1Introduction IddoBen-Ari CouplingfordriftedBrownianmotiononanintervalwithredistributionfromtheboundary

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ISSN:1083-589X in PROBABILITY

Coupling for drifted Brownian motion on an interval with redistribution from the boundary

^∗

Iddo Ben-Ari

^†

Abstract

We answer a question by Kolb and Wubker [7] on the threshold drift for Brownian Mo- tion on an interval with redistribution from the boundary. We do that by constructing an efficient coupling.

Keywords: Brownian Motion;Brownian motion; coupling; efficient coupling; redistribution;

jump-boundary; spectral gap property; jump-process; speed of convergence; spectral gap.

AMS MSC 2010:35P15; 60G40; 60J65.

Submitted to ECP on April 22, 2013, final version accepted on March 4, 2014.

1 Introduction

Consider an elliptic diffusion on a bounded domainD⊂R^d, with infinitesimal gener- atorL, that upon hitting the boundary∂Dis redistributed inDaccording to some pre- scribed probability measure (that could depend on the point of exit), restarting afresh, and repeated indefinitely. The redistribution is independent of the past. Under some standard smoothness assumptions on the boundary (and the redistribution measure as a function of the boundary point), the process is ergodic and converges exponentially fast to its invariant distribution in total variation. We call the resulting process diffusion with redistribution, or DR in short. This process has been studied by several authors: [4] [5] (analysis for BM with fixed deterministic redistribution), [3] [2](ergodicity, characterization, comparison), [8] (analysis of spectrum), [9] (holding times at boundary), and most recently [6][7] (coupling approach). The DR is never reversible and the problems of characterizing and estimating the exponential rate of convergence are typ- ically non-trivial. Unsurprisingly, the exponential rate is equal to the “spectral-gap" for

−L with the nonlocal boundary conditions imposed by the redistribution, that is, the minimal real part among all non-zero eigenvalues, yet this result is far from trivial to prove. As for estimation and comparison with other quantities, the main obstacle is that the corresponding eigenvalue may not be real. Yet, if it is real, then the spectral gap is bounded below by the principal eigenvalue for−Lwith the Dirichlet boundary condition. This realness condition was shown to hold under certain conditions and in some concrete examples. This has led to the question whether the principal Dirichlet eigenvalue is a lower bound when the realness condition is relaxed, at least when the underlying diffusion process is reversible. The question was formulated by Pinsky and the author of this note in [2].

∗This work was partially supported by NSA grant H98230-12-1-0225

†University of Connecticut, USA. E-mail:[email protected] http://homepages.uconn.edu/benari

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Recently, Kolb and Wubker [7] answered the question negatively by finding a counterexample. Their counterexample is obtained from drifted Brownian Motion on an interval(0, `), generated byL=Lσ,µwhere

Lσ,µ= σ² 2

d² dx² +µ d

dx,

for some constantσandµ, and deterministic redistribution to the center of the interval,

`

2. The underlying diffusion is reversible, and a straightforward calculation shows that the principal eigenvalue for−Lσ,µwith Dirichlet boundary condition tends to infinity as

|µ| → ∞. Kolb and Wubker showed that for all sufficiently large|µ|the spectral gap is constant, thus answering the question negatively. Their approach is essentially proba- bilistic and the core of the argument is construction of efficient coupling for the DR for large values of the drift coefficient. A problem they left open is the threshold value for µabove which the spectral gap is constant, and the authors conjectured it is equal to

√ 3^2πσ_`².

LetX = (X_t:t≥0)be the DR process on the interval(0, `), with underlying diffusion generated byLσ,µand redistribution from the boundary{0, `}to ^`₂. We refer the reader to [3] for the construction of the process. We denote the corresponding probability and expectation with initial distributionρby Pρ and Eρ, and ifρ = δx for somex ∈ (0, `), then we abbreviate and writePxandEx, respectively. LetPρ(t)denote the distribution ofX_tunderP_ρ, and for distributionsµ₁, µ₂, letd_t(µ₁, µ₂) =kPµ1(t)−P_µ₂(t)kT V, where k · k_{T V} is the total variation norm,

kµ1−µ2kT V = sup

f≥0,kfk∞=1

Z

f dµ1− Z

f dµ2

.

We writexforδ_xwhen it appears as a parameter ofd_t(·,·), and define d_t:= sup

µ₁,µ₂

d_t(µ₁, µ₂) = sup

x,y

d_t(x, y).

Let

D`=

u∈C²((0, `))∩L^∞((0, `)), lim

ζ→0⁺u(ζ) = lim

ζ→`⁻u(ζ) =u(` 2)

, and

Σσ,µ,`={λ∈C:∃u∈ D`, Lσ,µu=−λu}.

We define the spectral gap

γ1(σ, µ, `) = infn

Re(λ) :λ∈Σσ,µ,`\ {0}o . Similarly, let

D_`^D={u∈C²((0, `))∩C([0, `]), u(0) =u(`) = 0}, and

Σ^D_σ,µ,`={λ∈C:∃u∈ D^D_` , Lσ,µu=−λu}.

It is easy to see that the three parametersσ, µ, `can be reduced to one parameter by scaling. Indeed, it follows directly from the definition ofΣσ,µ,` that for`1>0,

Σσ,µ,`= `²₁σ²

`² Σ_1, `µ

`1σ2,`₁. (1.1)

The analogous identity holds for Σ^D_· as well. In fact, we will prove our results for σ= 1, `= 2π, and derive the general statements from these scaling identities.

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As it is well-known, Σ^D_σ,µ,`=

λ^D_k(σ, µ, `) =σ²π²(k+ 1)² 2`² + µ²

2σ² :k∈Z⁺

.

The eigenvalue

λ^D₀(σ, µ, `) =σ²π² 2`² + µ²

2σ² ^(1.2)

is called the principal eigenvalue.

Here is a summary of the relevant results.

Theorem 1.1([3],[7]).

a. X has a unique stationary distributionν, and

t→∞lim 1

t lndt(x, ν) =−γ1(σ, µ, `)<0, x∈(0, `).

b. There exists a constantµ₀=µ₀(σ, `)such that ifµ≥µ₀, then γ1(σ, µ, `) = 8σ²π²

`² ,

and there exists an efficient coupling.

We briefly recall the notion of coupling and efficient coupling. In this paper we will only consider Markovian couplings. A Markovian coupling forX with initial distributions µ₁ and µ₂ is a Markov process ((X_t¹, X_t²) : t ≥ 0) on(0, `)×(0, `), such that the marginals(X_t¹:t≥0)and(X_t²:t≥0)are copies ofX, the distribution ofX₀¹isµ1, and the distribution ofX₀²isµ2. Given a coupling, the coupling time (or meeting time)τCis defined through

τC= inf{t≥0 :X_t¹=X_t²}.

As it is well known and easy to verify, the tail of the coupling time dominatesd_t(µ₁, µ₂). That is,

d_t(µ₁, µ₂)≤P(τ_C> t).

As a result, couplings provide lower bounds onγ1(σ, µ, `). Coupling for X with initial distributionsµ₁, µ₂ is efficient if the coupling time decays at an exponential rate equal toγ₁(σ, µ, `):

t→∞lim 1

tlnP(τC> t) =−γ1(σ, µ, `).

The first part of the theorem (for a general domain, diffusion and redistribution) was proved in [3], and the second part was proved in [7]. We refer the reader to [7] for the explicit formula for theν. The proof of part b. of the theorem has led Kolb and Wubker to conjecture that

µ0(σ, `) =√ 32πσ²

` . (1.3)

We are ready to state our main result.

Theorem 1.2.

γ₁(σ, µ, `) =λ^D₀(σ, µ, `/2)∧λ^D₀(σ,0, `/4), (1.4) and there exists an efficient coupling.

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As it turns out, the first statement follows from a straightforward eigenvalue calculation, see Proposition 1.4 below. But this does not provide any insight or explanation.

The more substantial result is the existence of efficient coupling. This coupling both proves (1.4) and explains the origin of each of the terms. In addition, the proof gives upper and lower bounds ondt(x, y)for anyx, y∈(0, `)in terms of tails of exit times of BM or drifted BM from an interval.

The coupling used to prove the theorem is fairly simple. When the two copies of the DR are exactly `/2 units apart, they have the same increments. This guarantees that they meet when the copy in(0, `/2)exits this interval. When the distance between the two copies is different than`/2, then the (non-drifted) Brownian components of the increments are of the same magnitude but with opposite signs. The main idea is then to exploit the symmetry of the model to show that with this coupling, the first time when either the copies meet or are`/2 apart (note that both events can occur after a large number of redistributions), coincides with the exit time for BM from an interval of length`/4.

Suppose that underP_x, the processY = (Y_t:t≥0)is BM with diffusivityσand drift µ, withY₀=x, and letτ_`= inf{t≥0 :|Y_t| ≥ ^`₂}. Recall [10], that for|x|< `/2

t→∞lim 1

t lnPx(τ`> t) = lim

t→∞

1 t sup

|y|<`/2

lnPy(τ`> t) =−λ^D₀(σ, µ, `). (1.5) Since for allθ,Exe^θτ^` = 1 +θR∞

0 e^θtPx(τ`> t)dt, it follows from (1.5) that λ^D₀(σ, µ, `) = inf

θ:Exe^θτ^` =∞for somex∈(−` 2,`

2)

. (1.6)

From (1.2) we obtain

λ^D₀(σ, µ, `/2) = 2σ²π²

`² + µ²

2σ² ^andλ^D₀(σ,0, `/4) = 8σ²π²

`² . Combining this with Theorem 1.2 gives:

Corollary 1.3.

γ₁(σ, µ, `) = (_2σ2π²

`² +_2σ^µ²2 |µ| ≤√ 3^2πσ_`²;

8σ²π²

`² otherwise.

This proves (1.3).

The analytic proof to (1.3) is an immediate consequence to the following standard calculation.

Proposition 1.4. Letb= _2πσ^`µ₂. Then Σσ,µ,`= 4π²σ²

`²

0,2k²+ibk,b²+k²

2 :k∈Z− {0}

.

2 Proof of Theorem 1.2

We will reduce the problem from three parameters,σ, µ, `to one parameter by scaling, using the identity (1.1) and its analog forΣ^D_σ,µ,`. It follows that

γ1(σ, µ, `) =`²₁σ²

`² γ1(1, `µ

`1σ², `1), andλ^D₀(σ, µ, `) =`²₁σ²

`² λ^D₀(1, `µ

`1σ², `1).

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Therefore if we prove the theorem forσ= 1 and`= 2π, then for the general case we obtain:

γ1(σ, µ, `) = (2π)²σ²

`² γ1(1, `µ 2πσ²,2π)

= (2π)²σ²

`²

λ^D₀(1, `µ

2πσ², π)∧λ^D₀(1,0,π 2)

= (2π)²σ²

`² min

(`/2)²

σ²π² λ^D₀(σ, µ, `/2), (`/4)²

σ²(π/2)²λ^D₀(σ,0, `/4)

=λ^D₀(σ, µ, `/2)∧λ^D₀(σ,0, `/4).

In light of the above, in the remainder of this section we will assume the diffusivity σ= 1, and`= 2π. We will usebfor the drift coefficient, and without loss of generality, assume b ≥ 0. We let P_x^b denote the probability measure under which the process Y = (Yt : t ≥ 0)is BM on Rstarting from xwith diffusivity1 and drift b. Recall that τ_`= inf{t≥0 :|Yt| ≥ ^`₂}. We also denote byB= (B_t:t≥0)standard BM onR^.

2.1 Lower bound ondt

The lower bound is very simple, but it suggests the couplings to be used for the upper bound.

Lemma 2.1. dt≥P₀^b(τπ> t)∨P₀⁰(τ_π/2> t).

Thus from Theorem 1.1, Lemma 2.1 and (1.5), we obtain the following:

Corollary 2.2.

γ₁(1, b,2π)≤λ^D₀(1, b, π)∧λ^D₀(1,0, π/2).

Proof. Define the coupled processesX¹andX²on(0,2π)by letting X_t¹=π

2 +Bt+bt, X_t²=π+X_t¹, t≥0.

Let X be a DR starting at π. Let τ = inf{t : X_t¹ ∈ {0, π}}. Then τ has the same distribution asτπ (defined above (1.5)), under P₀^b. Observe that at timeτ, X_τ² =π or 2π according to whether X_τ¹ = 0orX_τ² = π. We continue the coupling fort ≥ τ by redefiningX¹andX²through:

X_t²=X_t¹=Xt−τ, t≥τ.

ThusX¹and X² are copies of the DR process, meeting at timeτ. Letf =1_[0,π). This gives

d_t≥E^π

2f(X_t)−E3π

2 f(X_t) =E f(X_t¹)−f(X_t²)

=P₀^b(τ_π> t).

To prove the second bound, observe that iff :R →Ris piecewise continuous andπ- periodic, then underP_x,f(X_t)has the same distribution asf(x+B_t+bt). Furthermore, for a fixedt, letg(u) =f(u+bt). Note that this transformation is one-to-one and onto from the set of piecewise continuousπ-periodic functions to itself, and that underP_x, the distribution off(Xt)coincides with the distribution ofg(x+Bt). Thus, if in addition 0 ≤ f ≤ 1, then dt(x, y) ≥ Exf(Xt)−Eyf(Xt) = Eg(x+Bt)−Eg(y−Bt). Choose now x= ^π₂, y = π, and letg be the π-periodic function equal to 1_[^π

4,^3π₄] on[0, π). Let τ = inf{t:|Bt|= ^π₄}. Thenτhas the same distribution asτπ/2underP₀⁰. Observe that fort < τ,g(x+B_t) = 1, andg(y−B_t) = 0. We define the coupling

B_t¹=x+B_t, B₀²=y, dB_t²=

(−dBt t < τ; dB_t t≥τ.

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Observe that eitherB_τ¹=B_τ² = ^3π₄ , orB¹_τ = ^π₄ andB_τ² =π+^π₄. Thus, according to the construction of the coupling and the choice ofg, it follows thatg(B¹_t) = g(B²_t)for all t≥τ. In particular,

dt≥Eg(x+Bt)−Eg(y−Bt) =E 1_{{τ >t}} g(B_t¹)−g(B²_t)

=P₀⁰(τ_π/2> t).

2.2 An upper bound ondt

We will use the following well-known lemma, which is an immediate corollary to the eigenvalue expansion for the transition function of drifted BM (e.g. [1, p. 94, Theorem 5.9]). The lemma could be avoided, but helps us obtain tighter upper bounds ondt. Lemma 2.3. LetY be the drifted BM generated byL1,bon(−^`₂,₂^`). Then

sup

|x|<^`₂

e^λ^D⁰^(1,b,`)tP_x^b(τ_`> t)−2`

πC²e^−bxcos(πx

` )

=e^−(λ^D¹^(1,b,`)−λ^D⁰^(1,b,`))tO_t(1),

whereC= q 2b

1−e^−b`

(b`)²+4π²

4π² ,λ^D₀(1, b, `) = ¹₂

π²

`² +b²

andλ^D₁(1, b, `)−λ^D₀(1, b, `) =^3π_2`₂². If(X, Y)is a coupling for the DR starting fromxandyrespectively, then we denote the joint distribution byPx,y and the corresponding expectation byEx,y. LetτCdenote the coupling time,

τ_C= inf{t≥0 :X_t=Y_t}.

Then for f ≥ 0, Exf(Xt)−Eyf(Xt) = Ex,y f(Xt)−f(Yt)

≤ kfk∞Px,y(τC > t). In particular,

dt(x, y)≤Px,y(τC > t).

From the triangle inequality,dt(x, y)≤dt(x, π) +dt(π, y)≤2 sup_xdt(x, π). Thus, dt≤2 sup

θ

Pθ,π(τC> t). (2.1)

We will now obtain an upper bound onP_θ,π(τ_C > t)through coupling of two copies of the DR. As will be shown below, we only need to considerθ ∈(π,2π). The coupling is constructed and analyzed in the proof of Lemma 2.4 below. This coupling consists of two stages.

Stage 1. The two copies have Brownian increments which are of the same magnitude and opposite signs. We begin this stage with one copy atπ and another copy atθ ∈ (π,2π). We stop this stage at timeT1, which is when either:

(a) The two copies meet, and thenτC=T1; or

(b) The distance between the copies is π. In this case we move to the second stage.

The first stage continues as long as none of the above conditions is met. While in the first stage, the first redistribution event, if occurs, can only occur when the copy starting in (π,2π)hits 2π⁻ (this will be explained below). Since the initial distance is strictly less than π, and condition (b) was not met, at this time the second copy is in (π,2π). As a result, at the redistribution, we again have one copy (the redistributed one) atπ and another copy in(π,2π). By induction, this holds for all redistributions during the first stage.

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Stage 2. In the second stage, the two copies share the same Brownian increments, and in particular, their distance remains πuntil the first redistribution event, at which they meet. This is the coupling timeτC.

Here are the details of the coupling. Fixθ∈(π,2π). LetX¹andX²be given by X_t¹=θ−Bt+bt, X_t²=π+Bt+bt, t≥0.

Define the stopping times

σ₁= inf{t:X_t¹=X_t²}, σ₂= inf{t:X_t¹−X_t²=π}, andσ=σ₁∧σ₂. Observe that

X_t¹=X_t²if and only ifBt=θ−π 2 . Similarly,

X_t¹−X_t²=πif and only ifBt=θ−2π 2 . Therefore,σis the exit time forBfrom the interval(^θ−2π₂ ,^θ−π₂ ). Next, let

τ₁= inf{t:X_t¹= 2π}, τ₂= inf{t:X_t²= 0}, andτ=τ₁∧τ₂. We have that

X_t¹= 2πif and only ifBt=θ−2π+bt.

Similarly,

X_t²= 0if and only ifBt=−π−bt.

Since−π < ^θ−2π₂ andb≥0, it follows thatτ₂≥σ₂. Thereforeσ∧τ =σ∧τ₁.

We continue the construction inductively. For this we need a definition. Forj ∈N^, letJj = 1ifj is odd andJj = 2ifj is even. Also, letτ¹=τ andσ¹ =σ. Starting from j= 1, ifσ^j≤τ^j, then the first stage ends, and we letT1=σ^j. Otherwise, that is on the event{τ^j < σ^j}, we letθ^j =X_τ^Jj^j+1 and redefine

X_t^J^j =π+ (−1)^j(Bt−B_τj) +b(t−τ^j), X_t^J^j+1 =θ^j+ (−1)^j+1(Bt−B_τj) +b(t−τ^j), t≥τ^j We also let

σ₁^j+1= inf{t > τ^j :X_t^J^j+1−X_t^J^j = 0}, σ^j+1₂ = inf{t > τ^j:X_t^J^j+1−X_t^J^j =π}, σ^j+1=σ₁^j+1∧σ^j+1₂ ; and

τ₁^j+1= inf{t > τ^j:X_t^J^j+1 = 2π}, τ₂^j+1= inf{t > τ^j :X_t^J^j = 0}, τ^j+1=τ₁^j+1∧τ₂^j+1. Similar to the argument given in the paragraph above,σ^j+1∧τ^j+1 is attained byσ^j+1₁ , σ₂^j+1 or byτ₁^j+1. This completes the construction of the coupling.

Lemma 2.4. IfX₀¹ =θ ∈(π,2π)and X₀² =π, thenT₁ = inf

t≥0 :B_t∈ {^θ−2π₂ ,^θ−π₂ } . That is, the distribution ofT₁coincides with the distribution ofτ_π/2underP⁰3π−2θ

4

. Proof. Letθ⁰=θ,τ⁰= 0andσ⁰=∞. In terms ofB, on the eventT

i≤j{τⁱ< σⁱ}, j ≥0, we have

τ^j+1= inf

t≥τ^j:Bt−B_τj = (−1)^j+1(2π−θ^j−b(t−τ^j)) , and

σ^j+1= inf

t≥τ^j:Bt−B_τj ∈(−1)^j{θ^j−π

2 ,θ^j−2π

2 }

.

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We will show that

σ^j+1= inf

t > τ^j :Bt∈ {θ⁰−2π

2 ,θ⁰−π 2 }

.

From this it follows thatT1, the time the coupling is stopped, coincides with the exit time ofB from(^θ⁰^−2π₂ ,^θ⁰₂^−π). To prove the claim, we first derive some identities. On {τ^j< σ^j}, we have

2π=θ^j+ (−1)^j+1(B_τj+1−B_τj) +b(τ^j+1−τ^j) θ^j+1=π+ (−1)^j(B_τj+1−B_τj) +b(τ^j+1−τ^j).

It follows that2(−1)^j(Bτ^j+1−Bτ^j) = −3π+θ^j+1+θ^j. Multiplying both sides by(−1)^j and summing overj= 0, . . . , k−1, it follows that on the eventTk−1

j=0{τ^j< σ^j}we have 2B_τk =−3πδ_k_odd+θ⁰−(−1)^kθ^k.

On rewriting this, we have

(−1)^kθ^k =−3πδkodd+θ⁰−2B_τk, and consequently,

(−1)^k

θ^k−π

2 ,θ^k−2π 2

=

θ⁰−2π

2 ,θ⁰−π 2

−B_τk.

That is, in terms of the BM, the stopping timeσ^k+1is the first timet≥τ^k such that

B_t−B_τk∈

θ⁰−2π

2 ,θ⁰−π 2

−B_τk,

completing the proof.

Lemma 2.5. Letθ∈(0,2π). Then

dt=Ot(1)e^−(λ^D⁰^(1,0,^π²^)∧λ^D⁰^(1,b,π))t

(t λ^D₀(1, b, π) =λ^D₀(1,0,^π₂);

1 otherwise, and for allt >1, the functionOt(1)is independent ofθ.

Proof. We first prove the lemma forθ∈(π,2π). At timeT1either the two copies coincide and coupling is achieved or that they areπunits away. In the latter case, we continue the coupling by letting

X_t¹=X_T¹₁+ (Bt−BT₁) +b(t−T1), X_t²=X_T²₁+ (Bt−BT₁) +b(t−T1).

A coupling will occur when one of the copies hits0or2π. Since their distance isπand one, sayX¹, is in(0, π), the coupling will occur exactly whenX¹exits the interval(0, π). Summarizing,τC is bounded above by the sumT1+T2 whereT2 = inf{t≥0 :X_T¹₁_+t∈

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{0, π}}. In particular,P_θ,π(τ_C> t)≤P_θ,π(T₁+T₂> t). Therefore dt(θ, π)≤Pθ,π(T1+T2> t)

≤

bt−1c

X

k=0

Pθ,π(T1> t−k−1, T2∈(k, k+ 1])

≤

bt−1c

X

k=0

E_θ,π

1_{T₁_>t−k−1}P_X^b1 T1∧X²_T

1−π/2(τ_π> k)

≤Ot(1)

bt−1c

X

k=0

e^−λ^D⁰^(1,0,^π²^)(t−k−1)e^−λ^D⁰^(1,b,π)k

=O_t(1)e^−(λ^D⁰^(1,0,^π²^)∧λ^D⁰^(1,b,π))t







t λ^D₀(1, b, π) =λ^D₀(1,0,^π₂);

1−e^−|λ^D⁰^{(1,b,π)−λ}^D⁰^(1,0,^π²^)|−1

otherwise, (2.2) where the inequality on the fourth line follows from Lemma 2.3, withOt(1)independent ofθ. This completes the proof forθ∈(π,2π).

Suppose now thatθ∈(0, π). The Consider the coupling X_t¹=θ+Bt+bt, X_t²=π+X_t¹, t≥0.

Letτ = inf{t : X_t¹ ∈ {0, π}}. At timeτ, X_τ² =π or2πaccording to whetherX_τ¹ = 0 or X_τ²=π. We continue the coupling fort≥τby redefiningX¹andX²through

X_t²=X_t¹=X_t−τ, t≥τ.

Thus,X¹andX²are coupled by timeτ, andτis the exit time of BM with driftb, starting fromθfrom the interval(0, π). Lemma 2.3 gives

dt(θ, θ+π)≤P_θ−π/2^b (τπ> t)≤Ot(1)e^−λ^D⁰^(1,b,π)t. From the triangle inequality,

d_t(θ, π)≤d_t(θ, θ+π) +d_t(θ+π, π), and the result follows from (2.2).

3 Proof of Proposition 1.4

Proof. Similarly to Section 2 we will consider the DR with diffusivityσ= 1, driftb= _2πσ^`µ₂ on the interval`= (0,2π). The general case follows from the scaling identity (1.1) ap- plied to this result.

Suppose L1,bu = −λu, and u ∈ D2π, and let v(x) = ^u_u(x)⁰^(x)

. Then since (u⁰)⁰ =

−2bu⁰−2λu, we have

v⁰=AvwhereA=

−2b −2λ

1 0

.

Note that a priori λ may not be real. This is crucial. The solution is of the form v(x) =e^xAv(0). IfAis diagonalizable, with eigenvaluesλ1, λ2, then this impliesu(x) = Ae^λ¹^x+Be^λ²^x. Otherwise, u(x) = e^λ¹^x(A+Bx). The trace ofAgives λ1+λ2 =−2b.

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Sincebis real,=λ1+=λ2= 0. The determinant givesλ₁λ₂= 2λ. We continue according to the cases.

1. Inequality,λ16=λ2. ThenAis diagonalizable, and the boundary condition is A+B=Ae^πλ¹+Be^πλ²=Ae^2πλ¹+Be^2πλ².

There are two cases to consider.

AB=0.Without loss of generality,A = 0andB = 1. The boundary condition reduces to1 =e^πλ² =e^2πλ², thereforeπλ₂ = 2πikfor some k ∈ Z, and then the trace implies λ₁ =−2b−2ik, withb 6= 0ork 6= 0(otherwiseλ₁ = λ₂). The determinant then gives 2λ= (−2b−2ik)2ik. In particular,λ= 2k²−2bki, andk6= 0orb6= 0.

AB6=0.Letα=e^πλ¹and letβ=e^πλ². The boundary condition could be rewritten as A(1−α) =−B(1−β), andA(1−α²) =−B(1−β²).

Ifα= 1, then β = 1, and this is equivalent toλ1 = 2ik1, andλ2 = 2ik2 fork1, k2 ∈ Z^. The trace holds if and only ifb= 0andk₂=−k16= 0(again, becauseλ₁6=λ₂), in which it follows from the determinant thatλ= 2k₁². This eigenvalue already appeared for the previous case. Ifα6= 1, thenβ 6= 1and we can divide the second equation by the first to obtainα=β. That isπλ2=πλ1+ 2πikfor somek∈Z− {0}. From the trace, we obtain

−2b = 2λ1+ 2ki, henceλ1 =−b−ik. The determinant gives2λ= (−b−ik)(−b+ik). That is,λ=^b²^+k₂ ², k∈Z− {0}.

Summarizing, the eigenvalues obtained whenλ₁6=λ₂are2k²−2bkiwithk6= 0orb6= 0, and ^b²^+k₂ ² fork6= 0.

2. Equality, λ1 = λ2. This clearly holds if and only if λ1 = −b, and an eigenvector φ = ^φ_φ¹

2

forAmust satisfy φ1 = −bφ2. Thus, the eigenspace is one-dimensional and spanned by the vector(−b,1). In particular, the boundary condition reads

A=e^−πb(A+πB),andA=e^−2πb(A+ 2πB).

IfA= 0thenB = 0, therefore there is no loss of generality assumingA= 1. The first equation becomese^πb = 1 +πB, and the second equation becomese^2πb = 1 + 2πB = 1 + 2(e^πb−1) = 2e^πb−1. Clearly, this equation holds if and only ifb= 0.

Summarizing this case, the only eigenvalue obtained is0, corresponding tob= 0.

Acknowledgement

The author would like to thank an anonymous referee and the associate editor for their invaluable comments and help in improving the manuscript.

References

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[7] ,Spectral analysis of diffusions with jump boundary, J. Funct. Anal.261(2011), no. 7, 1992–2012. MR-2822321

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1Introduction IddoBen-Ari CouplingfordriftedBrownianmotiononanintervalwithredistributionfromtheboundary

Coupling for drifted Brownian motion on an interval with redistribution from the boundary

Iddo Ben-Ari

1 Introduction

2 Proof of Theorem 1.2

3 Proof of Proposition 1.4

Acknowledgement

References

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