partial D$ belongs to Hölder space $C^{k+1,\gamma}$ for some $k\in\mathbb{N},\gamma\in(0,1)$, then $P(D,x,\cdot)\in C^{k,\gamma}(\partial D)$. Since our domains are star-shaped, by an abuse of terminology we will call $F(r,x,\cdot)$ the Poisson kernel of $r$, if it is a probability density on $\mathbb{S}^{n-1}$ corresponding to $P(D,x,\cdot)$ with $r=\partial D$ up to a change of variables.

As for the reasoning behind the regularization $r\mapsto\widetilde{r}=r\star g$ in our applications, note that even for smooth domains, one cannot expect their Poisson kernel to be Lipschitz in $L^{2}$-norm with respect to boundary perturbations as (1.13), or in any other norm. Indeed, as explained in [22], one expects the regularity of $P(D,x,\cdot)$ to be one derivative order less than that of the domain $D$. However, if one forms the kernel based on a regularized domain, then the Lipschitz property can be true (as shown below in Proposition 4.3). Though to a lesser degree, similar issue arises also in the more explicit hitting rule $F$ of (1.35), where regularization is still the key to verifying Assumption (C) via condition (1.26).

Recall the $C^{k}(\Omega)$ and $C^{k,\gamma}(\Omega)$, $\gamma\in(0,1]$ norms of functions, taking in this article only $\Omega=\mathbb{S}^{n-1}$ or $\Omega=\overline{D}$ (the closure of a bounded domain $D\subset\mathbb{R}^{n}$), and using $C(\Omega)$ for $C^{0}(\Omega)$. Letting $\partial^{\alpha}u$ for multi-index $\alpha$ denote any $|\alpha|$-th order derivative of $u$, we equip the collection of $k$-times continuously differentiable functions, with the norm

[TABLE]

Further denoting the $\gamma$-Hölder semi-norm of a function $w$ by

[TABLE]

if finite, where $\mathsf{d}(\cdot,\cdot)$ is the geodesic distance when $\Omega=\mathbb{S}^{n-1}$, and the Euclidean distance when $\Omega=\overline{D}$, we define the $C^{k,\gamma}(\Omega)$-Hölder norm of any $u\in C^{k}(\Omega)$, by

[TABLE]

We proceed to verify the conditions needed for Theorems 1.4 and 1.9, starting with the following Lipschitz control on the regularization map $r\mapsto\widetilde{r}$, the proof of which is deferred to the appendix.

Lemma 4.1**.**

For any $r,r^{\prime}\in L^{2}(\mathbb{S}^{n-1})$, we have that

[TABLE]

for some $C=C(g)<\infty$ that depends only on the convolution kernel $g$. In particular, for any $a\in(0,1)$ there exists $\delta=\delta(a,\alpha,g)>0$ so the image of $\mathcal{A}^{\sf e}_{2}(a)$ under $(r,x)\mapsto(\widetilde{r},x)$ is within the set

[TABLE]

where $\mathbb{B}(x,\delta)$ denotes the open Euclidean ball of radius $\delta$ centered at $x\in\mathbb{R}^{n}$ and $D_{\widetilde{r}}$ denotes the domain of boundary $\widetilde{r}$.

Equipped with Lemma 4.1, we prove property (1.14) for $H$ of (1.34).

Proposition 4.2**.**

For every $a\in(0,1)$, the map $(r,z)\mapsto H(r,z)$ is Lipschitz from $(\mathcal{A}_{2}(a)\times\mathbb{S}^{n-1},\|\cdot\|_{2}\times|\cdot|)$ to $\mathbb{R}^{n}$.

Proof.

With $\alpha(\cdot,\cdot)$ Lipschitz on compacts, we have by Lemma 4.1 that for $H$ of (1.34), some finite $\widetilde{C}=\widetilde{C}(a,\alpha)$ and all $r,r^{\prime}\in\mathcal{A}_{2}(a)$,

[TABLE]

Also, for any $r\in\mathcal{A}_{2}(a)$, $z,z^{\prime}\in\mathbb{S}^{n-1}$,

[TABLE]

The uniform on $\mathcal{A}_{2}(a)$ control $\|\widetilde{r}\|_{C^{3}(\mathbb{S}^{n-1})}\leq C(g)a^{-1}$ from Lemma 4.1, translates into the same control over the bounded Lipschitz norm of $\widetilde{r}(\cdot)$. Hence, with $\alpha(\cdot,\cdot)$ Lipschitz on compacts, we deduce that $|H(r,z)-H(r,z^{\prime})|\leq C|z-z^{\prime}|$ for some finite $C=C(a,\alpha,g)$ and all $r,z,z^{\prime}$ as above. ∎

4.1. Smoothed harmonic measure

Recall the construction above (1.32) of the smoothed domain $\widetilde{r}\in C^{3}(\mathbb{S}^{n-1})$ for every $r\in L^{2}(\mathbb{S}^{n-1})$. Due to the preceding discussion on Poisson kernels, it is clear that the regularized (as in (1.32)) Poisson kernel $F(r,x,\cdot)$ belongs to $C_{+}(\mathbb{S}^{n-1})$ for any $(r,x)\in\mathcal{D}(F)$, where the probabilistic meaning of the definitions of $F$ in (1.32) and $H$ in (1.34) is as follows.

If the process $(R^{\epsilon},x^{\epsilon})$ is defined up to time $s$ and the state at that time is given by domain with boundary $R^{\epsilon}_{s}$ and particle position $x^{\epsilon}_{s}$, we wait for the next jump mark, denoted by $t>s$, and given by an independent Exponential($\epsilon^{-1}$) random variable. To choose a point at the current boundary $R^{\epsilon}_{s}$, the particle follows the law of a Brownian motion in $\mathbb{R}^{n}$, starting at $x^{\epsilon}_{s}$ till its first exit from the smoothed domain $\widetilde{R}^{\epsilon}_{s}=R^{\epsilon}_{s}\star g$. We record its exit angle $\xi_{t}\in\mathbb{S}^{n-1}$ and define the location for the center of the new bump on the original domain by $R^{\epsilon}_{s}(\xi_{t})\xi_{t}$. Hence the updated domain is formed by

[TABLE]

Observe that the bump is added to the original domain and not the smoothed one. Next, the particle is pushed towards the origin by a strictly positive quantity, along the radius, still in the smoothed domain $\widetilde{R}^{\epsilon}_{s}$, namely $x^{\epsilon}_{t}=\alpha(\widetilde{R}^{\epsilon}_{s}(\xi_{t}),\xi_{t})\xi_{t}$ and there it waits for the next jump mark. Continuing in this way we define the process at all times. Note that we have omitted the travel time of the Brownian motion inside the smoothed domain and only deal with its exit distribution. Further, since $R^{\epsilon}_{t}(\theta)\geq R^{\epsilon}_{s}(\theta)$, $t>s$ for all $\theta\in\mathbb{S}^{n-1}$, necessarily also $\widetilde{R}^{\epsilon}_{t}(\theta)\geq\widetilde{R}^{\epsilon}_{s}(\theta)$, with the particle $(x^{\epsilon}_{t})$ always contained in the smoothed domain, once we assume it is the case for $(\widetilde{R}_{0}^{\epsilon},x^{\epsilon}_{0})$.

We postpone to the appendix the proof of the following key proposition about Lipschitz regularity for Poisson kernels in our regularized domains (where the precise value of $\gamma\in(0,1)$ is unimportant).

Proposition 4.3**.**

For any $a\in(0,1)$, $C<\infty$, $\delta>0$, the map $(\widetilde{r},x)\mapsto F(r,x,\cdot)$ of (1.32) is Lipschitz from the set $\mathcal{K}_{a,C,\delta}$ of (4.2), equipped with the $C^{2,\gamma}(\mathbb{S}^{n-1})$-norm for the first variable and the Euclidean norm for the second one, to $C^{1}(\mathbb{S}^{n-1})$.

The map $(r,x)\mapsto F(r,x,\cdot)$ of (1.32) is a composition of $(r,x)\mapsto(\widetilde{r},x)$ and $(\widetilde{r},x)\mapsto F(r,x,\cdot)$. The former map is globally Lipschitz per Lemma 4.1, whereas the latter is per Proposition 4.3 globally Lipschitz on some $\mathcal{K}_{a,C,\delta}$ that contains the image of $\mathcal{A}^{\sf e}_{2}(a)$ under the first map, yielding the following corollary.

Corollary 4.4**.**

For every $a\in(0,1)$, the map $(r,x)\mapsto F(r,x,\cdot)$ is globally Lipschitz from $(\mathcal{A}^{\sf e}_{2}(a),\|\cdot\|_{2}\times|\cdot|)$ to $C^{1}(\mathbb{S}^{n-1})$.

Remark 4.5**.**

From Corollary 4.4 we get that $\|F(r,x,\cdot)\|_{\rm Lip}\leq C(a)$ for all $(r,x)\in\mathcal{A}^{\sf e}_{\infty}(a)$, hence Assumption (C) holds in our setting (see Proposition 1.6(d)).

We next show that $F(\cdot)$ of (1.32) is bounded above and below, uniformly in $\mathcal{A}^{\sf e}_{2}(a)\times\mathbb{S}^{n-1}$, thereby verifying (1.16) and as explained in Remark 1.5, allowing us also to dispense of the stopping time $\sigma^{\epsilon}(\delta)$ in Theorem 1.4.

Proposition 4.6**.**

For each $a\in(0,1)$ there exist $c=c(a,\alpha,g)>0$ such that

[TABLE]

Proof.

For each $r\in\mathcal{A}_{2}(a)$, the state space of $(x^{1,r}_{t})_{t\geq 0}$ is contained in the star-shaped domain enclosed by $\widetilde{r}\in C^{3}(\mathbb{S}^{n-1})$. In such domains, with $x$ in the interior, the Poisson kernel $F(r,x,\cdot)$ is pointwise positive (a consequence of Hopf lemma). By Proposition 4.3, $(\widetilde{r},x)\mapsto F(r,x,\theta)$ is continuous per fixed $\theta$ with $\theta\mapsto F(r,x,\theta)$ continuous per fixed $(\widetilde{r},x)\in\mathcal{K}_{a,C,\delta}$. Thanks to the compactness of $\mathbb{S}^{n-1}$, the joint continuity of $(\widetilde{r},x,\theta)\mapsto F(r,x,\theta)$ follows and we get (4.3) from Lemma 4.1 and the compactness of $\mathcal{K}_{a,C,\delta}\times\mathbb{S}^{n-1}$ under the $C^{2,\gamma}(\mathbb{S}^{n-1})\times\mathbb{R}^{n}\times\mathbb{S}^{n-1}$ norm (see proof of Proposition 4.3). ∎

Remark 4.7**.**

Propostion 4.6 further establishes Assumption (E) for our model. Indeed, the sufficient condition (1.25) of Proposition 1.6(c), then holds with $n_{0}=1$, $\delta=c$ and $m_{r}(\cdot)$ the push-forward under $H(r,\cdot)$ of the uniform measure on $\mathbb{S}^{n-1}$.

Remark 4.8**.**

With $\omega_{n}^{-1/2}\|\cdot\|_{2}\leq\|\cdot\|_{\infty}\leq\|\cdot\|_{C^{1}(\mathbb{S}^{n-1})}$, hence also $\mathcal{A}^{\sf e}_{\infty}(a)\subset\mathcal{A}^{\sf e}_{2}(a)$, Corollary 4.4 implies that (1.13) holds for both $p=2$ and $p=\infty$. Recall Proposition 4.2 that (1.14) holds here, and Proposition 4.6 that so do both (1.16) and (1.22). Combined, these in turn yield by Proposition 1.6(a) the last remaining Lipschitz condition required (namely, property (1.15) of $\overline{b}$), and thereby complete the verification of Assumption (L) in our setting.

Having verified Assumptions (C), (E) and (L) (see Remarks 4.5, 4.7 and 4.8, respectively), we can apply Theorem 1.4 to this model without the stopping time $\sigma^{\epsilon}(\delta)$ of (1.20). This amounts to proving Theorem 1.10(a) for this model, whereas our next proposition, considering special cases where we have explicit descriptions, constitutes the proof of Theorem 1.10(b) in this setting.

Proposition 4.9**.**

(a) If the function $\alpha(\ell,z)=\alpha(\ell)$ does not separately depend on $z$, then the centered Euclidean ball $\mathbb{B}$ is an invariant solution to (1.12).

(b) If the function $\alpha(\ell,z)=\alpha(z)\ell$ depends linearly on $\ell$, then Assumption (I) is satisfied.

(c) If $\alpha(\ell,z)=\gamma\ell$ for some fixed number $\gamma\in[0,1)$, then the unique invariant measure $\nu_{r}$ is explicitly given by the harmonic measure from the origin in the domain enclosed by $\gamma\widetilde{r}$, for every $r\in C_{+}(\mathbb{S}^{n-1})$.

(d) If $\alpha(\ell,z)\equiv 0$ and we force $\widetilde{r}=r$, then $\mathbb{B}$ is the unique attractive solution of (1.12), whenever $r_{0}\in\mathcal{C}=C_{+}^{1}(\mathbb{S}^{n-1})$.*

Proof.

(a). First note that $\mathbb{B}\star g=\mathbb{B}$, so regularization by $g$ has no effect here. Further, by the rotational invariance of the Brownian law, the uniform measure on $\alpha(1)\mathbb{S}^{n-1}$ is invariant for the process $(x_{t}^{1,\mathbb{B}})$ (i.e. a Brownian motion on $\mathbb{B}$ starting at $\alpha(1)\mathbb{S}^{n-1}$ and radially projected back to that set upon hitting $\partial\mathbb{B}=\mathbb{S}^{n-1}$). In addition, recall (1.2) that $y_{\mathbb{B},x}=\omega_{n}$ is constant, hence so is $\theta\mapsto\overline{b}(\theta)$, from which it directly follows that $\mathbb{B}$ is invariant for (1.12).

(b). The identity (1.28) is due to the scaling invariance of the Brownian motion, while (1.29) is satisfied by our choice.

(c). By the scaling invariance of Brownian motion, the harmonic measure from the origin on $\gamma\widetilde{r}$ and on $\widetilde{r}$, viewed as measures on spherical angles, are equal. Since the transition kernel of the Brownian motion from $\gamma\widetilde{r}$ to $\widetilde{r}$ is exactly given by $F(r,x,\cdot)$, we see that the harmonic measure from the origin is the unique (thanks to Assumption (E)), invariant measure for $(x_{t}^{1,r})$.

(d). Since $H\equiv 0$ (the origin), here $\overline{b}(r)=F(r,0,\cdot)/y_{r,0}$. By Corollary 4.4, $F(r,0,\cdot)\in C^{1}(\mathbb{S}^{n-1})$. Hence, for any $r_{0}\in C_{+}^{1}(\mathbb{S}^{n-1})$, the ode (1.12) admits $C^{1}$ solution $(r_{t})_{t\geq 0}$. By Proposition 3.5, for $\mathbb{B}$ to be attractive for (1.12) it suffices to show that for any $r\in C^{1}(\mathbb{S}^{n-1})$, the Poisson kernel from the origin to $\widetilde{r}$, is at angle $\operatorname{arg\,min}_{\theta}r$ no smaller than at $\operatorname{arg\,max}_{\theta}r$. To this end, upon forcing $\widetilde{r}=r$ consider two standard Brownian motions in $\mathbb{R}^{n}$, one in the domain enclosed by $r$, the other in the Euclidean ball $\mathbb{B}(0,\min_{\theta}r)$. Couple them to move together starting from the origin until the first hitting time by both of $\partial\mathbb{B}(0,\min_{\theta}r)$, where one Brownian motion is stopped and the other can continue to move until hitting $r$. This coupling yields that $F(r,0,\operatorname{arg\,min}_{\theta}r)\geq 1/\omega_{n}$. An analogous coupling, between a Brownian motion in the domain enclosed by $r$ and another in $\mathbb{B}(0,\max_{\theta}r)$, yields that $F(r,0,\operatorname{arg\,max}_{\theta}r)\leq 1/\omega_{n}$, thus verifying our claim. ∎

For anisotropic $\alpha(\ell,z)$ (that do not satisfy the condition of Proposition 4.9(a)), one may obtain other limiting shapes as invariant solutions to the ode (1.12), such as diamond, square etc (see Figure 4), implicitly determined as in Proposition 3.3 (and in the anisotropic case, the Euclidean ball is typically not an invariant shape).

4.2. Distance to particle

Following the approach of Section 4.1, in particular as in Remarks 4.5, 4.7 and 4.8, our next proposition establishes Theorem 1.10(a) (namely, shows that Theorem 1.4 applies without $\sigma^{\epsilon}(\delta)$ of (1.20)), for $F$ of (1.35), where the probability of choosing a boundary point is a fixed function of its distance to the particle.

Proposition 4.10**.**

For every $a\in(0,1)$, the map $(r,x)\mapsto F(r,x,\cdot)$ of (1.35) is Lipschitz from $(\mathcal{A}^{\sf e}_{\infty}(a),\|\cdot\|_{p}\times|\cdot|)$ to $(L^{p}(\mathbb{S}^{n-1}),\|\cdot\|_{p})$, for both $p=2$ and $p=\infty$. In addition, $\|F(r,x,\cdot)\|_{\rm Lip}$ is bounded, uniformly over $\mathcal{A}^{\sf e}_{\infty}(a)$, while $F(r,x,\theta)$ is bounded above and away from zero, uniformly over $\mathcal{A}^{\sf e}_{2}(a)\times\mathbb{S}^{n-1}$.

Proof.

Starting with the boundedness above and away from zero of $F(r,x,\cdot)$, we have in view of Lemma 4.1 that $\|H(r,\cdot)\|_{\infty}\leq\|\widetilde{r}\|_{\infty}\leq C(g)\|r\|_{2}\leq C(g)a^{-1}$. The argument of $\varphi(\cdot)$ in (1.35) is thus bounded, uniformly over $\mathcal{A}^{\sf e}_{2}(a)$. Now recall from Lemma 4.1 that for some $\delta_{\star}(a)>0$ and all $(r,x)\in\mathcal{A}^{\sf e}_{2}(a)$, the point $x$ is at least $\delta_{\star}$ away from $\widetilde{r}$, and thereby the argument of $\varphi(\cdot)$ in (1.35) is bounded away from zero uniformly over $\mathcal{A}^{\sf e}_{2}(a)$. The uniform over $\mathcal{A}^{\sf e}_{2}(a)\times\mathbb{S}^{n-1}$ bound above and away from zero on $F$ is then a direct consequence of our assumption that $\varphi(\cdot)$ is positive and continuous on $(0,\infty)$.

Next, with the numerator of (1.35) uniformly bounded above and the denominator uniformly bounded below, it suffices to separately prove the Lipschitz property for the numerator and denominator in (1.35). To this end, note that $\varphi(|\cdot|)$ is globally Lipschitz on compact subsets of $(0,\infty)$. Dealing simultaneously with $p=2$ and $p=\infty$, it follows that for some $C=C(\varphi)$ finite and all $(r,x),(r^{\prime},x^{\prime})\in\mathcal{A}^{\sf e}_{\infty}(a)$,

[TABLE]

where all $L^{p}$-norms are with respect to $\theta$. The difference between the denominator values at $(r,x)$ and $(r^{\prime},x^{\prime})$ is at most the $L^{1}$-norm of the difference between the corresponding numerators, hence its Lipschitz property follows from the preceding bound.

Finally, note that $\|\widetilde{r}\|_{\rm Lip}\leq\|g^{\prime}\|_{\infty}\|r\|_{1}$ is uniformly bounded on $\mathcal{A}^{\sf e}_{\infty}(a)$, hence the same applies for $\theta\mapsto\widetilde{r}(\theta)\theta-x$. With $\varphi(|\cdot|)$ Lipschitz on compacts, it thus follows that $\|F(r,x,\cdot)\|_{\rm Lip}$ is uniformly bounded on $\mathcal{A}^{\sf e}_{\infty}(a)$. ∎

We turn to get the analog of Proposition 4.9 for this model, thereby establishing Theorem 1.10(b) in this setting.

Proposition 4.11**.**

(a) If the function $\alpha(\ell,z)=\alpha(\ell)$ does not separately depend on $z$, then the centered Euclidean ball $\mathbb{B}$ is an invariant solution to (1.12).

(b) Assumption (I) holds whenever $\alpha(\ell,z)=\alpha(z)\ell$, $\varphi(t)=t^{\beta}$ for some $\beta\in\mathbb{R}$.

(c) If $\alpha(\ell,z)\equiv 0$, $\beta<0$ and we force $\widetilde{r}=r$, then $\mathbb{B}$ is the unique attractive solution of (1.12), whenever $r_{0}\in C_{+}^{1}(\mathbb{S}^{n-1})$.*

Proof.

(a). Following the proof of Proposition 4.9(a), the invariance of $|\theta-\alpha(1)z|$ to a common rotation of $(z,\theta)\in\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}$ yields that the uniform measure on $\alpha(1)\mathbb{S}^{n-1}$ is invariant for $(x_{t}^{1,\mathbb{B}})$.

(b). It is easy to check that having $\varphi(t)=t^{\beta}$ in (1.35) results with (1.28) holding, whereas (1.29) is satisfied by the assumed linearity in $\ell$ of $\alpha(\ell,z)$. Thus, Assumption (I) holds in this case and from part (a) we know that when $\alpha(\ell,z)\equiv 0$ the ball $\mathbb{B}$ is an invariant solution to (1.12).

(c). For $H\equiv 0$ and $\varphi(t)=t^{\beta}$, one has that

[TABLE]

In particular, for any $r_{0}\in C_{+}^{1}(\mathbb{S}^{n-1})$, the ode (1.12) with $\overline{b}(r)$ of (4.4) admits $C^{1}$ solution $(r_{t})_{t\geq 0}$ and in view of Proposition 3.5, for $\mathbb{B}$ to be attractive for (1.12) when $\beta<0$, it suffices to have along such solution $r_{t}$, the regularized $\widetilde{r}_{t}$ no smaller at angle $\operatorname{arg\,max}_{\theta}r_{t}$ than at $\operatorname{arg\,min}_{\theta}r_{t}$. This indeed holds once we force $\widetilde{r}=r$. ∎

Appendix A

Proof of Proposition 1.6.

(a) Since $b(r,x)$ is a ratio of $F(r,x,\cdot)$ and the scalar $y_{r,x}$, with $y_{r,x}\geq\omega_{n}a^{n-1}$ on $\mathcal{A}_{\infty}(a)$ and $F$ assumed Lipschitz in $L^{p}$-norm on this set (by (1.13)), it suffices to show that in addition $(r,x)\mapsto y_{r,x}$ is Lipschitz in that norm (on $\mathcal{A}_{\infty}(a)$). To this end, note that for any $(r,x),(r^{\prime},x^{\prime})\in\mathcal{A}^{\sf e}_{\infty}(a)$ and $1/p+1/q=1$,

[TABLE]

by Hölder’s inequality and expanding $|r^{n-1}-(r^{\prime})^{n-1}|$. Considering either $p=q=2$, or $p=\infty$, $q=1$, our claim follows from (1.13) at $p$, since $p\mapsto\omega_{n}^{-1/p}\|\cdot\|_{p}$ is non-decreasing for functions on $\mathbb{S}^{n-1}$.

Turning next to $h$, for any $(r,x),(r^{\prime},x^{\prime})\in\mathcal{A}^{\sf e}_{\infty}(a)$, by (1.13), (1.14),

[TABLE]

as needed.

Fixing $a\in(0,1)$, and dealing simultaneously with $p=2$ and $p=\infty$, we proceed to show that $r\mapsto\overline{b}(r)$ is Lipschitz from $(\mathcal{A}_{\infty}(a),\|\cdot\|_{p})$ to $(L^{p}(\mathbb{S}^{n-1}),\|\cdot\|_{p})$. To this end note that the Markov chain on $\mathbb{S}^{n-1}$ of transition kernel density $\mathsf{K}_{r}(\xi,\theta)=F(r,H(r,\xi),\theta)$ with respect to the uniform measure, is, by (1.22), uniformly ergodic throughout $r\in\mathcal{A}_{2}(a)$, with ergodicity coefficient depending only on $a$ (via $\underline{F}(a)$), and as such, has a unique invariant measure $\mu_{r}$ on $\mathbb{S}^{n-1}$. Further, as the rule $H$ is non-random, the invariant measure $\nu_{r}$ of the Markov chain $\{x^{1,r}_{T_{i}}\}$ is merely the push-forward by $\xi\mapsto H(r,\xi)$ of the aforementioned $\mu_{r}$ with $\overline{b}$ of (1.12) represented alternatively as

[TABLE]

Next, recall the preceding where we saw that (1.13) results with (1.23) holding at the corresponding value of $p$. Combining the latter with (1.14), we deduce that the analogous to (1.23), uniform over $\mathcal{A}^{\sf e}_{\infty}(a)$ Lipschitz bounds, apply also for $b_{\star}(\cdot,\cdot)$. Thus, utilizing the characterization of total variation norm of finite signed measures (cf. [17, page 124]), we have for some finite constant $C=C(a)$ and all $r,r^{\prime}\in\mathcal{A}_{\infty}(a)$,

[TABLE]

Turning to the first term, recall that $b_{\star}(r,\cdot)\leq a^{-(n-1)}F(r,\cdot)$ throughout $\mathcal{A}_{\infty}(a)$, so in view of (1.16) it suffices to bound $\|\mu_{r}-\mu_{r^{\prime}}\|_{{\rm TV}}$. Denoting by $\mathcal{M}_{1}$ the space of signed Borel measures on $\mathbb{S}^{n-1}$ with total variation one, we find that (per notation in [32, (2.1)]), for some $K=K(a)$ and $C=C(a)$,

[TABLE]

(using in the above, also (1.13) and (1.14)). Consequently, by [32, Corollary 3.1], for some $\widetilde{C}=\widetilde{C}(a)$ which depends on the uniform ergodicity coefficient,

[TABLE]

thereby completing the proof.

(b) Consider the Banach space $(C(\mathbb{S}^{n-1}),\|\cdot\|_{\infty})$, and the subset $\mathcal{A}_{\infty}(a)$ for some $a<\inf_{\theta}r_{0}(\theta)\wedge\|r_{0}\|^{-1}_{\infty}$. Since per (1.15), the map $r\mapsto\overline{b}(r)$ is Lipschitz from $L_{\infty}$ to $L_{\infty}$ in $\mathcal{A}_{\infty}(a)$, there exists a unique continuous solution $(r_{t})$ to (1.12) locally in time, defined up to the first exit time of the set $\mathcal{A}_{\infty}(a)$, cf. [1, Theorem 7.3]. Hence, Proposition 1.7 holds as long as the solution is defined, and we can bound the growth of its $L^{2}$-norm by Hölder’s inequality

[TABLE]

This in turn implies the following bound on the growth of $F(r_{t},\cdot,\cdot)$,

[TABLE]

in terms of $\overline{F}(p(t))$ of (1.16), thereby leading to

[TABLE]

Thus, by the ode (1.12), as long as the solution is defined, we have control on the growth of its $L_{\infty}$-norm,

[TABLE]

Given any $T<\infty$, taking in the beginning $a<q(T)^{-1}\wedge\inf_{\theta}r_{0}(\theta)$, ensures the existence and uniqueness of a continuous solution up to time $T$.

(c) The minorisation (1.25) implies by standard theory of general state space Markov chains (see [37, Theorem 8]), that for any $r\in C_{+}(\mathbb{S}^{n-1})$ the embedded chain $\{x_{T_{i}}^{1,r}\}$ has a unique invariant measure $\nu_{r}$, with the uniform on $\mathcal{A}^{\sf e}_{\infty}(a)$ convergence

[TABLE]

The proof is by coupling, which extends to the process $(x^{1,r}_{t})_{t\geq 0}$ with $x^{1,r}_{0}=x$ and its stationary version $(\mathsf{x}^{1,r}_{t})_{t\geq 0}$ (i.e. starting at distribution $\nu_{r}$ and using the same jump times $\{T_{i}\}$ for both processes). It follows that the processes coalesce at the coupling time $\mathcal{T}_{x}$ with

[TABLE]

for some positive constant $c=c(\delta,n_{0})=c(a)$, any $t\geq 0$ and all $(r,x)\in\mathcal{A}^{\sf e}_{\infty}(a)$. By the triangle inequality, employing this coupling for proving (1.17), we separately bound

[TABLE]

and

[TABLE]

There is no contribution to (A.1) from $s\geq\mathcal{T}_{x}$ and a-priori $\|b(r,x)\|_{2}\leq C(a)<\infty$ for all $(r,x)\in\mathcal{A}^{\sf e}_{\infty}(a)$. Hence (A.1) is at most $4C(a)^{2}\mathbb{E}\mathcal{T}_{x}/t\leq 4C(a)^{2}/(ct)$. By stationarity the expectation in (A.2) is independent of $t_{0}$ and utilizing the Markov property, it equals

[TABLE]

where by Fubini

[TABLE]

Using the preceding coupling per value of $x$ in (A.3), we deduce that

[TABLE]

where by Cauchy-Schwarz, for the pre-compact $\mathcal{K}_{r}=\{x\in\mathbb{R}^{n}\colon(r,x)\in\mathcal{D}(F),\,x\in{\rm Image}(H(r,\cdot))\}$,

[TABLE]

Plugging into (A.3) this uniform bound on $\Gamma_{r,x}$ and the uniform tail bound on $\mathcal{T}_{x}$, bounds the term (A.2) by $4C(a)^{2}/(ct)$, thereby completing the proof.

(d) The processes $(R^{\epsilon}_{t\wedge\zeta^{\epsilon}},x^{\epsilon}_{t\wedge\zeta^{\epsilon}})$ are, for some $a=a(r_{0},\delta)$, within $\mathcal{A}^{\sf e}_{\infty}(a)$, whereby $y_{r,x}\geq\omega_{n}a^{n-1}$. Thus, given (1.4), the definition of $b(r,x)$ and the bound (A.21) on $\|f\star g_{\eta}-f\|_{2}$, it suffices for Assumption (C) to bound the rate of convergence to zero as $t\to 1$, for the rhs of (A.21) at $f=F(r,x,\cdot)$, uniformly over $(r,x)\in\mathcal{A}^{\sf e}_{\infty}(a)$. In particular, recall from (A.18) that $\|T_{t}f-f\|_{2}\leq 2\|f\|_{\rm Lip}(1-t)^{1/2}$ (since $\|x-y\|_{2}=\sqrt{2(1-\langle x,y\rangle)}$ when $x,y\in\mathbb{S}^{n-1}$), hence (1.26) suffices for Assumption (C) to hold.

(e) By our construction and (1.4), any bump added to some $R^{\epsilon}_{t}$ is supported on a spherical cap in $\mathbb{S}^{n-1}$ whose radius is at most

[TABLE]

Proceeding to cover $\mathbb{S}^{n-1}$ by spherical caps $\{S_{j}\}_{j=1}^{N_{\epsilon}}$ of radius $\eta_{\epsilon}$, in view of (A.4) the growth of $R^{\epsilon}_{T^{\epsilon}_{i}}$ somewhere within $S_{j}$ requires that $x^{\epsilon}_{T^{\epsilon}_{i}}$ hit the concentric cap of radius $2\eta_{\epsilon}$. Further, for $\kappa^{\epsilon}$ of (1.27) and $\mathfrak{a}=\mathfrak{a}(r_{0},T):=(1+\|r_{T}\|_{2})^{-1}\wedge\inf_{\theta}r_{0}(\theta)$,

[TABLE]

Thus, for any $T^{\epsilon}_{i}\leq\kappa^{\epsilon}$, the probability of $x^{\epsilon}_{T^{\epsilon}_{i}}$ hitting a spherical cap of radius $2\eta_{\epsilon}$, is by (1.16) and (A.5), at most $\overline{F}(\mathfrak{a})\omega_{n}(2\eta_{\epsilon})^{n-1}$. Consequently, the total number of changes in $R_{t}^{\epsilon}$ restricted to $S_{j}$ and time interval $[0,T\wedge\kappa^{\epsilon}]$, is stochastically dominated by a Poisson variable of mean

[TABLE]

with $C_{\star}=C_{\star}(r_{0},T,\mathfrak{a})$ finite. Recall (1.4) that we add to the domain the (local) bump $\epsilon y^{-1}_{r,x}g_{\eta}=\epsilon^{1/n}\eta^{n-1}g_{\eta}$, whose radial height is at most

[TABLE]

for some absolute finite constant $\Lambda$ and any $(r,x)$. Consequently, by a union bound over the $N_{\epsilon}$ spherical caps in our covering of $\mathbb{S}^{n-1}$,

[TABLE]

By volume considerations $N_{\epsilon}\leq c_{n}\eta_{\epsilon}^{-(n-1)}$ for some universal constant $c_{n}$. In view of (A.6) and the definition (A.4) of $\eta_{\epsilon}$, this translates to $N_{\epsilon}\leq C(n,r_{0})\lambda_{\epsilon}^{n-1}$ for some finite $C(n,r_{0})$. Thus, from the (super) exponential in $\lambda_{\epsilon}$ tail probabilities for a Poisson($\lambda_{\epsilon})$ law, we deduce that for $\delta_{n}=(\|r_{0}\|_{\infty}+2C_{\star}\Lambda)^{-1}$,

[TABLE]

and our claim then follows from the definition (1.18) of $\zeta^{\epsilon}(\cdot)$. ∎

Proof of Proposition 1.7.

For $C_{+}(\mathbb{S}^{n-1})$-solutions $(r_{t})_{t\geq 0}$, (1.12) is valid in pointwise sense and we can compute

[TABLE]

yielding $Leb(r_{t})=Leb(r_{0})+t$, for any $t\geq 0$.

We proceed to similarly verify (1.3) for any $(r,x)\in\mathcal{D}(F)$. Indeed,

[TABLE]

The second term gives exactly $\epsilon$. Upon applying Cauchy-Schwarz inequality to the first term and using the $L^{2}$-approximation property (A.21) of the spherical approximate identity as $\epsilon\to 0$, we see that the whole expression is $\epsilon+o(\epsilon)$. ∎

Proof of Lemma 4.1.

Recall the definition (1.1) of spherical convolution. For any multi-index $\alpha$ with $|\alpha|=k\in\{0,1,2,3\}$ and any $z\in\mathbb{S}^{n-1}$, we have that

[TABLE]

where $\partial^{\alpha}$ is any $k$-th order derivative with respect to $z$ variable on $\mathbb{S}^{n-1}$. Since $\mathbb{S}^{n-1}$ is compact, hence the supremum in the last line is finite and depends only on $g$, we arrive at the claimed bound (4.1) on $\|\widetilde{r}-\widetilde{r}^{\prime}\|_{C^{3}(\mathbb{S}^{n-1})}$. In particular, from (4.1) with $r^{\prime}=0$ (hence $\widetilde{r}^{\prime}=0$), we have that $\|\widetilde{r}\|_{C^{3}(\mathbb{S}^{n-1})}\leq C\|r\|_{2}\leq Ca^{-1}$ for any $r\in\mathcal{A}_{2}(a)$. Since convolution with $g\geq 0$ does not lower the minimal value of $r\in\mathcal{A}_{2}(a)$, it further follows that then $\inf_{\theta}\widetilde{r}(\theta)\geq a$. Finally, from (1.33) the continuous and positive $\ell-\alpha(\ell,z)$ is bounded away from zero on the compact $[a,Ca^{-1}]\times\mathbb{S}^{n-1}$. In particular, for some $\eta=\eta(a,C)>0$ and all $(r,x)\in\mathcal{A}^{\sf e}_{2}(a)$ the radial distance of $x$ from $\partial D_{\widetilde{r}}$ must be at least $\eta$. The uniform over $\mathcal{A}_{2}(a)$ control on $\|\widetilde{r}\|_{C^{3}(\mathbb{S}^{n-1})}$ implies a uniform bound on the Hessian of $\widetilde{r}(\cdot)$ and hence that $\overline{\mathbb{B}}(x,5\delta)$ be within the open, star-shaped domain $D_{\widetilde{r}}$ for some $\delta=\delta(Ca^{-1},\eta)>0$. In conclusion, if $(r,x)\in\mathcal{A}^{\sf e}_{2}(a)$ then $(\widetilde{r},x)\in\mathcal{K}_{a,C,\delta}$ of (4.2), as claimed. ∎

Proof of Proposition 4.3.

Fixing $a\in(0,1)$, $C<\infty$ and $\delta>0$, we write for brevity $\|(\widetilde{r},x)\|=\|\widetilde{r}\|+|x|$ and $\|\widetilde{r}\|$ for the $C^{2,\gamma}(\mathbb{S}^{n-1})$ norm of $\widetilde{r}$. By the Arzelà-Ascoli theorem, a $C^{3}(\mathbb{S}^{n-1})$ closed ball is $C^{2,\gamma}(\mathbb{S}^{n-1})$-compact, which clearly extends to the $(\widetilde{r},x)$-pairs with $\overline{\mathbb{B}}(x,5\delta)\subseteq D_{\widetilde{r}}$, hence also to $\mathcal{K}_{a,C,\delta}$. It thus suffices to prove that $(\widetilde{r},x)\mapsto F(r,x,\cdot)$ is locally Lipshitz on $\mathcal{K}_{a,C,\delta}$. Specifically, fixing $(\widetilde{r}_{\star},x_{\star})\in\mathcal{K}_{a,C,\delta}$, we denote by $\mathcal{N}_{\eta}$ the set of $(\widetilde{r},x)\in\mathcal{K}_{a,C,\delta}$ with $\|(\widetilde{r},x)-(\widetilde{r}_{\star},x_{\star})\|<\eta$ and proceed to show that

[TABLE]

where $\eta>0$ and $\kappa_{i}<\infty$ depend only on $\|\widetilde{r}_{\star}\|$ and $\delta$. To this end, let $D=D_{\widetilde{r}}$ and $D^{\prime}=D_{\widetilde{r}^{\prime}}$, noting that for $\eta<\delta$ necessarily $\overline{\mathbb{B}}(x^{\prime},3\delta)\subset D\cap D^{\prime}$. One can then construct a $C^{3}$-diffeomorphism $\phi:\mathbb{R}^{n}\to\mathbb{R}^{n}$ that maps $D$ to $D^{\prime}$ such that $\widetilde{r}^{\prime}(\theta)\theta=\phi(\widetilde{r}(\theta)\theta)$, while being the identity map on $\overline{\mathbb{B}}(x^{\prime},\delta)$, and such that for some $\kappa_{1}<\infty$,

[TABLE]

Here ${\rm Id}(y)=y$ is the identity map, whose Jacobian matrix is $I_{n}$. Since Green’s function is harmonic (in its second argument), away from its pole,

[TABLE]

where at $y=x^{\prime}$ this identity still holds, albeit in the distributional sense, since evaluating the lhs at any smooth function $f(\cdot)$ gives $f(\phi^{-1}(x^{\prime}))-f(x^{\prime})=0$ (by definition of the map $\phi$). Denoting by $\mathsf{D}\phi(y)$ the Jacobian matrix of $\phi$ at $y$ and

[TABLE]

(with a pole singularity at $y=x^{\prime}$), we have that

[TABLE]

Thus, upon combining (A.9) and (A.10) we arrive at

[TABLE]

We show next that for some $\kappa_{2}$ finite,

[TABLE]

Indeed, since $\mathsf{D}\phi=I_{n}$ throughout $\mathbb{B}(x^{\prime},\delta)$, there is no singularity on the rhs of (A.11). With the boundary condition $(\Gamma-G_{\widetilde{r}})(x^{\prime},\cdot)\equiv 0$ on $\partial D$, we have by the global Schauder estimate of [19, Theorem 5.26], applied to the Poisson equation (A.11), combined with the maximum principle for the same equation (see [19, Proposition 2.15]), that for some $\kappa_{3}$ finite,

[TABLE]

We arrive at (A.12) upon further bounding the preceding rhs by

[TABLE]

The latter inequality is due to (A.8), since $G_{\widetilde{r}^{\prime}}(x^{\prime},\phi(\cdot))$ is $C^{2,\gamma}$ away from its pole at $x^{\prime}$, up to the $C^{3}$-boundary (at least for $\eta$ small, thanks to (A.8)). Recall (1.32), that

[TABLE]

We thus consider (A.12) at $\widetilde{r}=\partial D$, and get (A.7) upon using (A.8) and the fact that the inward normal unit vectors of $\widetilde{r}$ and $\widetilde{r}^{\prime}$ at points $y$ and $\phi(y)$ respectively, are close to each other (since $\|\widetilde{r}^{\prime}-\widetilde{r}\|<2\eta$, possibly reducing the value of $\eta>0$ as needed).

In view of (A.7), we get the local Lipschitz property of $(\widetilde{r},x)\mapsto F(r,x,\cdot)$, upon showing that for $\delta>\eta>0$ as above and some $\kappa_{5}$ finite

[TABLE]

Indeed, by the preceding construction, $\overline{\mathbb{B}}(x^{\prime\prime},3\delta)\subset\overline{\mathbb{B}}(x,5\delta)\subset D$ for any $x^{\prime\prime}$ on the line segment connecting $x$ and $x^{\prime}$. Note that $\|\partial_{z_{i}}G_{\widetilde{r}}(z,\cdot)\|_{C^{2}({\overline{D}}\backslash(\mathbb{B}(z,\delta))}$ are bounded uniformly over $z\in D$ of distance at least $3\delta$ from $\partial D$. Thus, applying the mean value theorem to $x\mapsto G(x,y)$, $x\mapsto\partial_{y_{i}}G(x,y)$ and $x\mapsto\partial^{2}_{y_{i}y_{j}}G(x,y)$, we get for some finite $\kappa_{6}$,

[TABLE]

from which (A.13) follows (thanks to (1.32)). ∎

Lemma A.1**.**

Fixing $n\geq 2$, a collection $g_{\eta}$ forms a local spherical approximate identity (as in Definition 1.2), if and only if

[TABLE]

for some continuous $\phi_{\eta}(s)\geq 0$, supported on $[-1,1]$, with $\|\phi_{\eta}\|_{\infty}=1$ and for $\eta\in(0,1)$,

[TABLE]

are bounded away from zero. In particular, this applies whenever $\phi_{\eta}(\cdot)=\phi(\cdot)$ is independent of $\eta$, and not identically zero.

Proof.

The mapping from $g_{\eta}(\cdot)$ to $\phi_{\eta}(\cdot)$ is merely a change of argument, with $g_{\eta}(\cdot)$ continuous and non-negative iff $\phi_{\eta}(\cdot)$ are. Further, having the arbitrary constant $c_{\eta}>0$ allows us to set wlog $\|\phi_{\eta}\|_{\infty}=1$. Our requirement that $g_{\eta}(\langle z,\theta\rangle)$ be supported on the spherical cap $|\theta-z|\leq 2\eta$, $\theta\in\mathbb{S}^{n-1}$ translates into $g_{\eta}(\cdot)$ supported on $[1-2\eta^{2},1]$, or equivalently, to $\phi_{\eta}(\cdot)$ supported on $[-1,1]$, whereas under (A.14) the uniform boundedness of $\eta^{n-1}\|g_{\eta}\|_{\infty}$ amounts to the same for $c_{\eta}$.

Next, by a change of variable (see [9, (2.1.8)]),

[TABLE]

iff $c_{\eta}^{-1}$ are given by (A.15). Further, recall [9, (2.1.8)], that for every $z\in\mathbb{S}^{n-1}$ and $f\in L^{2}(\mathbb{S}^{n-1})$,

[TABLE]

where $\{T_{t}\}_{t\in[-1,1]}$ is a family of translation operators [9, (2.1.6)] defined by

[TABLE]

Here $d\ell_{x,t}$ denotes Lebesgue measure on $\{y\in\mathbb{S}^{n-1}\colon\langle x,y\rangle=t\}$. These operators satisfy

[TABLE]

see [9, Lemma 2.1.7]. Hence, by (A.16)-(A.19) and the convexity of the norm, for all $\eta\in(0,1)$,

[TABLE]

with the rhs of (A.21) converging to zero as $\eta\to 0$ (see (A.19)). Finally, note that when $\phi_{\eta}(\cdot)=\phi(\cdot)$ is independent of $\eta$, we have from (A.15) that $\eta\mapsto c_{\eta}^{-1}$ is monotone, with $c_{0}^{-1},c_{1}^{-1}>0$, for any non-zero $\phi$ and $n\geq 2$. ∎

Acknowledgments. We thank Julián Fernández Bonder and Luis Silvestre for useful conversations and Martín Arjovsky for pointing out the plausibility of Lemma 2.1 and its proof.