An ergodic theorem for the maximum of branching Brownian motion with absorption
Fan Yang

	Branching Brownian motion with absorption]An ergodic theorem for the maximum of branching Brownian motion with absorption
	F. Yang]Fan Yang
	Fan Yang
 School of Mathematical Sciences 
 Beijing Normal University
 Beijing 100875
 P. R. China
	fan-yang@bnu.edu.cn
	The research of this project is supported by the National Key R&D Program of China (No. 2020YFA0712900). The research of F. Yang is supported by China Postdoctoral Science Foundation (No. 2023TQ0033) and Postdoctoral Fellowship Program of CPSF (No. GZB20230068).

	
	§ ABSTRACT  
		In this paper, we study branching Brownian motion with absorption, in which particles undergo Brownian motions and are killed upon hitting the absorption barrier. We prove that the empirical distribution function of the maximum of this process converges almost surely to a randomly shifted Gumbel distribution.
	
	[2020]Primary: 60J80; Secondary: 60G70
	
	
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§ INTRODUCTION

	A classical branching Brownian motion (BBM) in  can be constructed as follows. Initially there is a single particle at the origin of the real line and this particle moves as a 1-dimensional standard Brownian motion denoted by B = {B(t), t≥ 0 }. 
	After an independent exponential time with parameter 1, the initial particle dies and gives birth to L offspring, where L is a positive integer-valued random variable with distribution {p_k: k≥ 1 }. Here we assume that the expected number of offspring is 2 (i.e., ∑_k=1^∞ kp_k = 2) and the variance of the offspring distribution is finite (i.e., ∑_k=1^∞ k(k-1)p_k < ∞). Each offspring starts from its creation position and evolves independently, according to the same law as its parent. We denote the collection of particles alive at time t as N_t. For any u∈ N_t and s≤ t, let X_u(s) be the position at time s of particle u or its ancestor alive at that time. 
	
	McKean <cit.> established a connection between BBM and the Fisher-Kolmogorov-Petrovskii-Piskounov (F-KPP) equation
	
    ∂ u/∂ t = 1/2∂^2 u/∂ x^2 + ∑_k=1^∞ p_k u^k - u.

	The F-KPP equation has received entensive attention from both analytic techniques (see, for instance, Kolmogorov et al. <cit.> and Fisher <cit.>) and probabilistic methods
	(see, for example, McKean <cit.>, Bramson <cit.>, Harris <cit.> and Kyprianou <cit.>). 
	
	Let's recall some classical results on BBM and F-KPP equation. Define
	
    𝐌_t := max{X_u(t): u∈ N_t }.

	Bramson <cit.> established that
	
    lim_t→∞ℙ(𝐌_t - m_t ≤ z)=lim_t→∞ u(t,m_t+z)=w(z),   z∈,

	where m_t:= √(2) t-3/2√(2)log t and w solves the ordinary differential equation 1/2w”+√(2)w'+∑_k=1^∞ p_k w^k - w=0.
	Such a solution w is known as the traveling wave solution.
	Lalley and Sellke <cit.> provided the following representation of w for dyadic BBM
	
    w(z):=[e^-C_* e^-√(2) zZ_∞],

	where C_* is a positive constant and Z_∞ is the limit of the derivative martingale of BBM. 
	Specifically, define 
	
    Z_t = ∑_u∈ N_t (√(2)t - X_u(t)) e^√(2)(X_u(t)-√(2)t) ,

	then Z_t serves as the derivative martingale of the BBM. We denote Z_∞ as the limit of Z_t ℙ_x-almost surely, as established by Lalley and Sellke <cit.> or Kyprianou <cit.>. In the same paper <cit.>, they conjectured that the empirical (time-averaged) distribution of maximal displacement converges almost surely, that is,
	
    lim_T→∞1/T∫_0^T 1_{𝐌_s - m_s ≤ x}d s = exp{-C_* Z_∞ e^-√(2)x},

	This conjecture was later confirmed by Arguin, Boiver and Kistler <cit.>.
	
	In this paper, we consider the similar problems of BBM with absorption, where the particle is killed when it hits the absorbing barrier. The process can be defined as follows. Initially there is a single particle at x>0 and this particle evolves as the classical BBM with branching rate 1. We also assume that the number of offspring L has distribution {p_k,k≥ 1} with L = 2 and L^2 < ∞. In addition, we add an absorbing barrier at the line {(y,t):y= ρ t } for some ρ∈, i.e. particles hitting the barrier are instantly killed without producing offspring  (see Figure <ref>). 
	
	
	We use N_t to denote the set of the particles of the BBM with absorption that are still alive at time t. For any particle u∈N_t and any time s≤ t, we continue to use X_u(s) to represent the position of either particle u itself or its ancestor at time s.  
	The extinction time of the BBM with absorption is defined as
	
    ζ:=inf{t>0: N_t = ∅}.
 
	Additionally, we define 𝐌_t as the maximum position among all particles u∈N_t. The law of the BBM with absorption, starting from a single particle at position x is denoted by ℙ_x, and its expectation is denoted by _x. 
	
	The asymptotic behavior of BBM with absorption has been extensively studied in the literature.
	Kesten <cit.> demonstrated that the process dies out almost surely when ρ≥√(2) while there is a positive probability of survival, that is, ℙ_x(ζ=∞)>0, when ρ < √(2). Therefore, ρ = √(2) is the critical drift separating the supercritical case ρ < √(2) and the subcritical ρ > √(2).
	In the subcritical case,  Harris and Harris <cit.> provided the large time asymptotic behavior for the survival probability.
	In the critical case, Kesten <cit.> obtained upper and lower bounds on the survival probability, which were subsequently improved by Berestycki et al. <cit.>. Maillard and Schweinsberg <cit.> have further enhanced these results and investigated the behavior conditioned to survive. For the BBM with absorption in the near-critical case, Berestycki et al. <cit.> and Liu <cit.> are good references.
	In the supercritical case, Harris et al. <cit.> studied properties of the right-most particle and the one-sided F-KPP traveling wave solution using probabilistic methods in the case of binary branching. Specifically, they proved that
	
    lim_t→∞𝐌_t/t=√(2)   on  {ζ=∞},  ℙ_x

	and g(x) := ℙ_x(ζ<∞) is the unique solution to the one-side F-KPP traveling wave solution
	
    {   1/2g”-ρ g'+ g^2 -g = 0,   x>0, 
        g(0+)=1,   g(∞)=0.
    .

	Louidor and Saglietti <cit.> showed that the number of particles inside any fixed set, normalized by the mean population size, converges to an explicit limit almost surely.
	In this paper, we focus on the supercritical case. In the remainder of this paper, we always assume ρ<√(2).
	
	In <cit.>, we studied the maximal displacement and the extremal proccess of BBM with absorption. More precisely, we established the following result:
	
    lim_t→∞ℙ_x(𝐌_t - m_t≤ z) = _x(e^-C_*Z_∞ e^-√(2)z),

	where _∞ is defined as the limit of _t and 
	
    _t := ∑_u∈N_t (√(2)t-X_u(t)) e^√(2)(X_u(t)-√(2)t) .

	It's important to note that {_t, t≥ 0, ℙ_x } is not a martingale. However, according to <cit.>, the limit _∞ := lim_t→∞_t exists ℙ_x-almost surely for any x>0 and ρ < √(2). 
	Similar to (<ref>), our paper focuses on the empirical distribution function of the maximum of branching Brownian motion with absorption. We prove that the limit of this empirical distribution converges almost surely to a Gumbel distribution with a random shift. Here's the statement of the main result.
	
		For any x>0, ρ<√(2) and z∈, we have
		
    lim_T↑∞1/T∫_0^T 1_{𝐌_t -m_t ≤ z}dt = exp{-C_* _∞ e^-√(2) z},   ℙ_x,

		where the positive constant C_* is given by (<ref>). 
	
	
	

§ PROOF OF THEOREM <REF>

	We can put a BBM and a BBM with absorption in the same probability space. More precisely, we can construct a BBM as described in Section <ref>. By considering only the particles that are never killed by the line {(y,t): y=ρ t }, we obtain a BBM with absorption. Therefore, 
	
    N_t = {u∈ N_t: ∀ s≤ t, X_u(s) > ρ s }.

		
	Furthermore, we define
	
    N_t^s := {u∈ N_t: ∃ v∈N_s,  u>v },

	where u>v indicates that u is a descendant of v (see Figure <ref>). Notice that the set N^s_t contains all the particles alive at time t that do not hit the line segment {(y,r):y =ρ r, 0≤ r≤ s }. 
	For convenience, define
	
    M_t := max{X_u(t): u∈ N_t } - m_t     M_t := max{X_u(t): u∈N_t } - m_t.
 
	Then M_t = 𝐌_t - m_t and M_t = 𝐌_t - m_t. Similarly, define
	
    M_t^s := max{X_u(t): u∈N^s_t} - m_t.

	
	In the proof of Theorem <ref>, we need the following two lemmas, whose proofs are postponed to Sections <ref> and <ref>. First, as in <cit.>, consider a compact interval = [d,D] with -∞<d<D<∞ and the time R_T>0. However, in this paper, we require that there exists some l>0 such that R_T/T^l↑∞ and R_T/√(T)↓ 0, as T↑∞. We truncate the absorption barrier at time R_T and will show that the empirical distribution of M_t^R_T converges almost surely.
	
	
		Let R_T/T^l ↑∞ as T↑∞ for some l>0 but with R_T = o(√(T)). Then for any x>0 and z∈,
		
    lim_T↑∞1/T∫_0^T 1_{M_t^R_T≤ z}dt = exp{ -C_* _∞ e^-√(2)z},   ℙ_x

	
	
	Next, we consider the difference between M_t and M_t^R_T. For s<t, define N_t^[s,t] = N_t^s - N_t, which represents the set of particles at time t whose ancestors are not absorbed before time s but hit the absorption barrier between time s and t. Define
	
    M_t^[s,t] := max{ X_u(t): u∈N_t^[s,t]} - m_t.

	The following lemma provides the convergence of the empirical distribution of M_t^[s,t].
	
		For ϵ>0, and R_T as in Lemma <ref>,
		
    lim sup_T↑∞1/T∫_ϵ T^T 1_{M_t^[R_T,t] > z}dt = 0,   ℙ_x

	
	
	
	
		Note that N_t ⊂N_t^R_T. Hence we have M_t ≤M^R_T_t and then 1_{M_t ≤ z }≥1_{M^R_T_t ≤ z }. By Lemma <ref>, it holds that
		
    lim inf_T↑∞1/T∫_0^T 1_{M_t≤ z}dt ≥exp{ -C_* _∞ e^-√(2)z},   ℙ_x

		Since 1_{M_t≤ z} + 1_{M_t > z} = 1, we have
		
    lim sup_T↑∞1/T∫_0^T 1_{M_t > z}dt ≤ 1-exp{ -C_* _∞ e^-√(2)z},   ℙ_x

		To prove Theorem <ref>, it suffices to show that
		
    lim inf_T↑∞1/T∫_0^T 1_{M_t > z}dt ≥ 1-exp{ -C_* _∞ e^-√(2)z},   ℙ_x
 
		Since M_t^s = max{M_t, M_t^[s,t]}, we have
		
    1_{M_t^s > z}≤1_{M_t > z} + 1_{M_t^[s,t] > z},

		that is,
		
    1_{M_t > z}≥1_{M_t^s > z} - 1_{M_t^[s,t] > z}.

		Set s=R_T, we have
		
    lim inf_T↑∞1/T∫_ϵ T^T 1_{M_t > z}dt ≥lim inf_T↑∞1/T∫_ϵ T^T 1_{M_t^R_T > z}dt - lim sup_T↑∞1/T∫_ϵ T^T 1_{M_t^[R_T,t] > z}dt.

		For any ϵ>0, by (<ref>) and Lemma <ref>, we get that
		
    lim inf_T↑∞1/T∫_0^T 1_{M_t > z}dt ≥lim inf_T↑∞1/T∫_ϵ T^T 1_{M_t > z}dt
    			≥lim inf_T↑∞1/T∫_ϵ T^T 1_{M_t^R_T > z}dt.

		Therefore,
		
    lim inf_T↑∞   1/T∫_0^T 1_{M_t > z}dt ≥lim inf_T↑∞(1/T∫_0^T 1_{M_t^R_T > z}dt - 1/T∫_0^ϵ T1_{M_t^R_T > z}dt )
       ≥lim inf_T↑∞1/T∫_0^T 1_{M_t^R_T > z}dt - ϵ≥ 1 - exp{ -C_* _∞ e^-√(2)z} - ϵ,

		where the last inequality follows from Lemma <ref>.
		Let ϵ↓ 0, then the inequality (<ref>) holds. This completes the proof.
	
	
	

§ PROOF OF LEMMA <REF>

	In this section, we prove Lemma <ref>, which states that for any x>0 and z∈ℝ, 
	
    lim_T↑∞1/T∫_0^T 1_{M_t^R_T≤ z}dt = exp{ -C_* _∞ e^-√(2)z},   ℙ_x

	Recall that = [d,D] is a compact set. Similar to <cit.>, this lemma follows from the following two lemmas.
	
	
		For ϵ>0, and R_T as in Lemma <ref>. Then for any s∈ [ϵ,1],
		
    lim_T↑∞ℙ_x[ M^R_T_T· s∈ | _R_T] = ∫_d( exp{-C_*_∞ e^-√(2)z}),  ℙ_x

	
	
	
		For ϵ>0, and R_T as in Lemma <ref>,
		
    lim_T↑∞1/T∫_ϵ T^T ( 1_{M^R_T_t ∈} - ℙ_x [ M^R_T_t∈ | _R_T] ) dt = 0,   ℙ_x

	
	
		First, we write
		
    ℙ_x[ M^R_T_T· s∈ | _R_T] = ℙ_x[ M^R_T_T· s≤ D  | _R_T] - ℙ_x[ M^R_T_T· s≤ d  | _R_T].

		We only need to show the almost surely convergence of the first term. Recall that the definitions (<ref>), (<ref>) and (<ref>), then
		
    ℙ_x [ M^R_T_T· s≤ D   | _R_T]    = ∏_u∈N_R_Tℙ_x [ X_u(R_T) + M_T· s - R_T(u) + m_T· s - R_T≤ D + m_T· s | _R_T]
    
    			= ∏_u∈N_R_T   (1 - ℙ_x [ M_T· s - R_T(u) > D - X_u(R_T) + √(2)R_T + o_T(1)  | _R_T])

		where given _R_T, {M_T· s - R_T(u), u∈N_R_T} are independent and have the same distribution as {M_t, ℙ_0}. Note that o_T(1)→ 0 as T→∞. After time R_T, there is no absorption barrier when M^R_T_T· s is considered. Therefore, we can use a similar argument as in <cit.>. By <cit.>, 
		
    lim_R_T→∞min_N_R_T (√(2)R_T - X_u(R_T)) = + ∞

		Let f(D, R_T) := D - X_u(R_T) + √(2)R_T + o_T(1). By <cit.>, we have
		
    ℙ_x [ M_T· s - R_T(u) > f(D, R_T)  | _R_T]
    			= C_* (1+o_r(1)) (1+o_T(1))f(D, R_T) e^-√(2) f(D, R_T) ,

		and by (<ref>), this probability tends to zero uniformly for u∈N_R_T as T→∞. 
		Hence, 
		
    ℙ_x [ M^R_T_T· s≤ D   | _R_T]
    			   = exp(∑_u∈N_R_Tlog(1 - ℙ_x [ M_T· s - R_T(u) > f(D, R_T)  | _R_T]) )
       = exp( - ∑_u∈N_R_T C_* (1+o_r(1)) (1+o_T(1))^2f(D, R_T) e^-√(2) f(D, R_T) )

		By <cit.> and <cit.>, we know that
		
    lim_T↑∞∑_u∈N_R_T D e^-√(2)(√(2)R_T-X_u(R_T))≤lim_T↑∞∑_u∈ N_R_T D e^-√(2)(√(2)R_T-X_u(R_T)) = 0

		and
		
    lim_T↑∞∑_u∈N_R_T (√(2)R_T-X_u(R_T)) e^-√(2)(√(2)R_T-X_u(R_T)) = _∞.

		Therefore, we get 
		
    lim_T↑∞ℙ_x [ M^R_T_T· s≤ D   | _R_T]    = exp( -  C_* (1+o_r(1)) lim_T↑∞∑_u∈N_R_T f(D, R_T) e^-√(2) f(D, R_T) )
       = exp( -  C_* (1+o_r(1)) e^-√(2)D_∞).

		Letting r↑∞ yields that 
		
    lim_T↑∞ℙ_x [ M^R_T_T· s≤ D   | _R_T] = exp( -  C_* e^-√(2)D_∞).

		This completes the proof of Lemma <ref>.
	
	
	
		Since after time _T, the particle behaves as a branching Brownian motion, most of the proof of <cit.> is valid for Lemma <ref>. Now we provide the details.
		
		Fix x>0. For γ>0, 0≤ s ≤ t, define
		
    F_γ,t(s) := x + s/tm_t - min{s^γ, (t-s)^γ}.

		Choose 0<α<1/2<β<1. We say that a particle u∈ N_t is localized in the time t-tube during the interval (r,t-r) if and only if
		
    F_β,t(s) ≤ X_u(s) ≤ F_α,t(s),  ∀ s∈ (r,t-r).

		Otherwise, we say that it is not localized. By <cit.>, for given =[d,D]. There exist r_0, δ>0 depending on α,β and  such that for r≥ r_0
		
    sup_t≥ 3rℙ_x [ ∃ u∈ N_t: X_u(t) - m_t ∈ (r,t-r) ] ≤exp{-r^δ}.

		Choose r_T = (20ln T)^1/δ. For any t∈ (R_T,T), define
		
    M_t,loc^R_T := max{X_u(t): u∈N_t^R_T, u  localized (r_T,t-r_T) } - m_t.

		Then, we have
		
    ℙ_x    ( {M_t^R_T∈}∖{M_t,loc^R_T∈})
       ≤ℙ_x [ ∃ u∈N_t^R_T: X_u(t) - m_t ∈ (r_T,t-r_T) ]
       ≤ℙ_x [ ∃ u∈ N_t: X_u(t) - m_t ∈ (r_T,t-r_T) ].

		Hence, (<ref>) implies that
		
    ℙ_x ( {M_t^R_T∈}∖{M_t,loc^R_T∈}) ≤1/T^20.

		Since {M_t^R_T≥ D }∩{M_t,loc^R_T∈}≠∅, the inequality ℙ_x (M_t^R_T∈) - ℙ_x (M_t,loc^R_T∈) ≥ 0
		is not true. The same issue exists in <cit.>. However, we have the following claim with a slight modification from their paper <cit.>.

		Claim A: There exist r_0, δ>0 depending on α,β and D such that for r≥ r_0
		
    sup_t≥ 3rℙ_x [ ∃ u∈ N_t: X_u(t) - m_t ≥ D  (r,t-r) ] ≤exp{-r^δ}.

		We prove this claim in Appendix <ref>.
		Therefore,
		
    ℙ_x    ( {M_t,loc^R_T∈}∖{M_t^R_T∈})
       ≤ℙ_x [ ∃ u∈ N_t: X_u(t) - m_t ≥ D  (r_T,t-r_T) ] ≤1/T^20.

		
		Let
		
    Rest_ϵ,(T) := 1/T∫_ϵ T^T ( 1_{M^R_T_t ∈} - ℙ_x [ M^R_T_t∈ | _R_T] ) dt,

		and
		
    Rest_ϵ,^loc(T) := 1/T∫_ϵ T^T ( 1_{M^R_T_t,loc∈} - ℙ_x [ M^R_T_t,loc∈ | _R_T] ) dt.

		Notice that
		
    _x | 1/T∫_ϵ T^T ( 1_{M^R_T_t ∈} - 1_{M^R_T_t,loc∈}) dt | ≤1/T∫_ϵ T^T _x | 1_{M^R_T_t ∈} - 1_{M^R_T_t,loc∈}| dt
       ≤1/T∫_ϵ T^T max{ℙ_x ( {M_t^R_T∈}∖{M_t,loc^R_T∈}),ℙ_x ( {M_t,loc^R_T∈}∖{M_t^R_T∈})  }dt
       ≤1/T^20.

		Using a similar argument to the proof of <cit.>, it holds that
		
    lim_T↑∞( Rest_ϵ,(T) - Rest_ϵ,^loc(T) ) = 0,   ℙ_x

		It is sufficient to prove that lim_T↑∞Rest_ϵ,^loc(T) = 0, ℙ_x
		
		Define
		
    X_t^(D) :=  1_{M^R_T_t,loc≤ D } - ℙ_x [ M^R_T_t,loc≤ D  | _R_T] C_T(t,t') := 𝔼_x[X_t^(D)· X_t'^(D)].

		By <cit.>, it suffices to verify the validity of <cit.> for this C_T(t,t'). Define
		
    ĉ_T(I,J) := ℙ_x [ M^R_T_I,loc≤ D, M^R_T_J,loc≤ D  | _R_T] - ℙ_x [ M^R_T_I,loc≤ D  | _R_T] ℙ_x [ M^R_T_J,loc≤ D  | _R_T].

		Here we use I and J to denote the two times t,t'. Then C_T(I,J) := 𝔼_x ĉ_T(I,J).
		
		Notice that R_T>r_T and ρ<√(2). A key observation is that for s>R_T and sufficiently large T
		
    F_β,t(s) - ρ s = x + (√(2)-ρ)s - 3/2√(2)s/tlog t - min{s^β, (t-s)^β} > 0.

		Hence, the absorption barrier does not affect the particle localized during (r_T,t-r_T) after time R_T. Therefore, conditioned on _R_T, the estimation of the joint distribution of M^R_T_I,loc and M^R_T_J,loc is the same as that in <cit.>. 
		
		Compared with M_loc(I) in <cit.>, our M^R_T_I,loc has an additional restriction before time R_T, namely, that the particles have not been absorbed. Therefore, conditioned on R_T, the sum of the terms is less than that in <cit.>. Furthermore, <cit.> is also valid for our ĉ_T(I,J) and <cit.> is also an upper bound for the ĉ_T(I,J). (Although <cit.> should be -a-a^2≤ln(1-a) ≤ -a (0≤ a≤ 1/2), <cit.> is still true because of the inequality -a≤-a+a^2/2.) The results of <cit.> yield Lemma <ref>.
	
	
	
	
	Now we turn to the proof of Lemma <ref> and the argument is similar to that in <cit.>.
	
		Consider a compact interval = [d,D] with -∞<d<D<∞. In order to prove (<ref>), it suffices to show that
		
    lim_T↑∞1/T∫_0^T 1_{M_t^R_T∈𝒟}dt = ∫_d( exp{-C_*_∞ e^-√(2)z}),  ℙ_x

		Note that
		
    1/T∫_0^T 1_{M_t^R_T∈𝒟}dt    = 1/T∫_0^ϵ T1_{M_t^R_T∈𝒟}dt + 1/T∫_ϵ T^T 1_{M_t^R_T∈𝒟}dt
       = 1/T∫_0^ϵ T1_{M_t^R_T∈𝒟}dt + 1/T∫_ϵ T^T  ℙ_x [ M^R_T_t∈ | _R_T]  dt
         + 1/T∫_ϵ T^T ( 1_{M^R_T_t ∈} - ℙ_x [ M^R_T_t∈ | _R_T] ) dt.

		By Lemma <ref> and the dominated convergence theorem, it holds that
		
    lim_ϵ↓ 0   lim_T↑∞1/T∫_ϵ T^T  ℙ_x [ M^R_T_t∈ | _R_T]  dt = lim_ϵ↓ 0lim_T↑∞∫_ϵ^1 ℙ_x [ M^R_T_T· s∈ | _R_T]  ds
       = lim_ϵ↓ 0∫_ϵ^1 lim_T↑∞ℙ_x [ M^R_T_T· s∈ | _R_T]  ds = ∫_d( exp{-C_*_∞ e^-√(2)z}),  ℙ_x

		Combining this with Lemma <ref> and lim_ϵ↓ 0lim_T↑∞1/T∫_0^ϵ T1_{M_t^R_T∈𝒟}dt = 0, we get (<ref>). This completes the proof.
	
	
	

§ PROOF OF LEMMA <REF>

	
		For any u∈ N_t, define 
		
    τ(u) := inf{s∈ [0,t]: X_u(s) ≤ρ s }.

		and inf∅ is defined by +∞. Then τ(u) represents the time when the particle u or its ancestor first hits the absorption barrier. Then
		
    N_t^[s,t] = {u∈ N_t: τ(u) ∈ [s,t] }.

		Similarly, for 0<s<s'<t, define
		
    N_t^[s,s'] = {u∈ N_t: τ(u) ∈ [s,s'] }.

		And let M_t^[s,s'] = max{X_u(t): u∈N_t^[s,s']} - m_t.
		
		Let p∈ (0,1). Notice that N_t^[R_T,t] = N_t^[R_T,pt]∪N_t^[pt,t]. Therefore, 
		
    M_t^[R_T,t]≤max{M_t^[R_T,pt],  M_t^[pt,t]},

		and moreover
		
    1_{M_t^[R_T,t] > z}≤1_{M_t^[R_T,pt] > z} + 1_{M_t^[pt,t] > z}.

		The proof of <cit.> gave an upper bound for both ℙ_x(M_t^[pt,t] > z) and ℙ_x(M_t^[R_T,pt] > z). In <cit.>, let A=-z, then I = ℙ_x(M_t^[pt,t] > z) and II = ℙ_x(M_t^[R_T,pt] > z).
		By the proof of <cit.>, we have
		
    ℙ_x(M_t^[pt,t] > z) ≤C/p^3/2∫_pt^∞ e^-( ρ/√(2) - 1)^2 r d r

		and
		
    ℙ_x(M_t^[R_T,pt] > z)    ≤ C  Π_x[τ_0^√(2)-ρ1_{τ_0^√(2)-ρ≥ R_T }]
       = C ∫_R_T^∞rx/√(2π r^3) e^-(x-(√(2)-ρ)r)^2/2rd r,

		where the positive constant C changes line by line and depends only on x,ρ,z. Here, {B_t, Π_x } is a standard Brownian motion starting from x, and τ_0^√(2)-ρ := inf{s≥ 0: B_s ≤ (√(2)-ρ)s }.
		
		For any δ>0, by Markov's inequality and Fubini's theorem, 
		
    ℙ_x( 1/T∫_ϵ T^T 1_{M_t^[pt,t] > z}dt ≥δ)    ≤1/δ1/T∫_ϵ T^T ℙ( M_t^[pt,t] > z) dt
       ≤C/δ T p^3/2∫_ϵ T^T ∫_pt^∞ e^-( ρ/√(2) - 1)^2 r d r dt
       ≤C/δ T p^5/2 e^-( ρ/√(2) - 1)^2 pϵ T,

		which is summable over T∈ℕ. Therefore, by Borel-Cantelli lemma,
		
    ℙ_x[ {1/T∫_ϵ T^T 1_{M_t^[pt,t] > z}dt ≥δ}] = 0.

		Hence,
		
    lim sup_T↑∞1/T∫_ϵ T^T 1_{M_t^[pt,t] > z}dt = 0,   ℙ_x

		Using the same argument as above,
		
    ℙ_x( 1/T∫_ϵ T^T 1_{M_t^[R_T,pt] > z}dt ≥δ)    ≤C/δ T∫_ϵ T^T  ∫_R_T^∞rx/√(2π r^3) e^-(x-(√(2)-ρ)r)^2/2rd r dt
       ≤C/δ∫_R_T^∞x/√(2π r) e^-(√(2)-ρ)^2/2r + (√(2)-ρ)xd r,

		which is summable in T∈ℕ when R_T/T^l→∞ as T→∞ for some l>0. Therefore,
		
    lim sup_T↑∞1/T∫_ϵ T^T 1_{M_t^[R_T,pt] > z}dt = 0,   ℙ_x

		By (<ref>), (<ref>) and (<ref>), we get (<ref>). This completes the proof.
	
	
	

§ PROOF OF CLAIM A

	
		We slightly modified the proof of <cit.>. By equations (5.5), (5.54), (5.62) and (5.63) in <cit.>, (<ref>) holds for any compact interval 𝒟. However, we need (<ref>) to hold when the compact interval  is replaced by [D,∞), and the proof of <cit.> remains valid with a slight modification.
		
		For any x∈, t>r>0 and interval 𝒜, define
		
    P(x,t,r,𝒜) = ℙ_x [ ∃ u∈ N_t: X_u(t) - m_t ∈𝒜 (r,t-r) ].

		Let 𝒟_b = [b,b+1] and θ∈(0,α). We write ⌊ s ⌋ for the largest integer less than or equal to s. First, we will show that there exist r_1,δ_1>0 depending on α,β and D such that for r≥ r_1, b∈ [⌊ D ⌋,r^θ], the following holds:
		
    sup_t≥ 3rP(x,t,r,_b)≤exp{-r^δ_1}.

		To prove (<ref>), we repeat the proof of <cit.> on pages 1668-1674 when the compact set 𝒟 is replaced by _b = [b,b+1]. 
		
		When _b = [b,b+1], we have D = b+1 and D = b in <cit.>. Notice that for b≥ D, <cit.> implies that
		
    e^t Π[B_t∈ m_t+_b] ≤κ t e^-√(2)D.

		The upper bound of this probability does not depend on b. In <cit.>, we need to choose r large enough so that F≤ 0 on [r,t-r] where F is given by <cit.>. For any δ,t>0, define f_t,δ(s) = min{s^δ,(t-s)^δ} for 0≤ s ≤ t. For b<r^θ<r^α and any s∈ [r,t-r],
		
    F(s) = bt-s/t - f_t,α(s) ≤ b - r^α < 0,

		which satisfies the condition. The remaining proof of <cit.> remains unchanged.
		
		In the proof of <cit.>, for sufficiently large r, we have
		
    diam(𝒟) = |D| + |D| ≤ 2b+1 ≤ 3r^θ.

		Notice that 0<θ<α<1/2<β<1 and let 0<a<1 such that 2aβ-1>0. We can find r = r(α,β,θ,D,a) such that for r≥r the following holds:
		
    3r^θ - f_t,α(s) ≤ 0   3r^θ - f_t,β(s) ≤ -f_t,aβ(s)  r≤ s≤ t-r.

		Therefore, the proof of <cit.> is also valid and (<ref>) holds for b∈ [D,r^θ].
		
		By <cit.>, for sufficiently large r and t≥ 3r, we have
		
    P(x,t,r,[r^θ,∞))
    			≤ℙ_x [ M_t ≥ m_t + r^θ]
    			≤ C r^θ e^-√(2)r^θ+1,

		for some constant C>0. Therefore, combining (<ref>) with (<ref>), we get that
		
    sup_t≥ 3rP(x,t,r,[D,∞)) 
    			≤    ∑_b = ⌊ D ⌋^⌊ r^θ⌋P(x,t,r,𝒟_b) + P(x,t,r,[r^θ,∞))
    ≤     (|D|+r^θ) e^-r^δ_1 + C r^θ e^-√(2)r^θ+1.

		Choose δ < min{δ_1,θ}, there exists a sufficiently large r_0 such that (<ref>) holds for r≥ r_0. This completes the proof.
	
	
	

	Acknowledgment:
	We thank Professor Xinxin Chen for many constructive suggestions, and, in particular, for providing the proof strategy of Claim A.
	
	
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