Default functions and Liouville type theorems Atsushi Atsuji Keio University 17. Aug. 2016 [Plan of my talk] x1 Default functions: definition and basic properties x2 Submartingale properties of subharmonic functions : Symmetric diffusion cases x3 L1- Liouville properties of subharmonic functions x4 Liouville theorems for holomorphic maps Fix a filtered probability space (Ω; F; Ft;P ). Let Mt be a continuous local martingale. Def. If Mt is not a true martingale, we say Mt is a strictly local martingale. Mt is a (true) martingale , E[MT ] = E[M0] for 8T : bounded stopping time. \Local" property of Mt : γT (M) := E[M0] − E[MT ]: is called a default function (Elworthy- X.M.Li-Yor(`99)). Default formula : Assume that E[jMT j] < 1;E[jM0j] < 1 for a f − g stopping time T and MT ^S ; S : stopping times is uniformly ∗ integrable. Set Mt := sup Ms. 0≤s≤t ∗ ∗ E[MT : MT ≤ λ] + λP (MT > λ) + E[(M0 − λ)+] = E[M0]: Letting λ ! 1, γT (M) = lim λP ( sup Mt > λ): !1 λ 0≤t≤T Another quantity: σT (M) Def. h i1=2 σT (M) := lim λP ( M T > λ): λ!1 Theorem (Elworthy-Li-Yor, Takaoka(`99)) Assume that E[jMT j] < 1;E[jM0j] < 1. r π 9γ (M) = σ (M): T 2 T T Moreover Mt := MT ^t is a uniformly integrable martingale iff γT (M) = σT (M) = 0. See also Azema-Gundy -Yor(`80), Galtchouk-Novikov(`97). [Example] Rt : d-dimensional Bessel process: d − 1 dRt = dbt + dt; R0 = r. 2Rt 2−d If d > 2, then Rt is a strictly local martingale. As for default function, if R0 = r, Z t 2 2−d 1 du r γt(R ) = exp(− ); ν 1+ν 2 Γ(ν) 0 u 2u where d = 2(1 + ν). If d = 2, log Rt is a strictly local martingale. Z 1 t du r2 γt(log R) = exp(− ): 2 0 u 2u [submartingale case] Let Xt = X0 + Mt + At where M is a local martingale and A is an adapted increasing process. Lem.(Default function for submartingale) If X is positive and E[AT ] < 1, lim λP ( sup Xt > λ) = lim λP ( sup Mt > λ) !1 !1 λ 0≤t≤T λ 0≤t≤T = E[X0] − E[XT ] + E[AT ]: Example (stochastic Jensen's formula). Let Zt : BM(C) with Z0 = o, τr = infft > 0 : jZtj > rg and f be a non-constant holomorphic function on C, f(o) =6 0. j − j−2 Set Xt := log f(Zτr ^t) a : a local martingale bounded below. X r lim λP ( sup Xt > λ) = 2 log : λ!1 0<t<τ jζj r f(ζ)=a; jζj<r From this we can see an essential relationship between Nevanlinna theory and complex Brownian motion ( Carne(86), A.(95)). Our question: When is a local submartingale u(Xt) a submartingale ? x2 Submartingale property of subharmonic functions. [Settings] Let M : a smooth manifold, m a Radon measure on M with suppm = M, (Xt;Px) be a symmetric diffusion process defined from the Dirichlet form (E; F) with a core C ⊂ F \ Co(M) where Z E(u; v) = Γ(u; v)dm (u; v 2 C): M We have there exists L a s.a.operator on L2(m) such that Z E(u; v) = − uLvdm for u; v 2 C: M L is the generator of the diffusion. Assume that • (E; F) is a strongly local, irreducible regular Dirichlet form. • the transition probability p(t; x; dy) is absolutely continuous w.r.t. m. • there exists a nonnegative exhaustion function r(x) (i.e. fr(x) < rg : rel.cpt for 8r ≥ 0) such that Γ(r(·); r(·)) is bounded a.e. • there exists x0 2 M and c1(x0); c2(x0) > 0 such that 2 2 c1(x0)jruj ≥ Γ(u; u) ≥ c2(x0)jruj for 8u 2 C on a neighborhood of x0. Note that the first assumption implies a diffusion process corresponds to the Dirichlet form. Typical Example : Brownian motion on a complete, connected Riemannian manifold M. 1 L = ∆; Γ(u; u) = jruj2; r(x) = d(o; x), 2 m = Riemannian volume dv, p(t; x; dy) = p(t; x; y)dv(y) where 1 p(t; x; y) is the heat kernel of @=@t − ∆. 2 E F 1 M 1 M 1 = H0 ( ) = C0 ( ) where E E jj jj2 C 1 M 1(u; u) = (u; u) + u L2(m). Note = C0 ( ). [subharmonic function] Def. u is (L-)subharmonic if u 2 Floc and E(ϕ, u) ≤ 0 for 8ϕ ≥ 0; ϕ 2 F with compact support. It is well-known that u(Xt) is a continuous local submartingale: − [u] [u] u(Xt) u(x) = Mt + At Px-a.s. Def. Default function of u(XT ) Nx(T; u) = lim λPx( sup u(Xs) > λ): !1 λ 0≤s≤T [u] 1 As before if u is positive subharmonic and Ex[At ] < , − [u] Ex[u(Xt)] u(x) + Nx(t; u) = Ex[At ]: We consider the condition for the default function to be vanishing. Let U := fu : a positive subharmonic function jEx[u(Xt)] < 1(8t > 0)a:e:xg. Theorem. Let B(r) := fr(x) < rg. If X is transient, u 2 U and Z 1 lim inf flog uαdm + log m(B(r))g < 1 !1 2 r r B(r)) for some α > 2, then u(Xt) is a submartingale under Px for a.e.x. sketch of proof. ◦ 1 . Let τr = infft > 0jXt 2= B(r)g. If lim Ex[u(Xτr ): τr < t] = 0; r!1 then Nx(t; u) = 0. ◦ 2 . Estimate Ex[u(Xτr )]. Lem. Let x0 a point appearing in the assumption. If X is transient and u is a positive subharmonic function, there exists a constant C(x0) such that Z Z ≤ f 2 1=2 g Ex0 [u(Xτr )] C(x0) ( u(x) dm) + u(x)dm : B(r+1) B(r) ◦ 3 . Estimate Px(τr < t). Lem. (Takeda's inequality) Fix 1 > r0 > 0. If r > r0, there exists c > 0 such that Z 2 m(B(r + 1)) − cr Py(τr < t)dm(y) ≤ const: e t ; B(r0) r ◦ 8 4 . Nx0 (t0; u) = 0 for some x0; t0 implies Nx(t; u) = 0 for t > 0 and a.e.x. [Brownian motion case] When M is a complete Riemannian manifold and (Xt;Px) is Brownian motion on M, the Ricci curvature controls the conditions in the above theorem. Theorem. If there exists a constant C > 0 such that Ric ≥ −Cr(x)2 − C and a positive subharmonic function u satisfies Z 1 lim inf log u(x)dv(x) < 1; !1 2 r r B(r) then u(Xt) is a submartingale. x3. L1 Liouville theorem. [Known results] 1-1. Lp-Liouville theorem: (Yau `76, cf. P.Li-Schoen `84) If M is a complete Riemannian manifold and a positive ∆-subharmonic function u is Lp-integrable for some p > 1, u is constant. 1-2. Generalization in the context of Dirichlet form (T.Sturm `94). Under our setting, if a positive L-subharmonic u satisfies Z 1 rdr R = 1 p B(r) u dm for some p > 1, then u is constant. 2. L1-Liouville theorem. Let M be a complete Riemannian manifold and u a positive ∆-subharmonic function. Ricci curvature condition (P.Li `84 ) If M is a complete Riemannian manifold satisfying Ric ≥ −Cr(x)2 − C for some C > 0 and u is L1, then u is constant. p 3.Z Weighted L -Liouville theorem. (Nadirashvili `85) f(u(x)) If dv(x) < 1 for a nonnegative function f on [0; 1) M r(x)2 + 1 Z 1 satisfying 1=f(t)dt < 1, then u is constant. 0 Z u(x)p 9p > 1 s.t. dv(x) < 1 ) u const. M r(x)2 + 1 [L1-Liouville theorem and submartingale property] Prop. If u is a positive, integrable L-subharmonic function and u(Xt) is a submartingale under Px for a.e. x, then u is constant. Namely vanishing of default function of u implies L1-Liouville theorem. Proof. u(x) ≤ Ex[u(Xt)] for all 0 < t and a.e. x. Z t tu(x) ≤ Ex[u(Xs)]ds: 0 If X is recurrent, ratio ergodic theorem for recurrent Markov processes implies 8 R Z <> M u(x)dx 1 t ( if m(M) < 1); Ex[ u(Xs)ds] ! m(M) t :> 0 0 ( if m(M) = 1) as t ! 1. In both cases u should be bounded. Then u is a constant. Z 1 t If X is transient, Ex[ u(Xs)ds] ! 0 as t ! 1 since Z t 0 ζ Ex[ u(Xs)ds] < 1 for an integrable function u where ζ is the 0 life time of X. [Example] The following example is originally due to Li-Schoen. We give a little modification. Let M be a compact 2-dim Riemannian manifold with 2 2 a metric ds0, ∆M is the Laplacian defined from ds0 and X 1 Brownian motion on M with its generator ∆ . Fix o 2 M. Set 2 M Z 1 1 g(o; x) = 2π (p(t; o; x) − )dt + C; 0 vol(M) where p(t; x; y) is the transition density of X and C is a positive constant such that g(o; x) > 0 for all x 2 M n fog. Remark that 1 g(x; y) ∼ log (d(x; y) ! 0). Note d(x; y)2 1 − 1 ∆M g(o; x) = 2πδo(x) + : 2 V ol(M) Let M be M n fog.