Behavior of the Euler scheme with decreasing step in a degenerate situation

June 2007, Vol. 11, p. 236–247 www.edpsciences.org/ps

DOI: 10.1051/ps:2007018

BEHAVIOR OF THE EULER SCHEME WITH DECREASING STEP IN A DEGENERATE SITUATION

Vincent Lemaire

Abstract. The aim of this short note is to study the behavior of the weighted empirical measures of the decreasing step Euler scheme of a one-dimensional diffusion process having multiple invariant mea- sures. This situation can occur when the drift and the diffusion coefficient are vanish simultaneously.

Mathematics Subject Classification. 60H10, 65C30, 37M25.

Received April 3, 2006. Revised October 16, 2006.

1. Introduction and framework

First, let us recall some results about the behavior of a one-dimensional diffusion process. LetI=]l, r[ denote an open (non-trivial) interval of the real lineR. We consider the following stochastic differential equation

dX_t=b(X_t)dt+σ(X_t)dB_t, (1)

whereX₀ is a random variable taking values inIand B_t

t0a standard Brownian motion on R. We assume that b andσare continuous functions on ¯Itaking values inR, and that σis not degenerate on I i.e. ∀x∈I, σ²(x)>0. Then there exists a unique solution

X_t

t0adapted to the completed Brownian filtration, such that t→X_tis continuous on [0, ζ[, whereζ= inf{t0, X_t=l orX_t=r} is the explosion time of the diffusion.

The classification of one-dimensional diffusion processes is due to Feller, in particular in [1] and [2]. An overview and a comparison with the Russian terminology are given in [4]. In this classification, there is essentially two concepts: “attractivity” and “attainability” to determine the behavior of the diffusion in a neighborhood of a boundary point. The notion ofattractivity of a boundary point (l orr) is defined using thescale function.

The scale function pis a strictly increasing function defined up to an affine transformation and such that the process

p(X_t^ζ)_t

t0 is a local martingale. In our framework (σ not degenerated on I), p is inC²(I,R) and satisfiesAp= 0 whereAis the infinitesimal generator. Theattainability of a boundary point is defined using thespeed measure of the processi.e. the measure with densitym= _σ2²p with respect to the Lebesgue measure.

A boundary point ∆ (∆ =l or ∆ =r) is said to beattractive if lim

b→∆b∈I

|p(b)|<+∞andrepulsive if not. From a probabilistic point of view, if ∆ is an attractive boundary point, then for all a ∈ I and all x in the open

Keywords and phrases. One-dimensional diffusion process, degenerate coefficient, invariant measure, Euler scheme.

1 Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, UMR 8050, Universit´e de Marne-la-Vall´ee, 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vall´ee Cedex 2, France;Vincent.Lemaire@univ-mlv.fr

c EDP Sciences, SMAI 2007

Article published by EDP Sciences and available at http://www.edpsciences.org/psor http://dx.doi.org/10.1051/ps:2007018

interval with endpointsaand ∆ we have P_x

b→∆lim

b∈I

T_bT_a

>0.

whereT_a denotes the hitting time of the one-point set{a}i.e. T_a= inf{t0, X_t=a}. If ∆ is attractive then

∆ is attainable if and only ifP_x[T_∆<+∞]>0. More generally, we say that a boundary point isattainable if for alla∈I andxin the open interval of endpoints aand ∆ we have

b→∆limE_x[T_b∧T_a]<+∞.

Furthermore, in this paper, we say that a repulsive boundary point ∆ isstrongly repulsiveif for everyc∈Iwe have^c

∆m(y)dy<+∞. The speed measure is thus finite in a neighborhood of a strongly repulsive point. Using these definitions, we are able to establish the following classification of the ergodic behavior of the diffusion.

Theorem 1.1. We recall thatζ= inf{t0, X_t=l orX_t=r}. Then

• if l is attractive andr is repulsive thenX_t^ζ −−→^a.s l;

• if l andrare attractive then P

t→+∞lim X_t^ζ =l

= 1−P

t→+∞lim X_t^ζ =r

=p(r⁻)−p(X₀) p(r⁻)−p(l⁺);

• if l and r are repulsive then the diffusion is recurrent and does not explode (ζ = +∞ a.s.). More precisely

– if l and r are strongly repulsive (i.e. the speed measure is finite) then the diffusion is positive recurrent andν_t⇒ν a.s.where ν is the normalized speed measure. Moreover

∀f ∈L¹(ν), 1 t

t 0

f(X_s)ds−−→

R

f(x)ν(dx) a.s.;

– ifl is strongly repulsive andris (simply) repulsive then the diffusion is null recurrent and if the empirical measures are tight we have

1 t

t 0

δ_Xsds⇒δ_r a.s.;

– if l and r are not strongly repulsive then the diffusion is null recurrent, and if the empirical measures are tight then any weak limit of

1 t

_t

0δ_Xsds

t1 is a measure with support {l, r}.

The first two items are proved in [3] (Prop. 5.22). The third item is proven in Appendix A.

Let us now recall the link between the concepts of attractivity, repulsivity and strong repulsivity, and the Lyapunov functions. This link given by the following proposition is useful to study the ergodic behavior of the decreasing step Euler scheme. In the sequel, we will denote by J_∆⊂I an open (non-trivial) interval included in Iwith endpoint ∆.

Proposition 1.2. Let ∆ a boundary point of I.

(1) ∆is a repulsive boundary point ofI if and only if there exists a neighborhoodJ_∆⊂I of∆ and a strictly monotone functionv∈ C²( ¯J_∆,R₊)satisfyingv(∆) = 0, such that

∀x∈J_∆, Av(x) 1

2σ²(x)(v(x))²

v(x) , (2)

(2) ∆is a strongly repulsive boundary point ofI if and only if there exists a neighborhoodJ_∆⊂I of∆and a strictly monotone functionv∈ C²( ¯J_∆,R₊)having a minimum at ∆, such that

∃ε >0, ∀x∈J_∆, Av(x)1

2σ²(x)(v(x))²

v(x) +εv(x), (3)

(3) ∆ is an attractive boundary point of I if and only if there exists a neighborhood J_∆ ⊂I of ∆ and a strictly monotone functionv∈ C²( ¯J_∆,R₊)having a minimum at ∆, such that

∀x∈J_∆, Av(x)< 1

2σ²(x)(v(x))²

v(x) · (4)

This proposition is proven in Appendix B.

Remark 1.3. If ∆ is such thatb(∆) =σ(∆) = 0, then ∆ is a critical point for the equationu =b(u). It is worth noting that there exists a link between the nature of the critical point ∆ for the ODEu=b(u) and the nature of the boundary point ∆ for the SDE.

Indeed, if ∆ is a stable critical point then there exists a Lyapunov functionF ∈ C²such thatFb(u)<0 for everyuin a neighborhood of ∆. IfF/F is decreasing, the above proposition implies that ∆ is an attractive boundary point for the SDE.

If ∆ is an unstable point for the ODEu=b(u), it may be repulsive, strongly repulsive or attractive for the SDE, as shown in Example 2.6.

Let us now suppose that the diffusion X_t

t0 on the real line has (at least) a point ∆ such that b(∆) = σ(∆) = 0. In this situation we have

∀x∈]− ∞,∆[, P_x[X_t∈]− ∞,∆]] = 1, and ∀x∈]∆,+∞[, P_x[X_t∈[∆,+∞[] = 1.

In fact, the process X_t

t0has an ergodic behavior inI₁=]−∞,∆[ or inI₂=]∆,+∞[ according to the starting pointx. The Euler scheme of this diffusion is not continuous and maya priori jump above the boundary point

∆. In this note, we show that in some particular cases the boundary ∆ becomes a border for the scheme after an almost surely finite time. This is the principal result given by Theorem 2.1. We next prove in Proposition 2.5 that the empirical measures of the scheme have the same behavior than the empirical measures of the diffusion.

Finally, a numerical example is given to illustrate these results.

2. Behavior of the Euler scheme with decreasing step

The Euler scheme X_n

n0 with decreasing step γ_n

n1is defined as follows. Consider a positive sequence γ_n

n1 satisfying lim_nn

k=1γ_k = +∞ and denote Γ_n =_n

k=1γ_k. The Euler scheme is the inhomogeneous Markov chain defined as

X_n+1=X_n+γ_n+1b(X_n) +√γ_n+1σ(X_n)U_n+1, with

U_n

n1a real white noisei.e. a sequence ofi.i.d. random variables such thatE[U₁] = 0 and var(U₁) = 1.

Furthermore, we assume thatU₁ is a generalized Gaussian (cf. [9])i.e. such that

∃κ >0, ∀θ∈R, E

exp(θU₁)

exp

κ|θ|² 2

.

For example U₁ is a standard Gaussian or a Bernoulli random variable. A consequence of the generalized Gaussian property is the following

∀a0, P

|U₁|a

exp

−a² 2κ

. (5)

We consider a diffusion X_t

t0 on the real line withb andσcontinuous onR. Moreover we assume thatb andσhave sublinear growthi.e.

∃C_b>0, |b|²C_b(1 +|x|), and ∃C_σ >0, |σ|²C_σ(1 +|x|). (6) In the sequel ∆ denotes a finite point ofRsuch that b(∆) =σ(∆) = 0, andJ_∆ denotes an open interval with endpoint ∆ (J_∆=]∆,∆ +ε[ if ∆ is a left endpoint ofI=]∆,+∞[ and J_∆=]∆−ε,∆[ if it is a right endpoint ofI=]− ∞,∆[).

2.1. Euler scheme

We prove the following theorem which gives the behavior of one trajectory of the Euler scheme X_n

n0in this degenerate situation.

Theorem 2.1. We assume that σ satisfies σ(x) = 0 for every x ∈]− ∞,∆[∪]∆,+∞[ and that there exists U∆=]∆−ε,∆ +ε[ withε >0, and a convex function v∈ C²(U∆,R₊)satisfying v(∆) = 0and

∀x∈ U_∆, (vb)(x)0 and ∃c_σ>0,∀x∈ U_∆, |(vσ)(x)|c_σv(x). (7) If the step sequence

γ_n

n1 satisfies∀C >0,

n1exp −_γ^C_n

<+∞then the Euler scheme jump above ∆a finite number of times i.e.

P

∃n₀0, ∀nn₀, X_n∈]− ∞,∆[

+P

∃n₀0, ∀nn₀, X_n ∈]∆,+∞[

= 1.

We first prove the following lemma.

Lemma 2.2. We assume there exists a convex function v∈ C²( ¯J_∆,R₊)satisfyingv(∆) = 0and

∀x∈J¯_∆, (vb)(x)0 and ∃c_σ>0, ∀x∈J_∆, |(vσ)(x)|c_σv(x). (8) Then, on the event

X_n ∈J¯_∆ P

∆∈(X_n, X_n+1)Fn

exp

− 1 c²_σγ_n+1

.

Proof. We assume that ∆ is a left endpoint of I and we denote by A_n+1 the event {∆∈(X_n, X_n+1)} (the geometric segment with endpointsX_n andX_n+1). Sincev is continuous andv(∆) = 0 it is clear that

A_n+1={∃t∈[0,1], X_n+t(X_n+1−X_n)) = ∆},

⊂ {∃t∈[0,1], v(X_n+t(X_n+1−X_n)) = 0}. (9) Considert∈[0,1] such that ∆ =X_n+t(X_n+1−X_n). AsvisC² on ¯J_∆, Taylor’s formula gives

v(X_n+t(X_n+1−X_n)) =v(X_n) +v(X_n)t(X_n+1−X_n) +v(ξ_n+1)

2 t²(X_n+1−X_n)², withξ_n+1∈]∆, X_n[. The convexity ofv implies

0 =v(X_n+t(X_n+1−X_n))v(X_n) +tγ_n+1(vb)(X_n) +t√γ_n+1(vσ)(X_n)U_n+1. Sincevb0 on ¯J_∆we have from (9)

A_n+1∩

X_n ∈J¯_∆

⊂

∃t∈[0,1], v(X_n) +t√γ_n+1(vσ)(X_n)U_n+10

∩

X_n∈J¯_∆ ,

⊂

∃t∈[0,1], t√γ_n+1|(vσ)(X_n)U_n+1|v(X_n)

∩

X_n∈J¯_∆ .

Hence for anyn0, we have on the event

X_n∈J¯_∆ P[A_n+1| F_n]P

|U_n+1| v(X_n)

√γ_n+1|(vσ)(X_n)|

F_n

,

P

|U_n+1| 1 c_σ√γ_n+1

F_n

,

by the domination assumption (8) onvσ. We conclude using property (5) of the random variableU₁. Proof. By Lemma 2.2 we prove easily that for everyn0

P

∆∈(X_n, X_n+1)F_n

exp

− 1 c²_σγ_n+1

onX_n∈ U_∆. (10)

We now consider the event {X_n ∈ U/ _∆}. Then we have {∆∈(X_n, X_n+1)}=

∃t∈[0,1], U_n+1= ∆−X_n

t√γ_n+1σ(X_n)−√γ_n+1b(X_n) σ(X_n)

,

⊂

|U_n+1| |∆−X_n|

√γ_n+1|σ(X_n)| −√γ_n+1|b(X_n)|

|σ(X_n)|

.

As the driftbis dominated byC_b(1 +|x|), we have {∆∈(X_n, X_n+1)} ⊂

|U_n+1|

|∆−X_n|

√γ_n+1C_b(1 +|X_n|)−√γ_n+1

C_b(1 +|X_n|)

|σ(X_n)|

, and using the triangular inequality and |X_n−∆| ε we prove that ^|∆−X_1+|X^n|

n| ₁₊1+|∆|¹ ε

. We also deduce that there exists n₁0 andC >0 such that for anynn₁,

P[{∆∈(X_n, X_n+1)} ∩ {X_n∈ U/ _∆} | F_n]P

|U_n+1| C

√γ_n+1

C_b(1 +|X_n|)

|σ(X_n)|

∩ {X_n ∈ U/ _∆} F_n

,

P

|U_n+1| CC_b C_σ√γ_n+1

∩ {X_n∈ U/ _∆} F_n

, (11)

using|σ|√ C_σ√

V.

From (10) and (11) combined with (5), we get

∃n₁0, ∃C >0, ∀nn₁, P

∆∈(X_n, X_n+1)Fn

exp

− C γ_n+1

. By the condition on the sequence

γ_n

n1 and the conditional Borel-cantelli lemma we deduce that the event

{∆∈(X_n, X_n+1)}occurs a finite number of times.

Remark 2.3. The condition on the step sequence γ_n

n1 is not restrictive. Indeed, it is satisfied for γ_n

n1

defined byγ_n=γ₀n^−rwith γ₀>0 andr∈]0,1], orγ_n= log(n)^−r withr >1.

The technical assumption “v convex” is not very restrictive in practice. The important point to note is the conditionvb0 which implies that ∆ is unstable for the ODEu =b(u). But ∆ may be repulsive as well as attractive for the SDE (cf. Rem. 1.3). The conditionvσ=O(v) in a neighborhood of ∆ is very important and it seems difficult to relax it.

Remark 2.4. It is easy to extend the above theorem to the case of finitely many boundary points ∆_i. If for every ∆_i, there exists a neighborhood U∆i and a convex function v_i ∈ C²(U∆i,R) such that v(∆) = 0 and satisfying (7), then

P[∃i∈ {0, . . . , l}, ∃n₀0, ∀nn₀, X_n∈]∆_i,∆_i+1[] = 1, with ∆₀=−∞and ∆_l+1= +∞.

2.2. Weighted empirical measures

η_n

n1 a positive sequence, called weight sequence, such thatH_n=_n

k=1η_k increases to +∞whenn tends to +∞. We define the weighted empirical measures

ν_n^η

n1by

∀n1, ν_n^η(dx) = 1 H_k

n k=1

η_kδ_Xk−1.

In this section we assume that the diffusion satisfies a stability conditioni.e.

∃α >0, ∀|x|M, xb(x) +1

2σ²(x)−αx².

This condition implies that the points −∞and +∞ are strongly repulsive and that the empirical measures ν_t

t0 are tight. Sincebandσhave sublinear growth, this condition implies also the tightness of the weighted empirical measures

ν_n^η

n1 of the scheme and that any weak limit is an invariant probability for the diffusion (cf. [5] or [6]).

A consequence of Theorem 2.1 is the following proposition which describe the convergence of ν_n^η

n1 ac- cording to the behavior of bandσin a neighborhood of ∆. For more clearness, we parametrizeband σ.

Proposition 2.5. Let∆the unique point ofRsuch thatb(∆) =σ(∆) = 0. We assume that in a neighborhood of ∆we have b(x) = sgn(x−∆)ρ_b(x)andσ(x) =ρ_σ(x)withρ_b0,

ρ_b(x)∼c_b|x−∆|^β and σ(x)∼c_σ|x−∆|^ς, (12) where β, ς, c_b and c_σ are positive real numbers and ς 1. If the step sequence

γ_n

n1 satisfies ∀C > 0,

n1exp(−C/γ_n)<+∞, then

• if 1 +β−2ς >0 then∆ is an attractive boundary point and ν_n^η⇒δ_∆,

• if 1 +β−2ς = 0 andc_σ>√2c_b then∆ is an attractive boundary point andν_n^η ⇒δ_∆,

• if 1 +β−2ς = 0,c_σ<√2c_b and β= 1 (which implies ς = 1) then ∆ is a strongly repulsive boundary point and ν_n^η ⇒ν₊ orν_n^η ⇒ν₋,

• if 1 +β−2ς <0,c_σ√2c_b andβ ∈]0,1]then∆ is a strongly repulsive boundary point andν_n^η ⇒ν₊ orν_n^η⇒ν₋,

whereν₊ is the invariant probability on]∆,+∞[andν₋ the invariant probability on]− ∞,∆[.

Proof. To simplify notation, we assume without loss of generality that ∆ = 0.

LetU∆=]∆−ε,∆ +ε[ a neighborhood of ∆ and the convex functionv(x) =x². Then for everyx∈ U∆ we have

vb= 2xsgn(x)ρ_b(x)0.

Moreovervσ∼c_σ|x|^ς+1 withς 1 hence there existsC >0 such that|vσ|Cv. By Theorem 2.1 we know then that the scheme lives in ]− ∞,∆[ or in ]∆,+∞[ after an almost-surely finite random time.

Furthermore, withv(x) =x² we have

∀x∈ U∆, Av(x) = 2|x|ρ_b(x) +σ²(x) = 2c_b|x|^1+β+c²_σ|x|^2ς+o

|x|^{(1+β)∨(2ς)}

, (13)

and 1

2σ²(x)(v(x))²

v(x) = 2σ²(x)∼2c²_σ|x|^2ς.

– If 1 +β >2ς, there exists a neighborhood of 0 in whichAv < ¹₂σ^(v_v⁾². Hence by Proposition 1.2, 0 is an attractive boundary point. The sequence of weighted empirical measures

ν_n^η

n1 of the scheme is tight on [0,+∞[ and on ]− ∞,0], and any weak limit is an invariant probability. However, δ₀ (the Dirac at 0) is the unique invariant probability on [0,+∞[ or on ]− ∞,0]. Thus any weak limit of

ν_n^η

n1isδ₀, which proves the first item.

– If 1 +β = 2ς,Av(x) = (2c_b+c²_σ)|x|^2ς+o

|x|^2ς

. Ifc_σ>√2c_b there exists a neighborhood of 0 in which Av <2σ² and we conclude as above.

Ifβ = 1 andc_b<√2c_b, we have for everyxin U∆,Av(x)2σ²(x) +εx². By Proposition 1.2, the point 0 is then strongly repulsive. Any weak limit of

ν_n^η

n0 is a probability on ]− ∞,0[ or on ]0,+∞[, therefore we haveν_n^η⇒ν₋ orν_n^η⇒ν₊(we recall that the boundary points−∞and +∞are strongly repulsive).

– The proof for the case 1 +β <2ς,c_ς√2c_b andβ∈]0,1] is similar.

In order to illustrate this result, let us consider the following example.

Example 2.6. We consider, like in [7], the functionV :R→R₊ defined by V(x) =

x−3 sgn(x)₂

if|x|3,

721(x²−9)² if|x|3 and b(x) = −2

x−3 sgn(x)

if|x|3,

−₁₈¹x³+¹₂x if|x|3 ,

and b=−V. The ordinary differential equationu =b(u) has 3 critical points: −3, 0 and 3. The points−3 and 3 are stable and the point 0 is unstable. Letc∈]0,2[ a parameter andσdefined byσ(x) =cx. We consider the process

X_t

t0 solution of the SDE dX_t=b(X_t)dt+σ(X_t)dB_t. It is clear that the point 0 is a boundary point for

X_t

t0. Moreover we check that AV(x) =−

4−c²

x²+ 12 sgn(x) if|x|3, and that the points−∞and +∞are then strongly repulsive.

On the other hand, we use Proposition 1.2 to determine the nature of the boundary point 0 according toc.

Assume thatI=]0,+∞[ and letv the function defined on [0,+∞[ byv(x) =x². We have

∀x∈]0,3[, Av(x) = (1 +c²)x²−1

9x⁴ and 1

2σ²(x)(v(x))²

v(x) = 2c²x², (14) and then

• ifc <1 the boundary point 0 isstrongly repulsive (for ]0,+∞[ and ]− ∞,0[ by symmetry). Indeed the condition (3) is satisfied withε= ^1−c₂² andJ_∆=

0,3 1−c²

2

;

• ifc >1 it is easy to check that the boundary point 0 isattractive;

• ifc= 1 we consider the functionv(x) =xexp(x) and we check that 0 is a repulsive boundary point.

Thus the nature of the boundary point 0 (which is always stable for the ODEu =b(u)) may change according toc. By Theorem 1.1 the ergodic behavior of

X_t

t0 is the following:

• ifc∈]1,2[ thenX_t−−→^a.s 0 (for every starting pointX₀);

• ifc= 1 then ¹_tt

0δ_Xs(dx)⇒δ₀;

0 0.8 1.6 2.4 3.2 4

-2 -1 0 1 2 3 4 5 6 7 8

(a)c= 0.1

0 0.2 0.4 0.6 0.8

-2 -1 0 1 2 3 4 5 6 7 8

(b)c= 0.5

0 0.4 0.8 1.2 1.6

-2 -1 0 1 2 3 4 5 6 7 8

(c)c= 0.75

0 4 8 12 16 20

-2 -1 0 1 2 3 4 5 6 7 8

(d)c= 1

Figure 1. Approximation of the stationary density for different values ofc.

• ifc <1 then for everyf ∈L¹(m)

1 t

t 0

f(X_s)ds−−→^a.s

⎧⎪

⎨

⎪⎩

fdν₋ ifX₀∈]0,+∞[

f(0) ifX₀= 0 fdν₊ ifX₀∈]− ∞,0[

withν₋ the invariant measure of X_t

t0on ]− ∞,0[ andν₊ the invariant measure on ]0,+∞[.

By the above Proposition, we know that the weighted empirical measure ν_n^η

n1weakly converge toδ₀when c∈]1,2[ and toν₊ orν₋ whenc∈]0,1[. Note that the convergence toν₊ orν₋ does not depend on the initial condition and is not previsible.

We give a representation of the density ofν approximated byν_n^η withn= 10⁶. More precisely, we discretize the interval [−2,8] using 200 intervalsI_i of length 0.05 and we computeν_n^η(1_Ii) for eachI_i withn= 10⁶. The step sequence

γ_n

n1 is defined byγ_n = n^−1/3 and the weight sequence η_n

n1 is defined byη_n = 1. The results of this approximation of the stationary density are given in Figure 1 for different values ofc.

We remark that for a small noise (c = 0.1), the invariant probability concentrates around a stable point of the ODEu=b(u) (the point 3), and the more coefficient of diffusion increases, the more the invariant measure is spread out. For c = 0.75 we show that the invariant measure is infinite at 0, and for c = 1 the invariant measures seems to be the Dirac mass at 0.

A. Proof of Theorem 1.1

We first prove the following lemma.

Lemma A.1. If the two boundary points l and r are repulsive then the diffusion is positive recurrent if and only if its speed measure is finite.

Proof. By definition, the diffusion is positive recurrent if and only if for all a and b in I, E_a[T_b] < +∞ and E_b[T_a]<+∞. Letl < a < b < r. By symmetry it is sufficient to prove that

E_a[T_b]<+∞ ⇔ ^a

l

m(y)dy <+∞. (15)

Firstly,l is repulsive andT_l= lim_x→lT_xthereforeE_a[T_b] =E_a[T_b∧T_l] = lim_x→lE_a[T_b∧T_x]. Moreover we have∀x∈]l, a[,

E_a[T_b∧T_x] =P_a[T_b< T_x]

b

a (p(b)−p(y))m(y)dy+P_a[T_xT_b]

a

x (p(y)−p(x))m(y)dy, and since_b

a(p(b)−p(y))m(y)dy is finite and does not depend onx, the limit whenxtends tolofE_a[T_b∧T_x] is finite if and only if

x→llim

P_a[T_xT_b]

a x

(p(y)−p(x))m(y)dy

<+∞. AsP_a[T_xT_b] = ^p(b)−p(a)_p(b)−p(x) we have

P_a[T_xT_b]

a x

(p(y)−p(x))m(y)dy= (p(b)−p(a)) ^a

x

p(y)−p(x)

p(b)−p(x)m(y)dy,

= (p(b)−p(a)) ^a

x

P_y[T_b< T_x]m(y)dy,

and it follows that E_a[T_b]<+∞if and only if lim

x→l a x

P_y[T_b< T_x]m(y)dy <+∞. Since T_x strictly increases to T_l, P_y[T_b< T_x] increases to P_y[T_b< T_l] = 1 because l is repulsive. The monotone convergence theorem

yields (15).

Proof of Theorem 1.1. The first two items are proved in [3] (Prop. 5.22). We prove the third item. Suppose that landr are repulsive. By definition of the scale function we have for alll < a < x < b < r

P_x

0t<ζinf X_ta

P_x[x_Ta∧Tb=a] = p(b)−p(x) p(b)−p(a)·

Increasing bto rwe obtain P_x[inf_0t<ζX_ta] = 1 sinceris repulsive. The limit when adecreases tol also gives

P_x

0t<ζinf X_t=l

= 1.

In the same way we obtain P_x

sup_0t<ζX_t=r

= 1. The diffusion is thus recurrent onI and ζ = +∞ a.s.

Moreover, by Lemma A.1 we know that the recurrence is positive if and only if the speed measure is finite.

– If the two boundary points are repulsive then the speed measure is finite and by Theorem (53.1) in [8]

we have

∀f ∈L¹(ν), 1 t

t 0

f(X_s)ds⇒

R

fdν a.s.

whereν is the normalized speed measure.

– Ifl is strongly repulsive andris repulsive then the diffusion is null recurrent. Considering an increasing sequence of continuous functions with compact support (g_n(x))_n1 such thatg_n(x) →1 and ∀n1,

Rg_n(x)m(dx)= 0, we obtain by Theorem (53.1) in [8]

∀f ∈L¹(m), 1 t

t 0

f(X_s)ds−−→^a.s 0. (16)

On the other hand, we consider a sub-sequence ν_a(t)

t0 of ν_t

t0 converging to a measure ν (the empirical measures are tight). Letfbe a continuous function with compact support such that supp(f)⊂ [l, r[. Asν_a(t)⇒ν we have

1 a(t)

a(t) 0

f(X_s)ds−−→^a.s fdν.

The boundary pointl is strongly repulsive and supp(f)⊂[l, r[, thusf is integrable with respect tom.

Hence (16) implies

fdν = 0. The interval [l, r[ satisfies

∀f ∈ Cc( ¯I), supp(f)⊂[l, r[⇒ν(f) = 0, therefore supp(ν) ={r}. Sinceν is normalized we haveν=δ_r.

– In the same way, if the two boundary points are strongly repulsive then any weak limit of ν_t

t0 is a measure with support{l, r}.

B. Proof of Proposition 1.2

We first prove the following proposition which is equivalent to the Proposition 1.2.

Proposition B.1. Let ∆ be a boundary point (finite or infinite, left endpoint or right endpoint) of I. The following statements are equivalents

(1) ∆is a repulsive boundary point ofI if and only if there exists a neighborhoodJ_∆⊂I of∆ and a strictly monotone functionV ∈ C²(J_∆,R₊)such that

x→∆lim V(x) = +∞ and ∀x∈J_∆, AV(x)0.

(2) ∆is a strongly repulsive boundary point ofI if and only if there exists a neighborhoodJ_∆⊂I of∆and a strictly monotone functionV ∈ C²(J_∆,R₊)such that

∃ε >0, ∀x∈J_∆, AV(x)−ε.

(3) ∆is a attractive point ofIif and only if there exists a neighborhoodJ_∆⊂Iof∆and a strictly monotone function V ∈ C²(J_∆,R₊)such that

x∈Jsup∆

V(x) =V(∆)<+∞ and ∀x∈J_∆, AV(x)0.

Proof. We give the proof when ∆ is a right endpoint ofI. ThenJ_∆ is an interval ]c,∆[ withc∈I.

• – We suppose that there exists a neighborhood J_∆ of ∆ and a function V ∈ C²(J_∆,R₊) such that lim_x→∆V(x) = +∞andAV 0 onJ_∆. For everyx∈J_∆we have

AV(x) = 1 m(x)

V(x) p(x)

0

henceV/p is decreasing onJ_∆. There also existsC >0 such thatV(x)Cp(x) for everyx∈]c,∆[

sincep >0. It follows that lim

x→∆p(x)−p(c) = +∞because V tends to infinity in ∆.

– Conversely we must find the good Lyapunov function V. Let c >0 be such that p(c)>0 (c exists because ∆ is repulsive). Sincepis strictly increasing we have for everyx∈]c,∆[,p(x)> p(c)>0. We also define the function V on ]c,∆[ by

∀x∈]c,∆[, V(x) =p(x)−p(c).

The point ∆ is repulsive thus V increases to infinity whenxtends to ∆. MoreoverV ∈ C²(]c,∆[,R₊) andAV = 0.

• – LetV ∈ C²(J_∆,R₊) be such that lim_x→∆V(x) = +∞andε >0 such thatAV −ε. Thus we have

J∆

AV(x)m(x)dx−ε

J∆

m(x)dx, (17)

and sinceAV(x) = 1 m(x)

V(x) p(x)

we obtain for everyc∈J_∆,

∆

c AV(x)m(x)dx= lim

x→∆

V(x)

p(x) −V(c)

p(c)· (18)

By (17) we derive that

∆ c

m(x)dx 1 ε

V(c) p(c) − lim

x→∆

V(x) p(x)

.

As the functions V andpare increasing on J_∆ we have lim

x→∆

V(x)

p(x) 0, which gives _∆

c m(x)dxC (i.e. ∆ strongly repulsive).

– Conversely we assume that ∆ is strongly repulsive. Letc∈IandV the function defined on ]c,∆[ by

∀x∈]c,∆[, V(x) = ^x

c

p(y) ^∆

y

m(z)dz

dy.

It is clear that V ∈ C²(]c,∆[,R₊) and that for everyx∈]c,∆[,V(x) =p(x)_∆

x m(z)dz. Moreover

∀x∈]x,∆[, AV(x) =−1.

• We prove (3) in the same manner as (1). For the converse we consider the functionV(x) =p(x)−p(c) on ]c,∆[ withcsuch thatp(c)>0.

Proof of Proposition 1.2. LetJ_∆ a neighborhood of ∆ strictly included inI. We consider the case in which ∆ is the right endpoint ofI i.e. J_∆=]c,∆[ withc∈I. We first define forL0 the functionφ_L by

φ_L: [0,exp(L)[→R₊,

x→ −ln(x) +L.

The functionφ_L is a strictly decreasing one-to-oneC^∞ function. Moreover, for everyC² functionv with values in [0,exp(L)[ we have

A(φ_L◦v) =−Av v +1

2σ²(v)²

v² · (19)

• – We assume that there exists v ∈ C²([c,∆],R₊) strictly monotone satisfyingv(∆) = 0 and (2). We also define on ]c,∆[ the functionV by∀x∈]c,∆[,V(x) = (φ_L◦v) (x) withL= ln(v(c)). It is a strictly monotone function which tends to infinity in ∆. By (2) and (19) we obtain AV 0 on ]c,∆[. The proposition (B.1) implies that ∆ is repulsive.

– Conversely if ∆ is repulsive then there existsV ∈ C²(]c,∆[,R₊) strictly monotone which goes to +∞

when xtends to ∆. We also define v =φ⁻¹_L ◦V on ]c,∆[ with L= inf_x∈]c,∆[V(x), and we extend it by continuity on [c,∆] lettingv(c) = 1 andv(∆) = 0. ByAV 0 and (19) we haveAv¹₂σ^{2 (v}_v⁾² on ]c,∆[.

• – Let v a strictly monotone function on [c,∆] such that v(∆) = 0. For θ 0 andL = ln(v(c) +θ) we define V(x) = (φ_L,θ◦v)(x) for every x∈]c,∆[. This functionV is strictly monotone on ]c,∆[ and admits a limit (finite or not) whenxtends to ∆. From (3) and (19) we deduce that

∀x∈]c,∆[, AV(x)−ε.

– Conversely we consider the functionv =φ⁻¹_L ◦V on ]c,∆[ withL= inf_x∈]c,∆[V(x) and we extend it by continuity lettingv(c) = 1 andv(∆) = lim_x→∆exp(−V(x)).

• The proof is similar to the first two items.

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