Some Equivalent Definitions of Transience Jingbo Liu Mathematics - - PowerPoint PPT Presentation

Some Equivalent Definitions of Transience

Jingbo Liu

Mathematics Department jl3446@cornell.edu

May 9, 2016

Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 1 / 18

Introduction

The main goal of this presentation is to study the analytic and geometric background of the property of the Brownian motion to be recurrent or transient. We shall see that recurrence is related to various geometric properties

• f the underlying Riemann manifold such as the volume growth,

isoperimetric inequality, curvature etc. On the other hand, recurrence happens to be equivalent to certain analytic properties of the Laplace

• perator on the manifold.

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Introduction

Theorem

Let V (r) denote the Riemannian volume of a geodesic ball of radius r with a fixed center. Then the Brownian motion on a geodesically complete manifold M is recurrent provided ∞

rdr V (r) = ∞.

For example, this condition is satisfied if V (r) ≤ Cr2. In particular, this explains why the Brownian motion on R2 is recurrent—there is just not enough volume in the two-dimensional Euclidean space!

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The Main Theorem

Let M be a Riemannian manifold. Then the following properties are equivalent. (1) Brownian motion on M is transient; i.e. for any precompact set F ⊂ M and for some point x ∈ M, the process Xt eventually leaves F with the probability 1, i.e. Px{∃T : ∀t > T Xt / ∈ F} = 1. (2) There exists a non-constant positive superharmonic function on M. (3) If M is a simply connected Riemann surface, then (1) and (2) are equivalent to M being of hyperbolic type; i.e. M is conformally equivalent to H2.

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Preliminaries

Definition 1

Given an open set Ω ⊂ M, we say that a function v ≥ 0 is an admissible subharmonic function for Ω if it is a bounded subharmonic function on M such that v = 0 on M \ Ω and supΩv = 1.

Definition 2

An open set Ω is called massive if there is at least one admissible subharmonic function for Ω.

Definition 3

The subharmonic potential bΩ of an open set is the supremum of all admissible subharmonic functions v for Ω. Remark: We could similarly define the admissible superharmonic function u and superharmonic potential sΩ for Ω.

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Hitting probabilities

The Px−probability that the Brownian motion Xt visits a set F ⊂ M

• ever. Denote it by

eF(x) := Px{∃t ≥ 0 such that Xt ∈ F} The Px−probability that the Brownian motion Xt visits F at a sequence of arbitrarily large time. Denote it by hF(x) := Px{∃{tk} such that tk → ∞ and Xtk ∈ F, for all k ∈ N}.

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Proof of (1)= ⇒(2)

Let us denote Ω = M \ F and consider the function v := 1 − P∞sΩ, where P∞sΩ := limt→∞ PtsΩ(x) We have the following proposition.

Proposition 1

Let Ω ⊂ M be a non-empty open set with smooth boundary, and denote F := M \ Ω. (i)(G. A. Hunt) For any x ∈ M we have eF(x) = sΩ(x). (ii) For any x ∈ M we have hF(x) = P∞sΩ(x).

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Proof of (1)= ⇒(2)

Figure: 1

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Proof of (1)= ⇒(2)

By hypothesis and the proposition above, we conclude v(x) > 0, for some

• x. Also, we have the following dichotomy statement:

Proposition 2

Let Ω ⊂ M be an open set with non-empty smooth boundary, and let F := M \ Ω be compact, then either Ω is not massive, sΩ ≡ 1 and P∞sΩ ≡ 1,

• r Ω is massive, sΩ ≡ 1 and P∞sΩ = 0.

whence P∞sΩ ≡ 1 and Ω is massive.

Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 9 / 18

Proof of (1)= ⇒(2)

Since massiveness is preserved by increasing the set Ω, so by slightly enlarging Ω, we may assume that Ω has smooth boundary. Now choose an exhaustion {Ek} of M such that the boundaries ∂Ek and ∂Ω are transversal, and solve, for any set Ω ∩ Ek, the following Dirichlet problem      ∆bk = 0 bk|∂Ω∩Ek = 0 bk|∂Ek∩Ω = 1.

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Proof of (1)= ⇒(2)

Figure: 2

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Proof of (1)= ⇒(2)

Propositon 3

Let Ω ⊂ M be a non-empty open set. Assume that Ω has non-empty smooth boundary. Then bΩ = limk→∞ bk in Ω The function bΩ is continuous, subharmonic on M and harmonic in Ω. Respectively, the function sΩ is continuous, superharmonic on M and harmonic in Ω. Since Ω is massive and M \ Ω is non-empty, sΩ is a non-trivial bounded superharmonic function on M.

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Proof of (2)= ⇒(1)

Let v > 0 be a non-constant superharmonic function on M. For any number c ∈(inf v, sup v), the set Ω = {v < c} is proper and massive because (c − v)+ is an admissible subharmonic function for Ω. By Proposition 2, we have that P∞sΩ ≡ 0 by the massiveness of Ω. And by Proposition 1, it shows hF(x) ≡ 0 for any precompact set F. It is equivalent to say , with Px-probability 1, the Brownian trajectory Xt leaves F after some time forever.

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(2) ⇐ ⇒ (3)

To prove the second part, we need the following uniformization theorem of

• F. Klein, P. Koebe and H. Poincar´

e.

Theorem

Any simply connected Riemann surface is conformally equivalent to one of the following canonical surfaces:

• 1. the sphere (surface of elliptic type)
• 2. the Euclidean plane (surfce of parabolic type)
• 3. the hyperbolic plane (surface of hyperbolic type).

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(2) ⇐ ⇒ (3)

Conformal mapping in dimension 2 preserves superharmonic

• functions. Since H2 possesses a non-constant positive superharmonic

function whereas R2 or S2 does not, hyperbolicity of M is equivalent to the presence of a non-constant positive superharmonic function.

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Further Results

More equivalent statements

The Green function G(x, y) = 1

2

∞

0 p(t, x, y)dt on M is finite for all

x = y. For all x ∈ M, ∞

1 p(t, x, x)dt < ∞.

The capacity of any precompact open set is positive. There exists a non-zero bounded solution on M to the equation ∆u − q(x)u = 0 for any function q(x) ∈ C ∞

0 (M), which is non-negative and not

identically 0.

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References

William H. M. , Joaquin P. , Antonio R. (2006) Liouville type properties for embedded minimal surfaces

• Comm. in Ana. and Geom. 14.4(2006):703
• A. Grigor’yan (1999)

Analytic and geometric background of recurrence and non-explosion of Brownian motion on Riemannian manifolds

• Bull. of A.M.S. 36(2):135-249, 1999.

Ahlfors L. V. (1952) On the characterization of hyperbolic Riemann surfaces

• Ann. Acad. Sci. Fenn. Series A I. Math. 125 (1952) .

Chavel I., Karp L. (1991) Large time behavior of the heat kernel: the parabolic λ-potential alternative

• Comment. Math. Helvetici 66(1991) 541-556.

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References

Kakutani S. ,(1961) Random walk and the type problem of Riemann surfaces Princeton Univ. Press (1961) 95-101 Varpoulos N. Th. , (1984) Brownian motion and random walks on manifolds

• Ann. Inst. Fourier, 34 (1984) 243-269.

Coulhon T. , Grigor’yan A. , Random walks on graphs with regular volume growth

• Geom. and Funct. Analysis. 8 (1998) 656-701.

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