Transience, recurrence and speed of diffusions with a non-markovian two-phase “use it or lose it” drift

www.imstat.org/aihp 2014, Vol. 50, No. 4, 1198–1212

DOI:10.1214/13-AIHP549

© Association des Publications de l’Institut Henri Poincaré, 2014

Transience, recurrence and speed of diffusions with a non-Markovian two-phase “use it or lose it” drift

Ross G. Pinsky

Department of Mathematics, Technion – Israel Institute of Technology, Haifa, 32000, Israel. url:http://www.math.technion.ac.il/~pinsky/

Received 18 October 2012; revised 3 February 2013; accepted 6 February 2013

Abstract. We investigate the transience/recurrence of a non-Markovian, one-dimensional diffusion process which consists of a Brownian motion with a non-anticipating drift that has two phases – a transient to+∞mode which is activated when the diffusion is sufficiently near its running maximum, and a recurrent mode which is activated otherwise. We also consider the speed of a diffusion with a two-phase drift, where the drift is equal to a certain non-negative constant when the diffusion is sufficiently near its running maximum, and is equal to a certain positive constant otherwise.

Résumé. Nous étudions la transience/récurrence d’un processus de diffusion non-Markovien à une dimension, consistant en un mouvement brownien avec une dérive non anticipative qui a deux phases – un mode transitoire à+∞qui est activé quand la diffusion est suffisamment proche du processus de son maximum, et un mode récurrent qui est activé dans le cas contraire. On considère également la vitesse d’une diffusion avec une dérive à deux phases, où la dérive est égale à une certaine constante positive lorsque la diffusion est suffisamment proche du processus de son maximum, et est égale à une certaine constante strictement positive dans le cas contraire.

MSC:60J60

Keywords:Diffusion process; Transience; Recurrence; Non-Markovian drift

1. Introduction and statement of results

Over the past fifteen years or so, there has been much interest in the study of the long term behavior of various random walks with non-Markovian transition mechanisms, such as random walks in random environment, self-avoiding ran- dom walk, edge or vertex reinforced random walks, and excited (so-called “cookie”) random walks. See, for example, the monograph [13], and the survey articles [9] and [7], which include many references. Non-Markovian diffusion processes analogous to excited random walks have also been studied (see [2,3,12]), as well as so-called Brown- ian polymers, which are non-Markovian self-repelling diffusions, analogous to certain negatively reinforced random walks (see [1,4,5,8]). In this paper we investigate the transience/recurrence of a non-Markovian, one-dimensional dif- fusion process which consists of a Brownian motion with a non-anticipating drift that has two phases – a transient to +∞mode which is activated when the diffusion is sufficiently near its running maximum, and a recurrent mode which is activated otherwise. We also consider the speed of a diffusion with a two-phase drift, where the drift is equal to a certain non-negative constant when the diffusion is sufficiently near its running maximum, and is equal to a certain positive constant otherwise.

Letb^T(x)andb^R(x)be continuous functions onRwhich satisfy

−∞exp

− _x

0

2b^T(y)dy

dx= ∞,

_∞ exp

− _x

0

2b^T(y)dy

dx <∞; (1.1)

−∞exp

− x

0

2b^R(y)dy

dx= ∞,

_∞ exp

− x

0

2b^R(y)dy

dx= ∞.

As is well known [10], the one-dimensional diffusion processes corresponding to the operatorsL^T ≡¹_2dx^d22+b^T(x)_dx^d andL^R≡¹_2dx^d22 +b^R(x)_dx^d are respectively transient to+∞and recurrent. Letγ:[0,∞)→(0,∞)be a continuous function satisfying

γ >0, γ<1 and lim

x→∞

x−γ (x)

= ∞.

For a continuous trajectoryx(·):[0,∞)→R, letx^∗(t)=max0≤s≤tx(s)denote its running maximum. We define a non-anticipating, non-Markovian driftb(t, x(·))by

b t, x(·)

= b^T

x(t)

, ifx(t) > x^∗(t)−γ x^∗(t)

; b^R

x(t)

, ifx(t)≤x^∗(t)−γ x^∗(t)

. (1.2)

We consider the diffusion processX(t )that satisfies the stochastic differential equation X(t )=x₀+W (t )+

_t

0

b s, X(·)

ds, (1.3)

where W (·)is a Brownian motion. Existence and uniqueness for this stochastic differential equation follow from the standard theory for classical diffusion processes (see Section3). We call the solution to (1.3) a diffusion with a two-phase “use it or lose it” drift.

For example, the processX(t )might represent the price of a stock. As prices rise, people are encouraged to buy, creating a certain trend represented by the transient drift. In addition to this underlying trend, there is a random fluctuation represented by the Brownian motion. These random fluctuations might cause prices to slump. If the slump becomes sufficiently large, it discourages buying, which creates a new weaker trend, represented by the recurrent drift.

When random fluctuations eventually result in prices rising to levels close to the previous high, the original stronger trend reasserts itself.

We callγ the “down-crossing” function. For the majority of the paper, we will consider the case that the down- crossing functionγ is a constant. In this case, at any timet, the diffusionX(t )will run in the transient mode if and only ifX(t ) > X^∗(t)−γ, or equivalently, if and only if by timet the pathX(·)has not down-crossed an interval of lengthγwhose left hand endpoint is larger than or equal toX(t ).

The various equivalent definitions of transience and recurrence for classical non-degenerate diffusion processes hold for the diffusion with the two-phase drift. (This will be clear from the construction in Section3.) We state here the standard definitions, although we will use other equivalent definitions in the proofs. The diffusion with the two- phase drift isrecurrentif for any pair of pointsx₀andx₁, the process starting atx₀ almost surely returns tox₁ at arbitrarily large times. The diffusion with the two-phase drift istransient to+∞if starting at any x₀, the process almost surely satisfies limt→∞X(t )= ∞.

Our first result concerns the case in which the transient drift is constant:b^T(x)≡b >0, and the recurrent driftb^R satisfies a regularity condition; namely, that the driftb^R∨0 is also a recurrent drift. In this case, the diffusion with the two-phase drift is always recurrent.

Theorem 1. Assume that the down-crossing functionγis constant.Let the transient drift be constant:b^T(x)≡b >0, and assume that the recurrent driftb^Ris such that the driftb^R∨0is also recurrent.That is,assume that the condition satisfied byb^Rin(1.1)is also satisfied byb^R∨0.Then the diffusion with the two-phase drift is recurrent.

Remark. Note that the diffusion with the two-phase drift is recurrent even if the recurrent driftb^Ris just border line recurrent – for example,if for sufficiently large x,b^R(x)=_2x¹ is the drift of the radial part of a two-dimensional Brownian motion.

Maintaining the constant transient drift, but choosing the recurrent driftb^Rto take on very large positive values at most locations, and compensating to insure recurrence by having it take on even much larger negative values at other locations, we can construct a diffusion with such a two-phase drift that is transient.

Theorem 2. Assume that the down-crossing functionγis constant.Let the transient drift be constant:b^T(x)=b >0.

There exists a recurrent driftb^Rsuch that the corresponding diffusion with the two-phase drift is transient.

We continue to assume that the down-crossing functionγ is constant. As noted above,b(t, X(·))=b^T(X(t))if and only if by timet, the pathX(·)has not down-crossed an interval of lengthγ whose left hand endpoint is larger than or equal toX(t ). Now if the transient diffusion corresponding toL^T is such that it almost surely eventually stops making down-crossings of lengthγ, then the diffusion in the two-phase environment will eventually stop making down-crossings of lengthγ and will eventually be driven just by the transient drift; consequently, it will be transient.

In [11] we proved the following result.

Theorem P2. Consider the diffusion process corresponding to the operatorL^T.

(i) Ifb^T(x)≤_2γ¹ logx+_γ¹log⁽²⁾x for sufficiently largex,then the diffusion almost surely makesγ-down-crossings for arbitrarily large times;

(ii) Ifb^T(x)≥ _2γ¹ logx+ _γ^klog⁽²⁾x,for somek >1 and for sufficiently large x,then the diffusion almost surely eventually stops making down-crossings of sizeγ.

In light of TheoremP2and the paragraph preceding it, in the case of a constant down-crossing functionγ, ifb^T satisfies the condition in part (ii) of the theorem, then the diffusion with the two-phase drift is transient, regardless of what the recurrent driftb^Ris.

Continuing with a constant down-crossing functionγ, we now let the recurrent diffusion be Brownian motion, that is,b^R≡0, and determine what the threshold growth rate is onb^T that distinguishes between transience and recurrence for the diffusion with the two-phase drift.

Theorem 3. Assume that the down-crossing functionγ is constant.Let the recurrent diffusion be Brownian motion:

b^R≡0.

(i) Ifb^T(x)≤_2γ¹ log⁽²⁾x+_2γ¹ log⁽³⁾x,for largex,then the diffusion with the two-phase drift is recurrent;

(ii) Ifb^T(x)≥_2γ¹ log⁽²⁾x+_2γ^k log⁽³⁾x,for largex,wherek >1,then the diffusion with the two-phase drift is tran- sient.

Remark. In light of Theorem P2 and Theorem3,it follows that if the down-crossing function γ is constant, the recurrent diffusion is Brownian motion,and the transient diffusion has driftb^T satisfying

1

2γ log⁽²⁾x+ k

2γ log⁽³⁾x≤b^T(x)≤ 1

2γlogx+1 γ log⁽²⁾x

for largex and somek >1, then the diffusion with the two-phase drift will be transient,and will alternate infinity often between the recurrent and the transient regimes.

We now consider the case that the recurrent diffusion is Brownian motion:b^R≡0, that the transient drift is con- stant: b^T(x)≡b, but we allow the down-crossing functionγ to grow withx. We determine the threshold growth rate on the down-crossing function that distinguishes between transience and recurrence for the diffusion with the two-phase drift.

Theorem 4. Let the recurrent diffusion be Brownian motion: b^R ≡0, and let the transient drift be constant:

b^T(x)≡b.

(i) If the down-crossing functionγ satisfiesγ (x)≤_2b¹ log⁽²⁾x+_2b¹ log⁽³⁾x,for largex,then the diffusion with the two-phase drift is recurrent;

(ii) If the down-crossing function γ satisfies γ (x)≥ _2b¹ log⁽²⁾x +_2b^k log⁽³⁾x, for large x,where k >1, then the diffusion with the two-phase drift is transient.

We now make one minor and one more significant change in the setup we have used until now. The minor change is that the diffusion coefficient will be a constanta >0 instead of 1. The more significant change is that the two-phase driftb(t, x(·))will be given by (1.2), withγ >0 constant, with the transient driftb^T replaced by the constantb≥0, and with the recurrent driftb^Rreplaced by the constantc >0. (The caseb >0 andc∈(0, b)is more in keeping with the main theme of this paper; namely introducing a slowdown when the process has strayed too far from its running maximum.) The limiting casec= ∞, described below, is interesting. Thus, we will consider the operator

L=1 2a d²

dx²+b

t, X(·) d dx, whereb

t, X(·)

=

b≥0, ifX(t ) > X^∗(t)−γ;

c >0, ifX(t )≤X^∗(t)−γ. (1.4)

The next theorem gives the speed of the diffusion in this two-phase drift.

Theorem 5. Consider the diffusion in the two-phase drift corresponding to the operatorLin(1.4).The speed of the diffusionX(t )with the two-phase drift is given by

tlim→∞

X(t )

t =

c(e^{2bγ /a}−1) c(e^{2bγ /a}−1)+(b−c)(1−e−2bγ /a)

b a.s., ifb >0;

ca

2γ c+a a.s., ifb=0.

Remark. Letd(b, c, γ , a)≡_c(e2bγ /a−1)^c(e+^{2bγ /a}(b−c)(1⁻¹⁾−e^{−2bγ /a}).In the casec∈(0, b),we calld(b, c, γ , a)thedamping co- efficient. It gives the fractional reduction in speed between a classical diffusion with driftb and the slowed down diffusion with two-phase drift –b when the process is less than distanceγ from its running maximum,andcwhen the process is at least distanceγ from its running maximum.Of course,it is clear thatd(b, c, γ , a)must always fall between ^c_b and1whenc∈(0, b).We make the following observations:

1. Whenb→ ∞,the damping coefficientd(b, c, γ , a)converges to1exponentially inb;

2. Whenγ→ ∞,d(b, c, γ , a)converges to1exponentially inγ;

3. Whenc→0,the damping coefficientd(b, c, γ , a)converges to0linearly inc;

4. Whenc→0andγ→ ∞,the damping coefficientd(b, c, γ , a)converges to1 (respectively,converges to0,remains bounded away from0and1)if cexp(^2bγ_a )converges to∞(respectively,converges to0,remains bounded away from0and∞);

5. When c→0 and b→ ∞,the damping coefficientd(b, c, γ , a)converges to0 if cexp(^2bγ_a )remains bounded.

Otherwise,the damping coefficient converges to0 (respectively,converges to1,remains bounded away from0and 1)if ^cexp(2bγ /a)

b converges to0 (respectively,converges to∞,remains bounded away from0and∞).

6. The limiting casec= ∞corresponds to the situation in which the valueX^∗(t)−γ serves as a reflecting barrier forX(t );thus,the process can never stray farther than a distanceγ from its running maximum.The drift is always equal tob,but the reflecting barrier atX^∗(t)−γ speeds the process up.In the caseb >0,we have

d(b,∞, γ , a)= e^{2bγ /a}−1

e^{2bγ /a}+e^{−2bγ /a}−2=1+ 1−e^{−2bγ /a} e^{2bγ /a}+e^{−2bγ /a}−2.

In the caseb=0,we have a Brownian motion that is never allowed to stray farther than a distanceγ from its running maximum;its speed is_2γ^a .

In Section2, we present some preliminary information on down-crossings. In Section3, we give an explicit rep- resentation of the diffusion with a two-phase drift in terms of classical diffusions. In Section4we give a workable analytic criterion for transience/recurrence which depends on an auxiliary discrete time, increasing Markov process.

Sections5–9give the proofs of Theorems1–5.

2. Preliminaries concerning down-crossings

From the definition of the down-crossing functionγ, it follows thatx−γ (x)is increasing. Define the stopping time σ_γ on continuous pathsx(·):[0,∞)→Rwith the standard filtrationFt=σ (x(s): 0≤s≤t )by

σ_γ=inf t≥0: ∃s < twithx(t)≤x(s)−γ x(s)

=inf t≥0:x(t)=x^∗(t)−γ x^∗(t)

. The equality above follows from the fact thatx−γ (x)is increasing. Let

L^γ=x^∗(σ_γ);

K^γ=x(σγ)=x^∗(σγ)−γ x^∗(σγ)

.

In the case that the down-crossing functionγ is constant,σ_γ is the first time that the pathx(·)completes a down- crossing of an interval of lengthγ. The interval that was down-crossed is[K^γ, L^γ]. In [11], forγ constant,L^γ was called theγ-down-crossed onset location. In this paper, we will use this terminology also for variableγ. We will call σ_γ thefirstγ-down-crossed time.For use a bit later, letτ_a=inf{t≥0: x(t)=a}denote the first hitting time of the pointa.

Consider now the one-dimensional diffusion processY (t ) which corresponds to the operator L^T and which is transient to+∞. Denote probabilities for the process starting fromxbyP_x. Fixing a pointz₀, let

u_T(x)= _x

z₀

exp

− _y

z₀

2b^T(r)dr

dy. (2.1)

(The formula in the theorem below is independent ofz₀, but this specification ofz₀ will be useful later on.) The following result was proved in [11].

Theorem P1. For the diffusion process corresponding toL^T,and for constantγ,the distribution of theγ-down- crossed onset locationL^γ is given by

P_x

L^γ> x+y

=exp

− _x+y

x

u_T(z)

u_T(z)−u_T(z−γ )dz

, y >0.

Remark 1. In[11],where the notation is a bit different from here,the mathematical definition ofσ_γ,and consequently also ofL^γ,were written incorrectly.From the verbal description in[11],it is clear that the intended definition ofL^γ is the one given here.All the proofs in[11]are based on the correct definitions given here.

Remark 2. TheoremP2in Section1was proved in[11]as an application of TheoremP1.

The same method of proof used to prove TheoremP1can be used in the case of variableγto obtain a corresponding formula for the distribution ofL^γ.

Proposition 1. For the diffusion process corresponding toL^T,and for variableγ,the distribution of theγ-down- crossed onset locationL^γ is given by

P_x

L^γ> x+y

=exp

− x+y

x

u_T(z)

uT(z)−uT(z−γ (z))dz

, y >0.

3. A representation for diffusion with two-phase drift

Consider the diffusionX(t )with the two-phase drift starting fromx0. Up until the firstγ-down-crossed timeσγ, the process is exactly theY (·)-process corresponding to the operatorL^T and starting fromx0. We haveX(σγ)=Y (σγ)= K^γ andX^∗(σ_γ)=Y^∗(σ_γ)=L^γ, withL^γ distributed as in Proposition1. Let

ˆ

τL^γ =inf t≥0:X(σγ+t )=L^γ .

Thenσ_γ+ ˆτ_L^γ is the first time afterσ_γ that the processX(·)returns to its running maximumL^γ. LetZ^R,z,T(t)be the diffusion starting fromzand corresponding to the operatorL^R,z,T =¹_2dx^d22 +b^R,z,T(x)_dx^d, where

b^R,z,T(x)=

b^R(x), x≤z;

b^T(x), x > z.

Then the distribution of{X(σγ+t ),0≤t≤ ˆτL^γ}, conditioned onK^γ =X(σγ)=zandL^γ =X^∗(σγ)=a, is that of {Z^R,z,T(t),0≤t≤τ_a}. Of course,X(σ_γ+ ˆτ_L^γ)=X^∗(σ_γ + ˆτ_L^γ)=L^γ. Starting from timeσ_γ+ ˆτ_L^γ, when the process has returned to its running maximum,X again looks like theY process, until it again performs aγ-down- crossing, at which point it becomes aZ^R,z,T process for appropriatezuntil it returns to its running maximum, and everything is repeated again.

In light of the above description,X(t )can be represented as follows. For eachx∈R and eachn≥1, letY^n,x(·) be a diffusion process corresponding to the operator L^T and starting fromx. Make the processes independent for different pairs(n, x). Similarly, for eachz∈R and eachn≥1, letZ^{R,z,T ,n}(·)be a diffusion process corresponding to the operatorL^R,z,T and starting fromz. Make the processes independent for different pairs(n, z)and independent of theY^n,x processes. Letσ_γ^n,x denote the firstγ-down-crossing time for the processY^n,x, and letτa^n,z denote the first hitting time of a for the process Z^{R,z,T ,n}. LetL^γ₀ =x₀ and then by induction, forn≥1, defineL^γ_n to be the γ-down-crossed onset location forY^n,Lγn−1. Forn≥1, letK_n^γ correspond toL^γ_n asK^γ corresponds toL^γ. ThenX(·) can be represented as

X(t )=Y^1,x0(t), 0≤t≤σ_γ^1,x0;

(3.1) X(t )=Z^{R,K1γ,T ,1}(t), σ_γ^1,x0≤t≤σ_γ^1,x0+τ^1,K

γ 1

L^γ₁ , and forn≥2,

X _n−1

j=1

σ^j,L

γj−1

γ +

n−1

j=1

τ^j,K

γ j

L^γ_j +t

=Y^n,L

γ

n−1(t), 0≤t≤σ^n,L

γn−1

γ ;

(3.2) X

_n

j=1

σ^j,L

γ j−1

γ +

n−1

j=1

τ^j,K

γ j

L^γ_j +t

=Z^{R,Knγ,T ,n}(t), 0≤t≤τ^n,K

γ n

L^γ_n .

From the above representation, it is clear that existence and uniqueness for (1.3) follows from the standard theory for classical diffusion processes.

At those timess whenX(s) is running as aY^n,L

γ

n−1(·)-process, for somen≥1, we will say thatX(·)is in the Y-mode, and at those timesswhenX(s)is running as aZ^{R,Knγ,T ,n}(·)-process, for somen≥1, we will say thatX(·) is in theZ-mode. We denote byPx0 probabilities corresponding to the diffusionX(t )with the two-phase drift starting fromx₀, and byEx₀the corresponding expectation.

Note that{L^γ_n}^∞_n=0is a monotone increasing Markov process whose transition distribution from a state x is the distribution ofL^γ in Proposition1. That is, the transition probability measurep(x,·)is given by

p

x,[x+y,∞)

=exp

− _x+y

x

u_T(z)

uT(z)−uT(z−γ (z))dz

, fory≥0. (3.3)

We will use the notation P_x^L

0 to denote probabilities for this discrete time Markov process and will denote the corresponding expectation by E_x^L

0. The times {L^γ_n}^∞_n=0 will be called “regeneration” points for X(·)because the Px0-distribution of{X(n−1

j=1σ^j,L

γj−1

γ +n−1

j=1τ^j,K

γ j

L^γ_j +t ),0≤t <∞}given thatL^γ_n−1=a is the same as thePa- distribution of {X(t ),0≤t <∞}.

4. A transience/recurrence criterion

From the construction in Section3, the well-known equivalent alternative conditions for transience/recurrence, which hold for standard non-degenerate diffusion processes [10], are easily seen to hold for diffusions with a two-phase drift.

Since it is clear that the process is either transient to+∞or recurrent, we can use the following criterion:

Transience: for some pair of pointsz₀< x₀, one hasPx₀(τ_z0= ∞) >0;

(4.1) Recurrence: for some pair of pointsz₀< x₀, one hasPx0(τ_z0= ∞)=0.

We choose the pointz0so thatz0< x0−γ (x0). Then it follows from the representation of the processX(·)that at timeτ_z0 the process is in theZ-mode. Using this with the regeneration structure noted at the end of the previous section, it follows that

Px₀(τz₀= ∞)=E_x^L

0G L^γn

_∞

n=1

, (4.2)

where for any non-decreasing sequence{a_n}^∞_n=1satisfyinga₁≥x₀, we define G

{a_n}^∞_n=1

=^∞

n=1

P_a^{R,an−γ (an),T}

n−γ (an) (τ_an< τ_z0), (4.3)

and wherePz^R,z,T denotes probabilities for aZ^R,z,T-processes starting atz. From (4.2) we conclude thatPx₀(τz₀ =

∞)=0 if and only ifG({L^γ_n}^∞_n=1)=0 a.s.P_x^L

0; thus, from (4.3) we have Px₀(τ_z0= ∞)=0 if and only if

∞ n=1

P^R,L

γ

n−γ (L^γ_n),T

L^γ_n−γ (L^γ_n) (τ_z0 < τ_L^γ

n)

= ∞ a.s.P_x^L

0. (4.4)

As is well known [10], the functionv(x)≡P_x^R,z,T(τz₀< τz+c), forz0≤x≤z+c, satisfiesL^R,z,Tv=0 in(z0, z)∪ (z, z+c),v(z₀)=1, v(z+c)=0,v(z⁻)=v(z⁺), andv(z⁻)=v(z⁺). Solving this, one finds thatP_z^R,z,T(τ_z0 <

τ_z+c)=v(z)is given by P_z^R,z,T(τz₀< τz+c)

= exp(−_z

z0(2b^R)(y)dy)_z+c

z dyexp(−_y

z 2b^T(r)dr) z

z₀dyexp(−y

z₀2b^R(r)dr)+exp(−z

z₀(2b^R)(y)dy)z+c

z dyexp(−y

z 2b^T(r)dr). (4.5)

Analogous tou_T in (2.1), define the function u_R(x)=

_x

z₀

exp

− _y

z₀

2b^R(r)dr

dy. (4.6)

We then rewrite the somewhat unwieldy equation (4.5), which would become a lot more unwieldy below, in the form P_z^R,z,T(τz₀< τz+c)= u_R(z)(u_T(z+c)−u_T(z))exp(z

z₀2b^T(y)dy) u_R(z)+u_R(z)(u_T(z+c)−u_T(z))exp(_z

z02b^T(y)dy). (4.7)

From (4.1), (4.4) and (4.7), we obtain the following transience/recurrence criterion for the diffusion with the two- phase drift.

Proposition 2. Define

H (s)= u_R(s−γ (s))(u_T(s)−u_T(s−γ (s)))exp(s−γ (s)

z₀ 2b^T(y)dy) u_R(s−γ (s))+u_R(s−γ (s))(u_T(s)−u_T(s−γ (s)))exp(_{s−γ (s)}

z₀ 2b^T(y)dy)

. (4.8)

Let {L^γ_n}^∞_n=0 be the monotone increasing Markov process withL^γ₀ =x₀and transition probability measurep(x,·) given by(3.3).If

∞ n=0

H L^γ_n

= ∞ a.s., (4.9)

then the diffusion with the two-phase drift is recurrent.Otherwise,it is transient.

5. Proof of Theorem1

By the Ikeda–Watanabe comparison theorem [6], if we prove recurrence for the diffusion with the two-phase drift whose recurrent drift b^R is replaced by the drift b^R∨0, then original diffusion with the two-phase drift is also recurrent. By assumption, the driftb^R∨0 is also a recurrent drift. Thus, we may assume without loss of generality that the recurrent driftb^Ris non-negative.

Sinceb^T ≡b, we have from (2.1) that u_T(x)= 1

2b

1−exp

−2b(x−z₀) .

Using this along with the fact thatγis constant, we have u_T(s)−u_T

s−γ (s) exp

s−γ (s) z0

2b^T(y)dy

= 1 2b

1−exp(−2bγ )

≡c_b,γ, (5.1)

and thus the formula forH (s)in (4.8) simplifies to H (s)= cb,γu_R(s−γ )

u_R(s−γ )+c_b,γu_R(s−γ ). (5.2)

From (4.6), it follows that u_R is increasing, and by the assumption thatb^R is non-negative, it follows that u_R is non-increasing. Thus,H is non-increasing.

We also have u_T(z)

uT(z)−uT(z−γ (z))= 1

cb,γexp(2bγ )≡ 1 db,γ

, (5.3)

and thus the increment measurep(x, x+A),A⊂ [0,∞), corresponding to the transition probability measurep(x,·) in (3.3) for the Markov process{L^γn}^∞_n=0is independent of x and is equal to the exponential density with parame- ter _d¹

b,γ. Consequently,L^γ_n −L^γ₀ is the sum ofnIID exponential random variables with the above parameter, and thus

nlim→∞

L^γ_n

n =db,γ a.s. (5.4)

Using (5.4) along with the fact thatHis non-increasing, if we show that_∞

n=1H ((d_b,γ +1)n)= ∞, then it follows that_∞

n=1H (L^γ_n)= ∞a.s., and consequently, from Proposition2we conclude that the diffusion with the two-phase drift is recurrent. Sinceu_Ris non-increasing andu_R is increasing, it is easy to see from (5.2) that_∞

n=1H ((d_b,γ + 1)n)= ∞ if and only if _∞

n=1H ((dˆ _b,γ +1)n)= ∞, where Hˆ =^u_u^R_R^(s_(s⁻₋^{γ )}_{γ )}. SinceHˆ is monotone, it follows that _∞

n=1H ((dˆ b,γ +1)n)= ∞if and only if_∞u_R(s)

uR(s)ds= ∞, that is, if and only if lims→∞uR(s)= ∞. But this last inequality holds from (4.6) and (1.1) sinceb^Ris a recurrent drift.

6. Proof of Theorem2

As in the proof of Theorem1, the random variables{L^γ_n−L^γ_n−1}^∞_n=1are IID and distributed according to the exponen- tial distribution with parameter_d¹

b,γ, and the functionHis given by (5.2). In order to show transience, by Proposition2, we need to show that_∞

n=0H (L^γ_n) <∞with positive probability. Recalling (4.6) and (1.1), to complete the proof, we will construct a functionu_R which satisfiesu_R>0, u_R>0 and limx→∞u_R(x)= ∞, and for which the above sum is almost surely finite. (The corresponding driftb^Rwill then be given by−_2u^uR_R.)

Forj ≥2, define the intervalI_j= [x₀−γ+j, x₀−γ+j+_j¹2]. Without loss of generality, assume thatx₀−γ+2≥ 0. We now show that

P_x^L

0

L^γ_n −γ∈ ∞ j=2

Ij

≤ c

n² (6.1)

for somec >0. The distribution ofL^γ_n−x0is that of the sum ofnIID exponential random variables with parameter λ≡_db,γ¹ . Thus, its density function is ^λ_(n^nx₋ⁿ⁻¹_1)! exp(−λx),x≥0. For an appropriate constantC >1, we then have for n≥3,

P_x^L

0

L^γ_n −γ∈^∞

j=2

I_j

=^∞

j=2

Ij

λⁿxⁿ⁻¹

(n−1)!exp(−λx)dx

≤C _∞

0

λⁿxⁿ⁻³

(n−1)!exp(−λx)dx= Cλ³

(n−1)(n−2). (6.2)

Now (6.1) follows from (6.2).

We now construct a positive, strictly increasingC²-functionuR whose derivative on R−_∞

j=2Ij is uniformly bounded, and which satisfiesu_R(x)≥x², for x≥2. (Of course, to have this quadratic growth, u_R must get very large at certain places on_∞

j=2I_j.) By the law of large numbers, limn→∞L^γn

n =d_b,γ a.s. By (6.1) and the lemma of Borel–Cantelli,P_x^L

0(L^γ_n −γ∈_∞

j=2I_j i.o.)=0. Using the facts noted in this paragraph, we conclude that ∞

n=0

H L^γ_n

=^∞

n=0

c_b,γu_R(L^γ_n−γ )

uR(L^γ_n−γ )+cb,γu_R(L^γ_n−γ )<∞ a.s.

7. Proof of Theorem3

Sinceb^R=0, we have from (4.6) thatu^R(x)=x−z₀. Also,γ is constant. Thus, from (4.8), we have H (s)=

s

s−γdyexp(−y

s−γ2b^T(r)dr) s−γ−z0+_s

s−γdyexp(−_y

s−γ2b^T(r)dr). (7.1)

By comparison, it suffices to consider the case that b^T(x)= 1

2γ log⁽²⁾x+ k

2γ log⁽³⁾x

forx≥z₀, withz₀large enough so that log⁽³⁾z₀is defined. We need to show transience fork >1 and recurrence for k=1. In what follows, we will always assume thatk≥1.

We have

(y−s+γ )log^{(j )}(s−γ )≤ _y

s−γ

log^{(j )}rdr≤(y−s+γ )log^{(j )}s, s−γ≤y≤s. (7.2)

Thus,

γ−o(1/logs) log⁽²⁾s+klog⁽³⁾s ≤

_s

s−γ

dyexp

− _y

s−γ

2b^T(r)dr

≤ γ

log⁽²⁾(s−γ )+klog⁽³⁾(s−γ ), ass→ ∞. (7.3)

From (7.1) and (7.3) we conclude that there exist constantsC1, C2>0 such that C1

slog⁽²⁾s ≤H (s)≤ C2

slog⁽²⁾s for larges. (7.4)

We now investigate the growth rate of the Markov process{L^γ_n}^∞_n=0. Recall that givenL^γ_j =x, the distribution of L^γ_j+1−L^γ_j is the distribution given in (3.3). From (2.1) we have

u_T(x)

u_T(x)−u_T(x−γ ) = exp(−_x

z02b^T(r)dr) _x

x−γdyexp(−_y

z₀2b^T(r)dr)

= exp(−x

x−γ2b^T(r)dr) _x

x−γdyexp(−_y

x−γ2b^T(r)dr). (7.5)

From (7.2) and the definition ofb^T, we have c1

(logx)(log⁽²⁾x)^k ≤exp

− _x

x−γ

2b^T(r)dr

≤ c2

(logx)(log⁽²⁾x)^k for largex

for constantsc1, c2>0. Using this with (7.3) and (7.5), we conclude that there exist constantsC3, C4>0 such that C₃

(logx)(log⁽²⁾x)^k−1≤ u_T(x)

u_T(x)−u_T(x−γ ) ≤ C₄

(logx)(log⁽²⁾x)^k−1 for largex. (7.6) We now prove recurrence in the case thatk=1. Let{xn}^∞_n=0be a sequence of positive numbers withx₀sufficiently large so that the bound in (7.6) holds forx≥x₀, and lets_n=_n

j=0x_j. We have_sn

s_n−1 1

logzdz≥_logs^xn_n. Using this with (7.6) and (3.3), we have

P_x^L

0

L^γ_n−L^γ_n−1≤x_n|L^γ_n−1≤s_n−1

≥1−exp

−C₃x_n logs_n

, n≥1. (7.7)

Fix a large numberM. Withx₀as above, we wish to select the sequence{x_n}^∞_n=0so that C₃x_n

logs_n =2 log(n+M) forn≥1. (7.8)

We suppress the dependence of this sequence onMin the sequel. For the sequence{x_n}^∞_n=0satisfying (7.8), it follows from (7.7) that

P_x^L

0

L^γ_n≤snfor alln

≥1− ∞ n=1

1

(n+M)². (7.9)

Now (7.8) is a difference equation corresponding to the differential equation C3S(t)

logS(t)=2 log(t+M), t≥1. (7.10)