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 of 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 operator on the manifold. Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 2 / 18

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 ) ≤ Cr 2 . In particular, this explains why the Brownian motion on R 2 is recurrent—there is just not enough volume in the two-dimensional Euclidean space! Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 3 / 18

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 X t eventually leaves F with the probability 1, i.e. P x {∃ T : ∀ t > T X t / ∈ 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 H 2 . Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 4 / 18

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 sup Ω v = 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 b Ω 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 s Ω for Ω. Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 5 / 18

Hitting probabilities The P x − probability that the Brownian motion X t visits a set F ⊂ M ever. Denote it by e F ( x ) := P x {∃ t ≥ 0 such that X t ∈ F } The P x − probability that the Brownian motion X t visits F at a sequence of arbitrarily large time. Denote it by h F ( x ) := P x {∃{ t k } such that t k → ∞ and X t k ∈ F , for all k ∈ N } . Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 6 / 18

Proof of (1)= ⇒ (2) Let us denote Ω = M \ F and consider the function v := 1 − P ∞ s Ω , where P ∞ s Ω := lim t →∞ P t s Ω ( 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 e F ( x ) = s Ω ( x ) . (ii) For any x ∈ M we have h F ( x ) = P ∞ s Ω ( x ) . Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 7 / 18

Proof of (1)= ⇒ (2) Figure: 1 Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 8 / 18

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, s Ω ≡ 1 and P ∞ s Ω ≡ 1, or Ω is massive, s Ω �≡ 1 and P ∞ s Ω = 0 . whence P ∞ s Ω �≡ 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 {E k } of M such that the boundaries ∂ E k and ∂ Ω are transversal, and solve, for any set Ω ∩ E k , the following Dirichlet problem  ∆ b k = 0   b k | ∂ Ω ∩E k = 0  b k | ∂ E k ∩ Ω = 1 .  Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 10 / 18

Proof of (1)= ⇒ (2) Figure: 2 Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 11 / 18

Proof of (1)= ⇒ (2) Propositon 3 Let Ω ⊂ M be a non-empty open set. Assume that Ω has non-empty smooth boundary. Then b Ω = lim k →∞ b k in Ω The function b Ω is continuous, subharmonic on M and harmonic in Ω. Respectively, the function s Ω is continuous, superharmonic on M and harmonic in Ω. Since Ω is massive and M \ Ω is non-empty, s Ω is a non-trivial bounded superharmonic function on M . Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 12 / 18

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 ∞ s Ω ≡ 0 by the massiveness of Ω. And by Proposition 1, it shows h F ( x ) ≡ 0 for any precompact set F . It is equivalent to say , with P x -probability 1, the Brownian trajectory X t leaves F after some time forever. Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 13 / 18

(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). Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 14 / 18

(2) ⇐ ⇒ (3) Conformal mapping in dimension 2 preserves superharmonic functions. Since H 2 possesses a non-constant positive superharmonic function whereas R 2 or S 2 does not, hyperbolicity of M is equivalent to the presence of a non-constant positive superharmonic function. Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 15 / 18

Further Results More equivalent statements � ∞ The Green function G ( x , y ) = 1 0 p ( t , x , y ) dt on M is finite for all 2 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. Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 16 / 18

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. Jingbo Liu (Math Department) Presentation for MATH 6720 May 9, 2016 17 / 18

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