Long time asymptotics of mixed-type Kimura diusions Guillaume BalBinglu ChenZhongjian Wang Contents

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Long time asymptotics of mixed-type Kimura diﬀusions

Guillaume Bal∗Binglu Chen†Zhongjian Wang‡

Contents

1 Introduction 2

2 The C0Semigroup in one-space dimension 4

3 Invariant Measures in dimension one 5

3.1 Functional settings and index associated to L......................... 5

3.2 Null Space of L∗......................................... 7

3.3 ProofofTheorem3.1....................................... 10

3.3.1 Outlineofproof ..................................... 10

3.3.2 Reduction to a ﬁrst order system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3.3 Representation of solutions of the modeling operator . . . . . . . . . . . . . . . . . 12

3.3.4 ProofofTheorem3.1 .................................. 16

4 Exponential convergence to invariant measures 18

4.1 Case of one/two tangent boundary points . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.1 Function Space C(α, β) ................................. 18

4.1.2 Estimation of λ0and rate of convergence . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Case with two transverse boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.1 (W)-Lyapunov-Poincar´e inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.2 Longtimebehaviors ................................... 24

5 Long time behaviour in two dimension 25

5.1 Lyapunovfunction ........................................ 25

5.2 Exponential convergence to invariant measure . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Numerical Experiments 34

6.1 One dimensional cases with mixed boundary conditions . . . . . . . . . . . . . . . . . . . 34

6.2 2Dwithtangentdiagonal .................................... 36

A Appendix 38

Abstract

This paper concerns the long-time asymptotics of diﬀusions with degenerate coeﬃcients at the

domain’s boundary. Degenerate diﬀusion operators with mixed linear and quadratic degeneracies

ﬁnd applications in the analysis of asymmetric transport at edges separating topological insulators.

In one space dimension, we characterize all possible invariant measures for such a class of operators

and in all cases show exponential convergence of the Green’s kernel to such invariant measures. We

generalize the results to a class of two-dimensional operators including those used in the analysis of

topological insulators. Several numerical simulations illustrate our theoretical ﬁndings.

Key Words: Degenerate Diﬀusion Operators; Invariant Measure; Long Time Asymptotics; Fred-

holm Index; Lyapunov Functions.

MSC: 35K65, 60J70, 47D06, 37C40.

∗Departments of Statistics and Mathematics and CCAM, University of Chicago, Chicago, IL 60637;

guillaumebal@uchicago.edu

†Department of Mathematics, University of Chicago, Chicago, IL 60637; blchen@uchicago.edu

‡Departments of Statistics and CCAM, University of Chicago, Chicago, IL 60637; zhongjian@uchicago.edu

arXiv:2210.10037v2 [math.AP] 2 Nov 2022

1 Introduction

This paper analyzes the long time behavior of diﬀusion processes with inﬁnitesimal generator given by

a mixed type Kimura operator Lon one-dimensional and two-dimensional manifolds with corners.

In two-space dimensions, the mixed type Kimura operator Lacts on functions deﬁned on a manifold

with corner P. A boundary point p∈bP has a relatively open neighborhood Vthat is homomorphic to

either R2

+or R+×R, where we assume that pmaps to (0,0). We say that pis a corner in the ﬁrst case

and an edge point in the second case. More details are presented in Section 5.

In this paper, Lis a degenerate second-order diﬀerential operator such that for each edge, the coeﬃ-

cients of the normal part of the second-order term vanish to order one or two. For an edge point p, when

written in an adapted system of local coordinates on R+×R,Ltakes the form:

L=a(x, y)xm∂xx +b(x, y)xm−1∂xy +c(x, y)∂yy +d(x, y)xm−1∂x+e(x, y)∂y,(1)

where we assume that a, b, c, d, e are smooth functions and that a(x, y)>0 and c(x, y)>0. Also

m∈ {1,2}. When m= 1, the edge x= 0 is of Kimura type in that the coeﬃcients vanish linearly

towards it. When m= 2, the edge x= 0 is of quadratic type as the coeﬃcients now vanish quadratically.

For a corner p, as an intersection point of two edges, Lhas the following normal form:

L=a(x, y)xm∂xx +b(x, y)xm−1yn−1∂xy +c(x, y)yn∂yy +d(x, y)xm−1∂x+e(x, y)yn−1∂y,(2)

when written in an adapted system of local coordinates on R2

+, where a(x, y)>0, c(x, y)>0, and

m, n ∈ {1,2}.

Associated to the operator Lis a C0semigroup Qt=etL solution operator of the Cauchy problem

∂tu=Lu

with initial conditions u(x, 0) = f(x) at t= 0. The operator etL and some of its properties are presented

in detail in [10]. The main objective of this paper is to analyze the long time behavior of etL, and in

particular convergence to appropriately deﬁned invariant measures.

It turns out that the number of possible invariant measures and their type (absolutely continuous

with respect to one-dimensional or two-dimensional Lebesgue measures or not) strongly depend on the

structure of the coeﬃcients (a, b, c, d, e). We thus distinguish the diﬀerent boundary types that inﬂuence

the long time asymptotics of transition probabilities.

Deﬁnition 1.1. A Kimura edge Eis called a tangent (Kimura) edge when d(0, y)=0and a transverse

(Kimura) edge when d(0, y)>0.

A quadratic edge Eis called a tangent (quadratic) edge when d(0,y)

a(0,y)<1, a transverse (quadratic) edge

when d(0,y)

a(0,y)>1, and a neutral (quadratic) edge when d(0,y)

a(0,y)= 1.

We assume that:

Assumption 1.1. Every edge is either tangent, transverse, or neutral.

Note that we do not consider the setting with d(0, y)<0 on a Kimura edge. In such a situation,

diﬀusive particles pushed by the drift term d(0, y)<0 have a positive probability of escaping the domain

P. We would then need to augment the diﬀusion operator with appropriate boundary conditions.

The long-time analysis in two-dimensions for a class of operators including a speciﬁc example of

interest in the ﬁeld of topological insulators [6] is given in section 5. In the application in [6], the

degenerate diﬀusion equation describes reﬂection coeﬃcients of waveﬁelds propagating in heterogeneous

media, which model the separation between topological insulators; see [5, 7]. Before addressing two-

dimensional operators, we consider the simpler one-dimensional setting, where we need to consider ten

diﬀerent scenarios, listed in table 1 below, depending on the form of the generator Lat the domain’s

boundary. These cases are analyzed in detail in sections 2-4.

To describe all invariant measure on the interval [0,1] in one dimensional, consider

L=a(x)xm(0)(1 −x)m(1) d2

dx2+b(x)xm(0)−1(1 −x)m(1)−1d

dx,(3)

with a(x), b(x)∈C∞([0,1]).

For i= 0,1, we say that x=iis of Kimura type when m(i) = 1 and of quadratic type when m(i) = 2.

When x=iis of Kimura type, we assume that the vector ﬁeld b(x)d

dx is inward pointing at x=i. For

brevity, we use eaand e

bto denote

ea(x) = a(x)xm(0)(1 −x)m(1),e

b(x) = b(x)xm(0)−1(1 −x)m(1)−1.(4)

Deﬁnition 1.2. When x= 0 (1 resp.)is a Kimura endpoint, we say it is a tangent point if b(0) =

0 (b(1) = 0 resp.)and a transverse point if b(0) >0 (b(1) <0resp.).

When x= 0 (1 resp.)is a quadratic endpoint, we say it is a tangent point if b(0)

a(0) <1 ( b(1)

a(1) >−1resp.),

a transverse point if b(0)

a(0) >1 ( b(1)

a(1) <−1resp.), and a neutral point if b(0)

a(0) = 1 ( b(1)

a(1) =−1resp.).

The quadratic endpoint and the tangent Kimura endpoint are sticky boundary points in the sense

that the Dirac measure supported on them is an invariant measure. When both endpoints are transverse,

there is another invariant measure µwith full support on the whole interval. By computing the index of

Lon an appropriate H¨older space, we characterize the kernel space of L∗composed of invariant measures

for the diﬀusion L.

In both cases, starting from a point in P, the corresponding transition probability of the diﬀusion

converges to the invariant measure at an exponential rate. In cases with at least one tangent boundary

point, we consider a functional space of functions that vanish at the tangent boundary points and show

that Lhas a spectral gap on such a space. In the absence of tangent boundary points, we prove that the

invariant measure µsatisﬁes an appropriate Poincar´e inequality so that Lalso admits a spectral gap in

L2(µ). Our main convergence results for Qt=etL, whose properties are described in Theorem 2.1, are

summarized in Theorems 4.1 and 4.2 below.

In two space dimensions, we do not consider all possible invariant measures as a function of the

nature of the drift terms dand ein the vicinity of edges or corners. Instead, we restrict ourselves to the

following case:

Assumption 1.2. For Lon a 2 dimensional compact manifold with corners P, there is exactly one

tangent edge H, and when restricted to H,L|His transverse to both boundary points.

This case involves exactly one tangent edge with two transverse boundary points so that, applying

results from the one-dimensional case, we ﬁnd that Lhas a unique invariant measure µfully supported

on (the one-dimensional edge) H. Starting from any point pnot on the quadratic edge, we show in

Theorem 5.2 that the transition probability converges to µat an exponential rate in the Wasserstein

distance sense. The main tool used in the convergence is the construction of a Lyapunov function in

Theorem 5.1.

The setting of Pa triangle with two transverse Kimura edges while the third edge is quadratic with

transverse endpoints as described in Assumption 1.2 ﬁnds applications in the analysis of the asymmetric

transport observed at an edge separating topological insulators [6]; see Remark 5.2 below. The corre-

sponding one-dimensional version with one transverse Kimura point and one tangent quadratic point (see

entry (1,4) in Table 1 below) also appears in the analysis of reﬂection coeﬃcients for one-dimensional

wave equations with random coeﬃcients [20].

There is a large literature on the analysis of the long-time behavior of etL when Lis non-degenerate

and when Lis of Kimura type. In the latter case, Lis the generalized Kimura operator studied in [16].

In that work, Lis analyzed on a weighted H¨older space, denoted by Lγ:C0,2+γ(P)→C0,γ (P). Lγis

then Fredholm of index zero, which is used to characterize the nullspace of an adjoint operator and show

that the non-zero spectrum lies in a half plane Re µ < η < 0 so that for f∈C0,γ (P), etLfconverges to

a stable limiting solution at an exponential rate.

However, in the presence of quadratic edge/point, Lγis not Fredholm. The reason is that near such

quadratic edges or points, the operator may be modeled by an elliptic (non-degenerate) operator on an

inﬁnite domain (with thus continuous spectrum in the vicinity of the origin). We thus need another

approach that builds on the following previous works. In [1], the growth bound of a strongly continuous

positive semigroup is associated with the Lyapunov function. In [4], for a time continuous Markov process

admitting a (unique) ergodic invariant measure, the rate of convergence to equilibrium is studied using

a weaker version of a Lyapunov function called a φ-Lyapunov function and an appropriate Poincar´e

inequality.

An outline of the rest of this paper is as follows. The semigroup etL is analyzed in section 2 in the

one-dimensional case. The space of invariant measures associated with a one-dimensional diﬀusion L,

which depends on the structure of the drift term at the two boundary points, is constructed in section

3; see Table 1 for a summary. The exponential convergence of the kernel of etL (the Green’s function)

to an appropriate invariant measure over long times is demonstrated in section 4.

The operator etL in the two-dimensional setting is analyzed in [10]. For the class of operators satisfying

Assumption 1.2, the construction of the (unique normalized) invariant measure and the exponential

convergence of the kernel of etL to this measure in the Wasserstein sense are given in section 5. Numerical

simulations of stochastic diﬀerential equations presented in section 6 illustrate the theoretical convergence

results obtained in dimensions one and two.

2 The C0Semigroup in one-space dimension

Let Lbe the one-dimensional mixed-type Kimura operator on [0,1] given in (3). Let Qt=etL be the

solution operator of the Cauchy problem for the generator Land denote by qt(x, y) its kernel. Its main

properties are summarized in the following result:

Theorem 2.1. The operator Qtdeﬁnes a positivity preserving semigroup on C0([0,1]). For f∈

C0([0,1]), the function u(x, t) = Qtf(x)solves the Cauchy Problem for Lwith initial condition f(x)

in the sense that

lim

t→0+||Qtf−f||C0= 0.(5)

Proof. If both endpoints are of Kimura type, Lis the 1D Kimura operator. By [15, Sec 9, Theorem 1],

Qtdeﬁnes a positive and strongly continuous semi-group on Cm([0,1]) for m∈N. If both endpoints

are of quadratic type, we can regard Las a uniform parabolic operator on R(using, e.g., a change of

variables x=ezin the vicinity of x= 0 as we do below), which is well studied (see, e.g., [21]).

For the remaining case, we might as well assume that x= 0 is a Kimura endpoint and x= 1 a

quadratic endpoint. We intend to build the global solution out of local solution near the boundary. Let

([0,1−η], φ0),([η, 1], φ1) for some 0 < η < 1

4small be the coordinate charts so that pulling back Lto

these coordinate charts gives two local operators

L0=x∂2

x+b0∂x+xc(x)∂x, x ∈[0, φ0(1 −η)),

L1=∂2

z+d(z)∂z, z ∈(φ1(η),∞).

We extend these two local operators to the whole sample space

L0=x∂2

x+b0∂x+xc(x)ϕ0(x)∂x,e

L1=∂2

z+d(z)ϕ1(z)∂z

where ϕ0(x) is a smooth cutoﬀ function so that

ϕ0(x) = (1 for x∈[0, φ0(1 −2η)]

0 for x>φ0(1 −η),ϕ1(z) = (1 for z∈[φ1(2η),∞)

0 for z < φ1(η).

Let e

t,e

tbe the solution operators of e

L0,e

L1and denote their kernels by eq0

t,eq1

trespectively. Deﬁne

smooth cutoﬀ functions 0 ≤χ, ψ0, ψ1≤1 so that

suppψ0⊂[0,1−2η],suppψ1⊂[2η, 1], ψ0|suppχ≡1, ψ1|supp(1−χ)≡1.(6)

Given f∈C0([0,1]) and g∈C0([0,1] ×[0, T ]), set the homogeneous and inhomogeneous solution

operator as

Qtf=ψ0e

t[χf] + ψ1e

t[(1 −χ)f],

Atg=Zt

Qt−sg(s)ds.

Then

(∂t−L)e

Qtf=E0

tf:= [ψ0, L]e

t[χf]+[ψ1, L]e

t[(1 −χ)f],

(∂t−L)Atg= (Id −Et)g:= g−[ψ0, L]A0

t[χg]−[ψ1, L]A1

t[(1 −χ)g].

Our choice of χ, ψ0, ψ1(6) ensures that dist(supp[ψ0, L],suppχ)>0, dist(supp[ψ1, L],supp(1 −χ)) >

0, which ensures that E0

t, Etare bounded operators with operator norms bounded by O(e−c

t) for some

constant c > 0 as t→0+. Hence for T > 0 small enough, there exists an inverse (Id −Et)−1, which

can be expressed as a convergent Neumann series in the operator norm topology of C0([0,1] ×[0, T ]).

Finally we can express the solution operator by

Qtf=e

Qtf−At(Id −Et)−1E0

tf.

Since both e

t,e

tare strongly continuous, then so is e

Qt:

lim

t→0||e

Qtf−f||C0= 0.

As (Id −Et)−1E0

tis a bounded map from C0([0,1] to C0([0,1] ×[0, T ]), we have

||At(Id −Et)−1E0

t||C0([0,1])→(C0([0,1]),t)=o(t).

Therefore (5) holds. Let eqt, htbe the heat kernels of e

Qt,(Id −Et)−1E0

t. We express the heat kernel of

Qtas

qt(x, y) = eqt(x, y)−Zt

0Z1

0eqt−s(x, z)hs(z, y)dzds.

3 Invariant Measures in dimension one

In this section, we aim to ﬁnd all invariant measures of Lin spatial dimension one. For convenience of

computation, in this section we ﬁrst choose a global coordinate φso that (W0, φ),(W1, φ) is a cover of

[0,1/3],[2/3,1] under which Ltakes the following normal form:

1. In ([0,1/3], φ), L0has two possible forms:

L0=x∂2

x+b(x)∂x,if 0 is a Kimura endpoint

L0=∂2

z+b(z)∂z,if 0 is a quadratic endpoint

2. In ([2/3,1], φ), L1has two possible forms:

L1= (1 −x)∂2

x+b(x)∂x,if 1 is a Kimura endpoint

L1=∂2

z+b(z)∂z,if 1 is a quadratic endpoint

where we use bto denote the ﬁrst-order term in all cases.

Notation 1. We call the global coordinate φon [0,1] heat coordinates if L0, L1have forms above under

φ.

Let

b±= lim

x→1,0b(x) or b±= lim

z→±∞b(z).(7)

A straightforward derivation shows that, if in the original coordinate, L0=x2∂2

x+ (b−+ 1)x∂x, then

after turning to heat coordinates x=ez,L0takes the form ∂2

z+b−∂z.

3.1 Functional settings and index associated to L

Our functional setting involves local H¨older spaces, which diﬀer from the usual H¨older space (with

|·|k+γto denote its norm) near the boundaries and are variations of those used in [16]. In the following

deﬁnition, we assume cis a point away from the interval’s boundaries.

1. For quadratic type boundaries, with U= (−∞, c],

(a) f∈C0(U) belongs to Dγ(U) if the function fcan be continuously extend to z=−∞ and

the following norm is ﬁnite:

||f||γ,U =|f|γ,U + sup

z1≤z2≤cZz2

fdz,when it is not neutral; (8)

||f||γ,U =|f|γ,U + sup

z1≤z2≤c|Zz2

fdz|+ sup

z1≤z2≤c|Zz2

z1Zz

−∞

fdxdz|(9)

,when it is neutral;

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Longtimeasymptoticsofmixed-typeKimuradiusionsGuillaumeBal*BingluChenZhongjianWangContents1Introduction22TheC0Semigroupinone-spacedimension43InvariantMeasuresindimensionone53.1FunctionalsettingsandindexassociatedtoL.........................53.2NullSpaceofL............................................

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