Diffuison processes on CR-manifolds

Transcription

1 Diffuison processes on CR-manifolds Setsuo TANIGUCHI Faculty of Arts and Science, Kyushu University September 5, 2014 Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

2 Introduction Introcuction To construct Bm {X x t } t 0 on an n-dim Rie mfd M starting at x M; Rolling M along the Brownian motion {B t } t 0 on R n ; (O(M), π): Orthon frame bndle /M {L 1,, L n }: fundamental v files on O(M) {r r t } t 0: dr t = n L α (r t ) db α t, r 0 = r O(M) α=1 X x = π(r r ) (π(r) = x) Bm on M t t Do the same thing on a strictly pseudoconvex CR-mfd! 1 CR-manifold 2 CR-Brownian motion 3 Heat kernel 4 Dirichlet problem 5 Shot time asymptotics Joint work with Hiroki Kondo Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

3 CR-manifold Strongly pseudoconvex CR-mfd M is a (2n + 1)-dim CR-mfd iff 1 real (2n + 1)-dimensional oriented C -mfd 2 complex n-dim subbundle T 1,0 of the complexified tangent bundle CTM st T 1,0 T 0,1 = {0}, where T 0,1 = T 1,0 3 Frobenius cond is fulfilled: [T 1,0, T 1,0 ] T 1,0 1-form θ 0 on M st θ(h) = {0} (H := Re(T 1,0 T 0,1 )) The Levi form L θ is defined by L θ (Z, W) = idθ(z, W) (Z, W Γ (T 1,0 T 0,1 )), where Γ (T 1,0 ) is the totality of C -sections to T 1,0 M is strongly pseudoconvex iff the Levi form is strictly positive (Assumed here after) Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

4 CR-manifold Examples Real submfd of complex mfd N: complex mfd of (complex) dim n + 1 M: real 2n + 1-dim submfd of N T 1,0 = T 1,0 N CTM, where T 1,0 N is the hol tangent bdl/n H n = C n R: Heisenberg gr (z, t) (w, s) = (z + w, s + t + 2Im z, w ) Z α = + iz α z α t T 1,0 = span C {Z 1,, Z n } n ( θ = dt + i z α dz α z α dz α) dθ = 2i n α=1 α=1 dz α dz α L θ (Z α, Z β ) = δ αβ Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

5 CR-manifold Kohn-Rossi Laplacian ψ := θ (dθ) n ; vol form L :=the dual on θ H of L θ on H = Re(T 1,0 T 0,1 ) u, v θ = uvψ, M ω, η θ = L (ω, η)ψ (u, v θ C (M), ω, η 0 Γ(H )) M d b := r 0 d, b := r 1 d b, where r 0 : T M H, r 1 : H T 0,1 : restrictions Sublaplacian b := d b d b, K-R Laplacian b := b b : b u, v θ = d b u, d b v θ, b u, v θ = b u, b v θ 1 T: v field transversal to H with T dθ = 0 and T θ = 1 Then b = b + int Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

6 CR-manifold Tanaka-Webster connection J : H H; the complex structure; J = i on T 1,0, = i on T 0,1 g θ : the Webster metric; for X, Y H g θ (X, Y) = dθ(x, JY), g θ (X, T) = 0, & g θ (T, T) = 1 Tanaka-Webster connection:! lin conn X (Γ (H)) Γ (H) ( X Γ (TM)), J = 0, g θ = 0 T (Z, W) = 0, T (Z, V) = 2iL θ (Z, V)T for Z, W Γ (T 1,0 ), V Γ (T 0,1 ) where T (Z, W) = Z W W Z [Z, W] T (T, JX) + J(T (T, X)) = 0 for X Γ (TM) n := {1,, n} & n := {0, 1,, n, 1,, n}, where α; Z α := Z α For a local orthon frame {Z α } α n (Z α Γ (U; T 1,0 )), Z A Z B = C n Γ C AB Z C, A, B n (Z 0 = T) Γ C AB = 0 if (B, C) {(β, γ), (β, γ) β, γ n } Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

7 CR-manifold Fundamental vector fields p : [a, b] T 1,0 is parallel along p : [a, b] M iff p(t) (T 1,0 ) p(t) and ṗ p = 0 U(T 1,0 ) := x M {r : C n (T 1,0 ) x : isometric}, π(r) = x U(n)-principal bundle p : [a, b] U(T 1,0 ) is a horizontal lift of p : [a, b] M iff π( p) = p, p(t)ξ is pararel along p ( ξ C n ) For v T x M, η T r (U(T 1,0 )) (π(r) = x) is a holizontal lift of v if p, a holizontal lift of p st p(0) = r, p(0) = η, π η = v For v T x M and r U(T 1,0 ) with π(r) = x,!η r (v) T r (U(T 1,0 )); holizontal lift of v (L α ) r := η r (re α ), α n ; v fields on U(T 1,0 ) where {e α } α n is the standard basis of C n Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

8 CR-manifold Local expression For a local orhon frame {Z α } α n of T 1,0, set {e β α (r)} Cn n by r(e α ) = β n e β α (r)(z β) π(r) L α = e β α Z β β n β,γ,δ,ε n Γ γ βδ eδ ε eβ α e γ ε Γ γ e δ βδ ε eβ α β,γ,δ,ε n e γ ε Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

9 CR-Brownian motion CR-Brownian motion {B t = (B 1 t,, Bn t )} t 0: C n -valud continuous martingale with B α, B β t = 0 & B α, B β t = δ αβ t {r r} t t 0: the unique sol to the SDE on U(T 1,0 ): n dr t = α (r t ) db α=1{l α + L t α (r t ) db α}, r t 0 = r U(T 1,0 ) Q r :=the distribution of {r r} t t 0 on C([0, ); U(T 1,0 )) P x := Q r π 1 (r π 1 (x)) (π : U(T 1,0 ) M: proj) Rem: Q r π 1 = Q r π 1 if π(r) = π(r ), since r(t, ur, ub) = r(t, r, B) (u U(n)) Let X t : C([0, ); M) M be the coord pr {({X t } t 0, P x ), x M} is the diffusion generated by 1 2 b (CR-Brownian motion) 1 n 2 b = 1 } Lα L 2 α + L α L α α=1{ C (M) Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

10 Heat kernel Partial hyperellipticity Let P, N be C -mfds, ϕ : P N be C, A 0,, A n be C -v fields on P X t be the sol to the SDE on P: n dx t = A α (X t ) db α + A t 0 (X t )dt α=1 Y t := ϕ(x t ) Theorem 0 (T83) Assume that (ϕ ) p (L p ) = T ϕ(p) N ( p P), where L = { } [A i1,, [A ik, A j ] ] 1 j n, 0 i l n, k Z 0 Then Y t admits a C -density function P = O(M), N = M heat kernel for 1 2 M Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

11 Heat kernel Heat kernel Coming back to the CR-mfd; Let {Z α } be a local orthonormal frame for T 1,0 Then (π )L α = n e β α Z β β=1 (π )[L α, L α ] = it mod {Z β, Z β ; β = 1,, n} Hence span R { (π ) r ReL α, (π ) r ImL α, (π ) r [ReL α, ImL α ] : 1 α n } = T π(r) M ( r U(T 1,0 )) Under a suitable non-explosion assumption (assumed hereafter), by Theorem 0, Theorem 1 p C ((0, ) M M) st P x (X t dy) = p(t, x, y)ψ(dy) Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

12 Dirichlet problem Dirichlet problem Let G be a rel cpt conn open set in M with C 3 -bdry τ = inf{t 0 : X t G} For f C( G), define u f (x) := E x [ f(x τ )] Theorem 2 (Probabilistically) u f C(G) u f, b v θ = 0 ( v C 0 (G)) & u f G = f Under local orthon frame {Z α }, b = n α=1 {Z 2 α + Z2 α } + b, [ReZ α, ImZ α ] = 1 2 T mod {ReZ β, ImZ β ; 1 β n} Corollary 3 ụ f C (G); u f is a classical sol to the Diriclet problem for b Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

13 Dirichlet problem Stroock-Varadhan (1972): It suffices to show sup x G E x [τ ] < P x (τ = 0) = 1 ( x G) dx t = 2n i=1 Stroock-T 1986: V i (X t ) db i t + V 0(X t )dt, where V 2i = 2 ReZ i, V 2i 1 = 2 ImZ i & V 0 = b Γ = {x G P x (τ = 0) = 1} Ψ L = { x G V i ϕ = 0 ( i < L), i =L(V i ϕ) 2 (x) 0 }, where, for i = (i 1,, i k ) {0, 1,, 2n} k, V i = V i1 V ik and i = k + #{ j i j = 0} L; odd Ψ L Γ L = 2; x Ψ L is in Γ if either (i) M := (V i V j ϕ(x)) 1 i, j n is not symmetric (ii) M is symmetric but admits at least 1 negative ev (iii) V 0 ϕ(x) < 0 Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

14 Short time asymptotics Short time asymptotics Theorem 4 It holds that p(t, x, x) 1 t n+1 C k (x)t k as t 0 1 ( 2τ ) ndτ Furthermore C 0 (x) = C 0 = (4π) n+1 n! R sinh(2τ) k=0 1 Localization around x M: {Z α }: a loc orthonormal frame of T 1,0 on a rel cpt coord nbd U V: an open rel cpt subset of U with x V X α := 2 ReZ α, X n+α := 2 ImZ α, α = 1,, n {(X t, P y ) : y U}: the diffusion generated by 2n k=1 X2 1 n { } k 2 α,β=1 Γ α α + Γ ββz α Z α ββ p(t, x, y): a density function of P y with respect to ψ 1 2 Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

15 Short time asymptotics 2 p(t, x, x) p(t, x, x) 3 Folland-Stein (1974): a coord sys u = (u 0, u 1,, u 2n ) around x st u(x) = 0 & X j = X n+ j = T = + 2u n+ j 2n u j u + 0 k=1 2u j 2n u n+ j u + 0 u 0 + 2n O 2n+1,1 k=1 k=1 u, k O j,1 O n+ j,1 u + O k j,2 u, 0 u + O k n+ j,2 u, 0 where O,i = O ({ u 0 1/2 + 2n 1 uk } i) as (u 0,, u 2n ) 0 Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16

16 Short time asymptotics Not yet The result reported here is just an entrance to the stochastic approach to CR-manifolds You may ask questions; Explositon? Ricci curvature? Carré de champ L θ (d bu, d b v)? Recurrence? Transience? Weitzenböck formula? The heat eq for p-forms? Fefferman metric? Off-diagonal asymptotics? Applicable to b = b + int? On CR-manidolds of CR-codimension more than 2? And more! Path space over a CR-manifold? Setsuo TANIGUCHI (Kyushu Univ) CR-Browninan motion September 5, / 16