Limiting behaviors for Brownian motion

reflected on Brownian motion

Xia Chen ^{1}

Department of Mathematics University of Tennessee Knoxville, TN 37996 xchen@math.utk.edu

Wenbo V. Li ^{2} Department of Mathematical Sciences University of Delaware Newark, DE 19716 wli@math.udel.edu

Abstract:

Suppose that g(t) and W_{t} are independent Brownian motions starting from g(0) = W_{0} = 0. Consider the Brownian motion Y_{t} reflected on g(t), obtained from W_{t} by the means of the Skorohod lemma. The upper and lower limiting behaviors of Y_{t} are presented. The upper tail estimate on exit time is computed via principal eigenvalue.

Abbreviated title: Brownian Motion(BM) Reflected on BM Keywords: Brownian Motion, Reflected Brownian Motion Limiting behaviors, deviation estimates

1Supported in part by NSF Grant DMS-0102238.

2Supported in part by NSF Grant DMS-0204513.

1 Introduction

Brownian motion reflected on Brownian motion appeared in recent papers by Soucaliuc, Toth and Werner (2000), Burdzy, Chen and Sylvester (2000) and Burdzy and Nualart (2002) in their study of reflected Bronian motion and corresponding heat equation in domains with space-time boundaries. In this paper, we study the upper and lower limiting behaviors of Brownian Motion reflected on Brownian Motion. Our starting point is the following beautiful result of Burdzy and Nualart (2002).

Suppose that g(t) and W_{t} are independent real Brownian motions starting from g(0) = W_{0} = 0. Consider the Brownian motion Y_{t} reflected on g(t), obtained from W_{t} by the mean of the Skorohod lemma. Here g should be thought of as a “fixed Brownian path.” Then

−Y_{t} = (W_{t} + C_{t})/\sqrt{2}, t ≥ 0

where C_{t} is a 3-dimensional Bessel process independent of W_{t} and starting from 0. A process with the same distribution as \{(W_{t} + C_{t})/\sqrt{2}, t ≥ 0\} is called a BMB-process in Burdzy and Nualart (2002) and many useful properties are given.

The main goal of this paper is to present some “global” results for BMB-process and a natural generalization. Namely

Theorem 1.1 Let X(t), X(0) = 0, be a d-dimensional (d ≥ 1) Bessel process independent of W. Then

lim sup_{t→∞} \frac{1}{\sqrt{t log log t}}(W (t) + X(t)) = 2 a.s.  (1.1)
lim inf_{t→∞} \frac{1}{\sqrt{t log log t}}(W (t) + X(t)) = −\sqrt{2} a.s.  (1.2)

and for d ≠ 2

lim inf_{t→∞} \sqrt{\frac{log log t}{t}} sup_{s≤t} |W (s) + X(s)| = \frac{π}{2} a.s.  (1.3)

In particular, we have for Y_{t}, Brownian motion reflected on Brownian motion,

lim inf_{t→∞} \frac{1}{\sqrt{t log log t}}Y (t) = −\sqrt{2} a.s.  (1.4)
lim sup_{t→∞} \frac{1}{\sqrt{t log log t}}Y (t) = 1 a.s.  (1.5)

and

lim inf_{t→∞} \sqrt{\frac{log log t}{t}} sup_{s≤t} |Y (s)| = \frac{π}{\sqrt{8}} a.s.  (1.6)

It is interesting to see that the behaviors in (1.4) and (1.6) for Y is exactly the same as those for W. Furthermore, due to the time reversibility, their behaviors near time zero are also the same.

Next we outline some of the tools we used. As it can be seen in the next section, the main part of this work is to estimate the upper tail of the exit time from a suitable domain. The approach we follow is to reduce our problem to the principal eigenvalue of the Markov process (W (t), X(t)) killed upon the exit from the domain. This approach has been effectively utilized by Donsker and Varadhan (1975-1983) in their fundamental work on Large deviations for Markov processes and its applications, and by Pinsky (1985, 1995) and Rémillard (1994) in various problems involving estimates of exponential type. In Donsker and Varadhan (1975), the principal eigenvalue is represented in terms of the I-function in large deviation theory. In Berestycki, Nirenberg and Varadhan (1994), the existence of the principal eigenvalue is discussed in general setting. Our results require exact evaluation of the principal eigenvalue, which is beyond the general theory. Fortunately, the generators we deal with are self-adjoint, in which case the principal eigenvalue can be written as a computable quadratic variation. Some techniques we use here are partially inspired by the work of Rémillard (1994). To be more precise, we state our main probability estimate.

Theorem 1.2 Let W and X be given as in Theorem 1.1. Then for d ≠ 2,

lim_{t→∞} \frac{1}{t} log ℙ\{ sup_{s≤t} |W (s) − X(s)| ≤ 1\} = −\frac{π^{2}}{4} .  (1.7)

Note that sup_{s≤t} |W (s) − X(s)| and sup_{s≤t} |W (s) + X(s)| have the same distribution and we use minus sign for convenience in our proofs. Furthermore, as it can be seen in the next section, we have the variation formula in the case d = 2, but could not evaluate it explicitly. We strongly believe that both (1.3) and (1.7) hold for d = 2.

There are two ways to view the estimate in (1.7). The first can be stated as

lim_{t→∞} \frac{1}{t} log ℙ\{τ_{Γ} ≥ t\} = −\frac{π^{2}}{4}  (1.8)

where τ_{Γ} is the first exit time of (d + 1)-dimensional Brownian motion from the unbounded domain

Γ = {(x, y) ∈ ℝ^{d} × ℝ : −1 < y − |x|_{2} < 1}.  (1.9)

In this setup, the generator is the half Laplacian on ℝ^{d+1} and the domain is the part between two parallel right corns. Our approaches detailed in the next section work in this setting as d = 1. Other related interesting problems and techniques on the first exit times of higher dimensional Brownian motion from unbounded domains can be found in Bañuelos, DeBlassie and Smits (2001), Li (2002).

The second way can be stated as

lim_{t→∞} \frac{1}{t} log ℙ\{τ_{G} ≥ t\} = −\frac{π^{2}}{4}  (1.10)

where τ_{G} is the first exit time of the diffusion process (X(t), W (t)) from the unbounded domain

G = {(x, y) ∈ ℝ^{+} × ℝ : |y − x| < 1}.  (1.11)

As d ≥ 2, the generator is

Lf(x, y) = \frac{1}{2}△f(x, y) + \frac{d − 1}{2x} \frac{∂}{∂x}f(x, y), (x, y) ∈ ℝ^{+} × ℝ.  (1.12)

These are two ways we handle the problem in the next section.

Next we make some simple observations. We assume throughout this paper that W (t), W_{j}(t), j = 1, 2, ⋯ d are independent standard Brownian motions and thus we can use the representation

X(t) = (∑_{j=1}^{d} W_{j}(t)^{2})^{1/2} = |B_{d}(t)|_{2}

where B_{d}(t) = (W_{1}(t), ⋯, W_{d}(t)) ∈ ℝ^{d} is the standard d-dimensional Brownian motion. It is well known and follows from rotation invariant that as process,

\{(W (t), W_{1}(t)) : t ≥ 0\} = \{(\frac{W (t) + W_{1}(t)}{\sqrt{2}}, \frac{W (t) − W_{1}(t)}{\sqrt{2}}) : t ≥ 0\}

in law and thus as process,

\{W (t) + |W_{1}(t)| : t ≥ 0\} = \{\sqrt{2} max(W (t), W_{1}(t)) : t ≥ 0\}

in law by using 2 max(a, b) = a + b + |a − b|. This allows us to obtain the following sharp lower bound in the case d = 1:

ℙ ( sup_{0≤s≤t} |W (s) + |W_{1}(s)|| ≤ 1)

= ℙ (\sqrt{2} sup_{0≤s≤t} |max(W (s), W_{1}(s))| ≤ 1)
≥ ℙ (−1 ≤ \sqrt{2}W (s) ≤ 1, \sqrt{2}W_{1}(s) ≤ 1 ∀0 ≤ s ≤ t)
= ℙ (\sqrt{2} sup_{0≤s≤t} |W (s)| ≤ 1) · ℙ (\sqrt{2} sup_{0≤s≤t} W_{1}(s) ≤ 1) .

For d ≥ 1, an easy upper and lower bounds for the probability estimate in Theorem 1.2 can be found by using the well known estimates

lim_{t→∞} \frac{1}{t} log ℙ ( sup_{0≤s≤t} X(s) ≤ 1) = −j_{ν}^{2}/2  (1.13)

where j_{ν} is the smallest positive zero of the Bessel function J_{ν}, ν = (d − 2)/2, and j_{−1/2} = π/2. The above estimate can be obtained either from the exact distribution result due to Ciesielski and Taylor (1962) or from a general principle eigenvalue approach detailed in Donsker and Varadhan (1976). Now by using the simple fact that

ℙ ( sup_{0≤s≤t} |W (s) + X(s)| ≤ 1) ≤ ℙ ( sup_{0≤s≤t} |W (s)| ≤ 1)

via Anderson’s inequality, and

ℙ ( sup_{0≤s≤t} |W (s) + X(s)| ≤ 1) ≥ ℙ ( sup_{0≤s≤t} |W (s)| ≤ λ) · ℙ ( sup_{0≤s≤t} |X(s)| ≤ 1 − λ)

for λ = j_{−1/2}^{2/3}/(j_{−1/2}^{2/3} + j_{ν}^{2/3}), we have

−(j_{−1/2}^{2/3} + j_{ν}^{2/3})^{3}/2 ≤ lim_{t→∞} \frac{1}{t} log ℙ ( sup_{0≤s≤t} |W (s) + X(s)| ≤ 1) ≤ −j_{−1/2}^{2}/2 = −π^{2}/8.

In particular, combining the above estimate with Theroem 1.2, we see that for −1 < ν < −1/2, j_{ν} ≥ (2^{1/3} − 1)^{3/2}J_{−1/2} = (2^{1/3} − 1)^{3/2}π/2.

Finally, we mention the following heuristic argument which is suggestive but seems impossible to produce a rigorous upper or lower bound. We observe that for fixed s ≥ 0,

W (s) + X(s) = W (s) + sup_{|x|_{2}=1} ∑_{j=1}^{d} x_{j}W_{j}(s) = sup_{|x|_{2}=1} (W (s) + ∑_{j=1}^{d} x_{j}W_{j}(s))

and for fixed x ∈ ℝ^{d} with |x|_{2} = 1,

W (s) + ∑_{j=1}^{d} x_{j}W_{j}(s) = \sqrt{1 + |x|_{2}^{2}}Ŵ(s) = \sqrt{2}Ŵ(s)

in distribution where Ŵ is a standard Brownian motion. Jointly, our Theorem 1.2 implies

lim_{t→∞} \frac{1}{t} log ℙ ( sup_{0≤s≤t} |W (s) + X(s)| ≤ 1) = lim_{t→∞} \frac{1}{t} log ℙ ( sup_{0≤s≤t} |\sqrt{2}W (s)| ≤ 1) = −\frac{π^{2}}{4}

where the last equality follows from (1.13).

The rest of the paper is organized as follows. In Section 2, we present the proof of Theorem 1.2 viewed as the large deviations of the first exit time of the diffusion process (X(t), W (t)) from the unbounded domain G in the case d ≥ 2, and the diffusion process (W_{1}(t), W (t)) from the unbounded domain Γ in the case d = 1. They are necessary for the proofs of Theorem 1.1 and important in their own. In Section 3, we give the proof of Theorem 1.1 which also requires some large deviation estimates.

2 The first exit time and principal eigenvalue

Recall that G is given in (1.11). We now prove that

lim_{t→∞} \frac{1}{t} log ℙ\{ max_{s≤t} |W (s) − X(s)| < 1\} = −\frac{1}{2} inf_{|f|_{π}=1, f∈C_{0}^{∞}(G)} ∫_{G} |∇f(x, y)|^{2}x^{d−1}dxdy (2.1)

where, by the notations we shall introduce below, C_{0}^{∞}(G) is the class of continuous functions f(x, y) on Ḡ which are infinitely differentiable and f(x, y) = 0 if y − x = ±1.

We first deal with the case d ≥ 2. Consider the diffusion process (X(t), W (t)) with state space R^{+} × R and generator given in (1.12). It is easy to see that as a linear operator on the Hilbert space L^{2}(R^{+} × R, π), L is self adjoint, where π is the measure on R^{+} × R given by

π(dx, dy) = x^{d−1}dxdy

and

< Lf, f >_{π}= −\frac{1}{2} ∫_{R^{+}×R} |∇f(x, y)|^{2}x^{d−1}dxdy

if f is smooth enough. Write

|f|_{π} = ( ∫_{R^{+}×R} f^{2}(x, y)x^{d−1}dxdy)^{1/2}.

For an open domain D (with respect to the relative Euclidian topology on R^{+} × R) in the space R^{+} × R we define

τ_{D} = inf{t ≥ 0; (X(t), W (t)) ∉ D}.

Define the semigroup T_{t} (t ≥ 0) by

T_{t}f(x, y) = 𝔼_{(x,y)}(f(X(t), W (t))I_{{τ_{D}≥t}}).

Note that ℙ_{(x,y)}{τ_{D} = 0} = 0 for each (x, y) ∈ D. We have T_{0} = id. Let C_{0}^{∞}(D) be the class of of functions f continuous on D̄, infinitely differentiable in D and f(∂D) = 0. Notice that R^{+} ×R is a whole space in our setting. In other words a open set D in R^{+} ×R may contain the vertical line segment {(x, y); x = 0 and − 1 < y < 1}, in which case the line segment should not be viewed as a part of ∂D. By a trivial extension, all functions f in C_{0}^{∞}(D) can be viewed as the functions defined on R^{+} × R with f = 0 outside D. For each f ∈ C_{0}^{∞}(D),

f(X_{τ_{D}∧t}, W_{τ_{D}∧t}) − ∫_{0}^{τ_{D}∧t} Lf(X(s), W (s))ds

is a martingale. Hence, using the fact that f(∂D) = 0,

\frac{1}{t}(T_{t}f(x, y) − f(x, y)) = \frac{1}{t} ∫_{0}^{t} 𝔼_{(x,y)}(Lf(X(s), W (s))I_{\{τ_{D}≥s\}})ds ⟶ Lf(x, y)

as t → 0 and (x, y) ∈ D. Thus the generator L_{D} of T_{t} coincides with L on C_{0}^{∞}(D). Since C_{0}^{∞}(D) is a core we have T_{t} = e^{tL_{D}} on C_{0}^{∞}(D).

To prove the lower bound we need only to show

lim inf_{t→∞} \frac{1}{t} log ℙ\{ sup_{s≤t} |W (s) − X(s)| < 1\} ≥ −\frac{1}{2} ∫_{G} |∇f(x, y)|^{2}x^{d−1}dxdy  (2.2)

for every f ∈ C_{0}^{∞}(G) with |f|_{π} = 1 and K = \overline{support(f)} ⊂ G being compact.

We fix an open domain D in R^{+} × R with compact closure such that K ⊂ D ⊂ D̄ ⊂ G. and (0, 0) ∈ D. Let p_{D}(t; (x, y)) be the density (with respect to the Lebesgue measure) of the measure

μ(A) = P {τ_{D} ≥ t; (X(t), W (t)) ∈ A}.

Then combining results of Azencott (1984) and Leandre (1987, e.g. Theorem 11.3), we have inf_{(x,y)∈K} p_{D}(t_{o}; (x, y)) > 0 for some t_{o} > 0, see also Stroock and Varadhan (1979).

By Markov property, for t ≥ t_{o}, and every f as above

P {τ_{D} ≥ t} = 𝔼_{(0,0)}[I_{{τ_{D}≥t_{o}}}P_{(X(t_{o}),W (t_{o}))}{τ_{D} ≥ t − t_{o}}]

= ∫ ℙ_{(x,y)}{τ_{D} ≥ t − t_{o}}p_{D}(t_{o}, (x, y))dxdy
≥ |f|_{∞}^{−1} sup{x^{d−1}; (x, y) ∈ K}^{−1} inf_{(x,y)∈K} p_{D}(t_{o}, (x, y)) ∫ f(x, y)P_{(x,y)}{τ_{D} ≥ t − t_{o}}x^{d−1}dxdy
≥ |f|_{∞}^{−2} sup{x^{d−1}; (x, y) ∈ K}^{−1} inf_{(x,y)∈K} p_{D}(t_{o}, (x, y)) ∫ fT_{t−t_{o}}fx^{d−1}dxdy
= c ∫ fe^{(t−t_{o})L_{D}}fx^{d−1}dxdy = c < f, e^{(t−t_{o})L_{D}}f >_{π}

where the third step follows from

ℙ_{(x,y)}{τ_{D} ≥ t − t_{o}} = 𝔼_{(x,y)}(I_{{τ_{D}≥t−t_{o}}})

≥ |f|_{∞}^{−1}𝔼_{(x,y)}(f(X_{t}, W_{t})I_{{τ_{D}≥t−t_{o}}}) = |f|_{∞}^{−1}T_{t−t_{o}}f(x, y)

and the constant c does not depend on t.

We now consider the spectral structure of the self adjoint operator L_{D}. By Jensen’s inequality

< f, e^{(t−t_{o})L_{D}}f >_{π}= ∫_{−∞}^{0} e^{(t−t_{o})λ}E_{f}(dλ) ≥ exp {(t − t_{o}) ∫_{−∞}^{0} λE_{f}(dλ)}
= exp \{(t − t_{o}) < L_{D}f, f >_{π} \} = exp \{ − (t − t_{o})\frac{1}{2} ∫_{G} |∇f(x, y)|^{2}x^{d−1}dxdy\}

Here we have used the fact that the spectral measure E_{f}(dλ) is a probability measure due to

∫_{−∞}^{0} E_{f}(dλ) = |f|_{π}^{2} = 1.

Hence (2.1) holds.

On the other hand, using the fact that ℙ{ max_{s≤t} X(s) ≥ t^{2}} ≤ e^{−δt^{2}} for any δ > 0 and t large, we have

lim sup_{t→∞} \frac{1}{t} log ℙ\{ max_{s≤t} |W (s) − X(s)| < 1\} ≤ lim sup_{t→∞} \frac{1}{t} log ℙ\{τ_{D_{t}} ≥ t\}

where D_{t} = {(x, y) ∈ R^{+} × R : 0 ≤ x < t^{2} and |y − x| < 1}.

Next we observe

ℙ{τ_{D_{t}} ≥ t} = 𝔼_{(0,0)}[I_{{τ_{D_{t}}≥1}}ℙ_{(X(1),W (1))}{τ_{D_{t}} ≥ t − 1}]
≤ 𝔼_{(0,0)}[I_{{((X(1),W (1))∈D_{t}}}ℙ_{(X(1),W (1))}{τ_{D_{t}} ≥ t − 1}]
= C ∫_{D_{t}} ℙ_{(x,y)}\{τ_{D_{t}} ≥ t − 1\}x^{d−1} exp \{ − \frac{x^{2} + y^{2}}{2} \}dxdy
≤ C ∫_{D_{t}} ℙ_{(x,y)}{τ_{D_{t}} ≥ t − 1}x^{d−1}dxdy.

Given ϵ > 0, let D_{t}^{ϵ} be the ϵ-neighborhood of D_{t} in R^{+} × R and choose a f_{0} ∈ C_{0}^{∞}(D_{t}^{ϵ}) such that f_{0} ≥ 0 is bounded and f_{0} ≥ 1 in D_{t}. Then

∫_{D_{t}} ℙ_{(x,y)}{τ_{D_{t}} ≥ t − 1}x^{d−1}dxdy
≤ ∫_{D_{t}} f_{0}(x, y)𝔼_{(x,y)}[f_{0}(X(t − 1), W (t − 1))I_{{τ_{D_{t}}≥t−1}}]x^{d−1}dxdy
≤ ∫_{D_{t}^{ϵ}} f_{0}(x, y)𝔼_{(x,y)}[f_{0}(X(t − 1), W (t − 1))I_{{τ_{D^{ϵ}_{t}}≥t−1}}]x^{d−1}dxdy

= < f_{0}, e^{(t−1)L_{D^{ϵ}_{t}}}f_{0} >_{π}≤ |f_{0}|_{π}^{2} exp {(t − 1) sup_{|f|_{π}=1, f∈C_{0}^{∞}(D_{t}^{ϵ})} < f, L_{D_{t}^{ϵ}}f >_{π} }
≤ C_{1}t^{2} exp \{ − (t − 1)\frac{1}{2} inf_{|f|_{π}=1, f∈C_{0}^{∞}(D_{t}^{ϵ})} ∫_{D_{t}^{ϵ}} |∇f(x, y)|^{2}x^{d−1}dxdy\}
≤ C_{1}t^{2} exp \{ − (t − 1)\frac{1}{2} inf_{|f|_{π}=1, f∈C_{0}^{∞}(G^{ϵ})} ∫_{G^{ϵ}} |∇f(x, y)|^{2}x^{d−1}dxdy\}.

Hence we have

lim sup_{t→∞} \frac{1}{t} log ℙ\{ max_{s≤t} |W (s) − X(s)| < 1\}

≤ −\frac{1}{2} inf_{|f|_{π}=1, f∈C_{0}^{∞}(G^{ϵ})} ∫_{G^{ϵ}} |∇f(x, y)|^{2}x^{d−1}dxdy.

Letting ϵ ⟶ 0 we have

lim sup_{t→∞} \frac{1}{t} log ℙ\{ max_{s≤t} |W (s) − X(s)| < 1\} ≤ −\frac{1}{2} inf_{|f|_{π}=1, f∈C_{0}^{∞}(G)} ∫_{G} |∇f(x, y)|^{2}x^{d−1}dxdy. (2.3)

Therefore, (2.1) follows from (2.2) and (2.3).

The case d = 1 is slightly different since the operator given by (1.12) is no longer the generator of (X(t), W (t)) (p.416, Revuz-Yor (1991)). In this case one can write X(t) = |W_{1}(t)| where W_{1}(t) is another 1-dimensional Brownian motion independent of W (t). Notice that the two dimensional Brownian motion (W_{1}(t), W (t)) has (1/2)Δ as its generator. By the argument we proceed in the case d ≥ 2,

lim_{t→∞} \frac{1}{t} log ℙ\{ max_{s≤t} |W (s) − X(s)| < 1\} = −\frac{1}{2} inf_{|f|_{π}=1, f∈C_{0}^{∞}(Γ)} ∫_{Γ} |∇f(x, y)|^{2}dxdy

where Γ is given by (1.9) (with d = 1, of course). A simple argument shows that the infimum can be taken only for those f ∈ C_{0}^{∞}(G_{1}) satisfying f(−x, y) = f(x, y) for all (x, y) ∈ Γ. Therefore,

inf_{|f|_{π}=1, f∈C_{0}^{∞}(G_{1})} ∫_{Γ} |∇f(x, y)|^{2}dxdy = inf_{|f|_{π}=1, f∈C_{0}^{∞}(G)} ∫_{G} |∇f(x, y)|^{2}dxdy

Hence (2.1) remains valid in the case d = 1.

It remains to show that

inf_{|f|_{π}=1, f∈C_{0}^{∞}(G)} ∫_{G} |∇f(x, y)|^{2}x^{d−1}dxdy = \frac{π^{2}}{2} .  (2.4)

Given a ϵ > 0, we take

f_{ϵ}(x, y) = C_{ϵ}e^{−ϵx} cos \frac{π}{2} (y − x)

where the constant C_{ϵ} > 0 is determined by |f_{ϵ}|_{π} = 1. One can easily check that

∫_{G} |∇f_{ϵ}(x, y)|^{2}x^{d−1}dxdy = \frac{π^{2}}{2} + ϵ^{2}

by using |f_{ϵ}|_{π}^{2} = 1. Since ϵ is arbitrary, we have proved

inf_{|f|_{π}=1, f∈C_{0}^{∞}(G)} ∫_{G} |∇f(x, y)|^{2}x^{d−1}dxdy ≤ \frac{π^{2}}{2} .  (2.5)

On the other hand, by the substitution g(x, y) = x^{(d−1)/2}f(x, y) we have that for each f ∈ C_{0}^{∞}(G),

|∇f(x, y)|^{2} = x^{−(d−1)}|∇g(x, y)|^{2} + \frac{(d − 1)^{2}}{4} x^{−(d+1)}g^{2}(x, y) − (d − 1)x^{−d}g(x, y)\frac{∂}{∂x}g(x, y).

For d ≥ 3, we have by definition of g that lim_{x→0^{+}} x^{−1}g^{2}(x, y) = 0. Note that

∫_{G} x^{−1}g(x, y)\frac{∂}{∂x}g(x, y)dxdy

= ∫_{−1}^{1} dy ∫_{0}^{y+1} x^{−1}g(x, y)\frac{∂}{∂x}g(x, y)dx + ∫_{1}^{∞} dy ∫_{y−1}^{y+1} x^{−1}g(x, y)\frac{∂}{∂x}g(x, y)dx.

By using integration by parts we see that

∫_{0}^{y+1} x^{−1}g(x, y)\frac{∂}{∂x}g(x, y)dx

= \frac{1}{2}x^{−1}g^{2}(x, y)|_{x=0}^{x=y+1} + \frac{1}{2} ∫_{0}^{y+1} x^{−2}g^{2}(x, y)dx
= \frac{1}{2} ∫_{0}^{y+1} x^{−2}g^{2}(x, y)dx.

Similarly,

∫_{y−1}^{y+1} x^{−1}g(x, y)\frac{∂}{∂x}g(x, y)dx = \frac{1}{2} ∫_{y−1}^{y+1} x^{−2}g^{2}(x, y)dx.

Therefore,

∫_{G} x^{−1}g(x, y)\frac{∂}{∂x}g(x, y)dxdy = \frac{1}{2} ∫_{G} x^{−2}g^{2}(x, y)dxdy.

Combining above calculation together, we have

∫_{G} |∇f(x, y)|^{2}x^{d−1}dxdy
= ∫_{G} |∇g(x, y)|^{2}dxdy + \frac{(d − 1)^{2}}{4} ∫_{G} x^{−2}g^{2}(x, y)dxdy − (d − 1) ∫_{G} x^{−1}g(x, y)\frac{∂}{∂x}g(x, y)dxdy
= ∫_{G} |∇g(x, y)|^{2}dxdy + (\frac{(d − 1)^{2}}{4} − \frac{d − 1}{2}) ∫_{G} x^{−2}g^{2}(x, y)dxdy.

And thus for d ≥ 3,

∫_{G} |∇f(x, y)|^{2}x^{d−1}dxdy ≥ ∫_{G} |∇g(x, y)|^{2}dxdy.

Hence we obtain

inf_{|f|_{π}=1, f∈C_{0}^{∞}(G)} ∫_{G} |∇f(x, y)|^{2}x^{d−1}dxdy

≥ inf { ∫_{G} |∇g(x, y)|^{2}dxdy; g ∈ C_{0}^{∞}(G) and ∫_{G} g^{2}(x, y)dxdy = 1}.  (2.6)

In the case d = 1, (2.6) is automatically holds with equality.

Next we consider the problem over a larger domain with some symmetry. Let G̃ = {(x, y) ∈ R^{2}; |y − x| < 1}. By symmetry between x and y,

inf { ∫_{G} |∇g(x, y)|^{2}dxdy; g ∈ C_{0}^{∞}(G) and ∫_{G} g^{2}(x, y)dxdy = 1}

= inf { ∫G̃ |∇g(x, y)|^{2}dxdy; g ∈ C_{0}^{∞}(G̃) and ∫G̃ g^{2}(x, y)dxdy = 1}
≥ 2 inf \{ ∫G̃ (\frac{∂}{∂x}g(x, y))^{2}dxdy; g ∈ C_{0}^{∞}(G̃) and ∫G̃ g^{2}(x, y)dxdy = 1\}
= 2 inf \{ ∫_{−∞}^{∞} dy ∫_{y−1}^{y+1} (\frac{∂}{∂x}g(x, y))^{2}dx; g ∈ C_{0}^{∞}(G̃) and ∫G̃ g^{2}(x, y)dxdy = 1\} (2.7)

where the first equality in (2.7) needs to be justified. Let

G′ = {(x, y) ∈ R^{2}; (−x, −y) ∈ G}.

Then for each g ∈ C_{0}^{∞}(G̃)

∫G̃ |∇g(x, y)|^{2}dxdy
= ∫_{G} |∇g(x, y)|^{2}dxdy + ∫_{G′} |∇g(x, y)|^{2}dxdy
= ∫_{G} |∇g(x, y)|^{2}dxdy + ∫_{G} |∇ḡ(x, y)|^{2}dxdy

where ḡ(x, y) = g(−x, −y). Clearly, ḡ ∈ C_{0}^{∞}(G). Let

λ_{G} = inf { ∫_{G} |∇g(x, y)|^{2}dxdy; g ∈ C_{0}^{∞}(G) and ∫_{G} g^{2}(x, y)dxdy = 1}.

Then

∫_{G} |∇g(x, y)|^{2}dxdy ≥ λ_{G} ∫_{G} g^{2}(x, y)dxdy

for every g ∈ C_{0}^{∞}(G). Therefore,

∫G̃ |∇g(x, y)|^{2}dxdy ≥ λ_{G}{ ∫_{G} g^{2}(x, y)dxdy + ∫_{G} ḡ^{2}(x, y)dxdy}

= λ_{G}{ ∫_{G} g^{2}(x, y)dxdy + ∫_{G′} g^{2}(x, y)dxdy} = λ_{G}

if ∫G̃ g^{2}(x, y)dxdy = 1.

On the other hand, if g ∈ C_{0}^{∞}(G) and ∫_{G} g^{2}(x, y)dxdy = 1 satisfies

∫_{G} |∇g(x, y)|^{2}dxdy < ϵ + λ_{G}

we define g̃(x, y) ∈ C_{0}^{∞}(G̃) by g̃(x, y) = g(x, y)/\sqrt{2} if (x, y) ∈ G and g̃(x, y) = g(−x, −y)/\sqrt{2} if (x, y) ∈ G′. Then

∫G̃ g^{2}(x, y)dxdy = 1

and

∫G̃ |∇g̃(x, y)|^{2}dxdy = ∫_{G} |∇g(x, y)|^{2}dxdy < ϵ + λ_{G}.

So

inf { ∫G̃ |∇g(x, y)|^{2}dxdy; g ∈ C_{0}^{∞}(G̃) and ∫G̃ g^{2}(x, y)dxdy = 1} ≤ λ_{G}

and we finish the justification of (2.7).

Back to the estimate of the lower bound for (2.7). We start with a well known variational identity (see, c.f., Strassen (1964))

inf \{ ∫_{−1}^{1} |h′(x)|^{2}dx; h ∈ C_{0}^{∞}(−1, 1) and ∫_{−1}^{1} |h(x)|^{2}dx = 1\} = \frac{π^{2}}{4} .

Under a simple substitution, this gives that for any y ∈ R and any h ∈ C_{0}^{∞}(y − 1, y + 1),

∫_{y−1}^{y+1} |h′(x)|^{2}dx ≥ \frac{π^{2}}{4} ∫_{y−1}^{y+1} |h(x)|^{2}dx.

Hence we have that for any g ∈ C_{0}^{∞}(G̃) with ∫G̃ g^{2}(x, y)dxdy = 1,

∫_{−∞}^{∞} dy ∫_{y−1}^{y+1} (\frac{∂}{∂x}g(x, y))^{2}dx ≥ \frac{π^{2}}{4} ∫_{−∞}^{∞} dy ∫_{y−1}^{y+1} g^{2}(x, y)dx = \frac{π^{2}}{4} .

In view of (2.7),

inf_{|f|_{π}=1, f∈C_{0}^{∞}(G)} ∫_{G} |∇f(x, y)|^{2}x^{d−1}dxdy ≥ \frac{π^{2}}{2} .  (2.8)

Finally, (2.4) follows from (2.5) and (2.8) and we finish the proof of Theorem 1.2.

We end this section with the following comment: From the proof of (2.8) one can see that except the case d = 1, the infimum can not be reached. This may suggest that the following eigenvalue problem

\frac{1}{2}△f(x, y) + \frac{d − 1}{2x} \frac{∂}{∂x}f(x, y) = −\frac{π^{2}}{2} f(x, y) (x, y) ∈ G and f(∂G) = 0

does not have a solution which is reasonably smooth in G (Recall that G contains the line segment x = 0 with −1 < y < 1) unless d = 1, in which case the function

f(x, y) = cos \frac{π}{2} (y − x)

solves the equation. In the case d = 5, one can see that the function

f(x, y) = x^{−1} cos \frac{π}{2} (y − x)

solves above eigenvalue problem but f fails to be continuous at x = 0 with −1 < y < 1.

3 Limiting behaviors

Let B_{d}(t) be a d-dimensional Brownian motion independent of W (t). Since X =^{d} |B_{d}|_{2}, we replace X(t) by |B_{d}(t)|_{2} in this section.

We first prove (1.3). By Theorem 1.2 and Borel-Cantelli Lemma, one can easily show that

lim inf_{t→∞} \sqrt{\frac{log log t}{t}} max_{s≤t} |W (s) + |B_{d}(s)|_{2}| ≥ \frac{π}{2} a.s.

On the other hand, let t_{k} = k^{k} and let λ > π/2 be fixed. By Theorem 1.2,

ℙ\{ max_{t_{k}≤s≤t_{k+1}} |(W (s) − W (t_{k})) + |B_{d}(s) − B_{d}(t_{k})|_{2}| ≤ λ\sqrt{\frac{t_{k+1}}{log log t_{k+1}}} \}

= ℙ\{ max_{s≤t_{k+1}−t_{k}} |W (s) + |B_{d}(s)|_{2}| ≤ λ\sqrt{\frac{t_{k+1}}{log log t_{k+1}}} \}
≥ ℙ\{ max_{s≤1} |W (s) + |B_{d}(s)|_{2}| ≤ λ\sqrt{\frac{1}{log log t_{k+1}}} \}
= exp \{ − \frac{π^{2}}{4(λ^{2} + o(1))} log log t_{k+1}\}.

Consequently

∑_{k} ℙ\{ max_{t_{k}≤s≤t_{k+1}} |(W (s) − W (t_{k})) + |B_{d}(s) − B_{d}(t_{k})|_{2}| ≤ λ\sqrt{\frac{t_{k+1}}{log log t_{k+1}}} \} = ∞.

Thus it follows from Borel-Cantelli lemma that,

lim inf_{k→∞} \sqrt{\frac{log log t_{k+1}}{t_{k+1}}} max_{t_{k}≤s≤t_{k+1}} |(W (s) − W (t_{k})) + |B_{d}(s) − B_{d}(t_{k})|| ≤ λ a.s.

Letting λ ⟶ π/2 yields

lim inf_{k→∞} \sqrt{\frac{log log t_{k+1}}{t_{k+1}}} max_{t_{k}≤s≤t_{k+1}} |(W (s) − W (t_{k})) + |B_{d}(s) − B_{d}(t_{k})|| ≤ \frac{π}{2} a.s.

Next note that

\sqrt{2t_{k} log log t_{k}} = o(\sqrt{\frac{t_{k+1}}{log log t_{k+1}}} )

and thus by the standard laws of iterated logarithm,

lim_{k→∞} \sqrt{\frac{log log t_{k+1}}{t_{k+1}}} max_{s≤t_{k}} |W (s)| = 0 a.s.

and

lim_{k→∞} \sqrt{\frac{log log t_{k+1}}{t_{k+1}}} max_{s≤t_{k}} |B_{d}(s)|_{2} = 0 a.s.

Therefore the upper bound of (1.3) in Theorem 1.1 follows from

max_{s≤t_{k+1}} |W (s) + |B_{d}(s)|_{2}|

≤ max_{t_{k}≤s≤t_{k+1}} |(W (s) − W (t_{k})) + |B_{d}(s) − B_{d}(t_{k})|_{2}| + max_{s≤t_{k}} |W (s)| + max_{s≤t_{k}} |B_{d}(s)|_{2}.

We finished the proof of (1.3) in Theorem 1.1.

To prove (1.1) and (1.2) in Theorem 1.1, we need some upper tail estimates. First we show that

lim_{λ→∞} λ^{−2} log ℙ{ max_{t≤1} (W (t) + |B_{d}(t)|_{2}) ≥ λ}

= lim_{λ→∞} λ^{−2} log ℙ\{W (1) + |B_{d}(1)|_{2} ≥ λ\} = −\frac{1}{4}  (3.1)

and

lim_{λ→∞} λ^{−2} log ℙ{ max_{t≤1} ( − W (t) − |B_{d}(t)|_{2}) ≥ λ}

= lim_{λ→∞} λ^{−2} log ℙ\{ − W (1) − |B_{d}(1)|_{2} ≥ λ\} = −\frac{1}{2}.  (3.2)

Consider (W, B_{d}) as a Gaussian random element in C{[0, 1], R^{d+1}}. It follows from standard large deviation estimate,

lim_{λ→∞} λ^{−2} log ℙ{λ^{−1}(W, B_{d}) ∈ F } ≤ − inf_{(x,y)∈F} I(x, y)

for each close set F ∈ C{[0, 1], R^{d+1}} and

lim_{λ→∞} λ^{−2} log ℙ{λ^{−1}(W, B_{d}) ∈ G} ≥ − inf_{(x,y)∈G} I(x, y)

for each open set G ∈ C{[0, 1], R^{d+1}}, where

I(x, y) = \frac{1}{2} ∫_{0}^{1} \{|ẏ(t)|^{2} + ∑_{j=1}^{d} |ẋ_{j}(t)|^{2}\}.

Thus by contraction principle we only need to show

inf ; max_{t≤1} (y(t)+|x(t)|_{2})=1 I(x, y) = inf_{y(1)+|x(1)|_{2}=1} I(x, y) = \frac{1}{4}  (3.3)

and

inf ; max_{t≤1} (−y(t)−|x(t)|_{2})=1 I(x, y) = inf_{−y(1)−|x(1)|_{2}=1} I(x, y) = \frac{1}{2}.  (3.4)

Note that

max_{t≤1} (y(t) + |x(t)|_{2}) ≤ max_{t≤1} \sqrt{2}(|y(t)|^{2} + ∑_{j=1}^{d} |x_{j}(t)|^{2})^{1/2} ≤ 2\sqrt{I(x, y)}.

On the other hand,

y(1) + |x(1)|_{2} = 2\sqrt{I(x, y)}

by taking y(t) = x_{1}(t) = ⋯ = x_{d}(t) = ct for some positive constant. Hence (3.3) holds. Similarly, we have

−y(1) − ||x(1)|| ≤ −y(t) ≤ ( ∫_{0}^{1} |ẏ(t)|^{2})^{1/2} ≤ \sqrt{I(x, y)}

and,

−y(1) − ||x(1)|| = \sqrt{I(x, y)}

by taking y(t) = −ct and x_{1}(t) = ⋯ = x_{d}(t) = 0. Hence (3.4) holds.

For any r > 1, the upper bounds in (3.1) gives that

∑_{k} ℙ\{ max_{s≤r^{k}} (W (s) + |B_{d}(s)|_{2}) ≥ (2 + ϵ)\sqrt{r^{k} log log r^{k}}\} < ∞

for any r > 1. So Borel-Cantelli Lemma gives that

lim sup_{k→∞} \frac{1}{\sqrt{r^{k} log log r^{k}}} max_{s≤r^{k}} (W (s) + |B_{d}(s)|_{2}) ≤ 2 a.s.

By making r > 1 arbitrarily close to 1 we obtain the upper bound of (1.1) by a standard deterministic estimate. Notice that the obvious relation W (t) + X(t) ≥ W (t) and the law of the iterated logarithm for Brownian motions give the lower bounds for (1.2). By the lower bounds in (3.1) and (3.2), given ϵ > 0,

∑_{k} ℙ\{(W (r^{k+1}) − W (r^{k})) + |B_{d}(r^{k+1}) − B_{d}(r^{k})|_{2} ≥ (2 − ϵ)\sqrt{r^{k+1} log log r^{k+1}}\} = ∞

∑_{k} ℙ\{(W (r^{k+1}) − W (r^{k})) + |B_{d}(r^{k+1}) − B_{d}(r^{k})|_{2} ≤ −(\sqrt{2} − ϵ)\sqrt{r^{k+1} log log r^{k+1}}\} = ∞

for sufficiently large r > 1. By independence and Borel-Cantelli Lemma,

lim sup_{k→∞} \frac{1}{\sqrt{r^{k+1} log log r^{k+1}}} [(W (r^{k+1}) − W (r^{k})) + |B_{d}(r^{k+1}) − B_{d}(r^{k})|_{2}] ≥ 2 − ϵ a.s.

lim inf_{k→∞} \frac{1}{\sqrt{r^{k+1} log log r^{k+1}}} [(W (r^{k+1}) − W (r^{k})) + |B_{d}(r^{k+1}) − B_{d}(r^{k})|_{2}] ≤ −\sqrt{2} + ϵ a.s.

¿From the classic law of the iterated logarithm,

lim sup_{k→∞} \frac{1}{\sqrt{r^{k+1} log log r^{k+1}}} (|W (r^{k})| + |B_{d}(r^{k})|_{2}) ≤ 2\sqrt{2}r^{−1/2}

Since r > 1 can be arbitrarily large and ϵ > 0 can be arbitrarily small, we have the lower bound of (1.1) and the upper of (1.2).

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