Grushin 2MA16039T

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1 Grushin 2MA1639T 3 2 2

2 R d Borel α i k (x, bi (x, 1 i d, 1 k N d N α R d b α = α(x := (αk(x i 1 i d, 1 k N b = b(x := (b i (x 1 i d X = (X t t x R d dx t = α(x t db t + b(x t dt ( 3 u t = Au + V u, u(, x = f(x, t >, x Rd x R d { } u(t, x = E x f(x t exp V (X s ds ( 3.7 a(x = (a ij (x 1 i,j d a ij := αk(t, i xα j k (t, x A A = 1 2 a ij 2 (x x i x + j i,j=1 b i (x x i α((x, y = ( 1 x 1, b((x, y = (

3 (x, y R 2 { ( A = 1 2 ( } 2 + x 2 2 x y Grushin X t t A 2 1 Brown 2 3 Feynman-Kac 4 Grushin 2

4 1 Brown Feynman-Kac Grushin 34 3

5 1 Brown 1.1. (Ω, F, P B = (B t (ω t Brown (i B = a.s. (ii ω Ω B t (ω t (iii = t < t 1 < < t n, n N {B ti B ti 1 } 1 i n t i t i 1 Gauss W := C(,, R, W := {w W ; w = } Borel B(W, B(W Ω = W, F = B(W B t (w = w t, w = (w t t W Brown (W, B(W P Wiener 1.3. d B t = (B i t 1 i d = (B 1 t,, B d t, t d Brown B i t Brown {Bi } 1 i d σ- {σ(b i t; t } 1 i d 1.4. (Brown B = (B t t (Ω, F, P 1 Brown F = σ(b s ; s t, F t = Ft N N = {N F; P (N = } (i EB 2p t = (2p 1!!, EB 2p 1 t =, p N. (ii s < t B t B s F s (iii EB t B s = t s (:= min{t, s}, t, s. (iv Brown (s >, γ > (a B a t := B t+s B s, t. 4

6 (b B b t := B t, t. (c B c t := γb t/γ 2, t. (v B t (ω t T 1, T 2, T 1 < T 2 a.s. ω (vi B = (B t t d Brown A d d AB t d Brown σ s (ω := inf{t > ; B t S} S := B(, r B σs(ω(ω S B(, r := { x r} 5

7 2 2.1 (F t t (Ω, F, P (F t t : (F t t t F t+ := s>t F s F t = F t+ N := {N F; P (N = } F t 2.1. X = (X t t (F t t (i t X t (ii X = (X t t (F t - (iii s t EX t F s = X s a.s. (iii EX t F s X s a.s. X EX t F s X s a.s. X 2.2. (Doob X = (X t t λ >, p > 1 (i ( P sup X s λ 1λ E X t, sup X s λ. s t s t (ii t X t a.s. ( P sup X s λ 1 s t λ EX t. (iii X E X t p <, t 1/p E sup X s p p s t p 1 E X t p 1/p. 6

8 2.3. σ : Ω, (F t t Markov t {σ t} {ω; σ(ω t} F t 2.4. (Doob-Meyer X = (X t t (F t - X (D ( σ Markov T > {X σ T } σ (F t - M = (M t t A = (F t - A = (A t t X X t = M t + A t, t A t > s A t A s a.s. T > t, T 2 (F t - M = (M t t,t M = a.s. M T Jensen M 2 t (D 2.4 A t M 2 t A t 2.5. A t M t M M, N M T M, N t = 1 2 ( M + N t M t N t, t, T M, N t a.s. M t N t M, N t M, N t M N 2 M, N M T P (M t = N t, t, T = (M t, E, T Hilbert 2.8. (Burkholder p > c p, C p > M M T : E M p t < c p E M p t E sup M t 2p C p E M p t. t T 7

9 (Ω, F, P Brown B = (B t t (F t -Brown (i B = (B t t (F t - (ii s t B t B s F s f s db s B = (B t t (F t -Brown f s = f s (ω Brown s σ(b r ; r s 2.1. ( f = (f t t,t f t = f1 (a,b (t, t, T f f = f(ω F a - a < b T f M t (f f s db s := f(b t b B t a, t, T f, g: M = (M t (f t,t M T M(f, M(g t = f s g s ds, t, T. (2.1. M = (M t (f t,t M T M 2 M = M s t EM t F s = M s EM t M s F s = E f{(b t b B t a (B s b B s a } F s s a ( = E f{(b t b B t a (B s B s } F s = EE f(b t b B t a F a F s = E feb t b B t a F a F s (2.2 8

10 a t b (2.2 = E feb t B a F a F s = (2.2 s > a ( = E f{(b t b B a (B s b B a } F s = feb t b B s b F s (2.3 s t b (2.3 = feb t B s F s = (2.3 EM t M s F s = ( s t E M t (fm t (g M s (fm s (g f r g r dr F s = (2.4 (2.4 f t = f1 (a,b (t, g t = g1 (c,d (t a c s c ((2.4 = E f(b t b B t a g(b t d B t c f g 1 (a,b (r1 (c,d (rdr F s s = E f ge (B t b B t a (B t d B t c 1 (a,b (r1 (c,d (rdr F c F s (2.5 s s a c t b d (2.5 = E f ge (Bt B a (B t B c (t c F c F s = E f ge (Bt B c + B c B a (B t B c (t c F c F s = E f ge (Bt B c 2 + (B c B a (B t B c (t c F c Fs = (2.5 s c ((2.4 = f ge (B t b B a (B t d B c (B s b B a (B s d B c 1 (a,b (r1 (c,d (rdr F s (2.6 s s 9

11 a c b s d t (2.6 = f ge(b b B a (B d B c (B b B a (B s B c F s = f ge(b b B a (B d B s F s = (2.6 ( ( f = (f t t,t f t = j=1 f j 1 (tj 1,t j (t, t, T n N, fj F tj 1 - { = t < t 1 < < t n = T } ω M t (f f s db s := f j (B t tj B t tj 1, t, T j= f, g: M = (M t (f t,t M T M(f, M(g t = f s g s ds, t, T. L 2 T := {f; f = f t (ω f L 2 T < }. T f 2 L := E f 2 2 t dt f (t, ω (, T T Ω, B(, T F f t (ω (R, B(R L 2 T = L 2 T (F t := {f L 2 T ; f (F t - } 1

12 f (F t - ( t, T (s, ω (, t Ω, B(, t F t f s (ω (R, B(R f L 2 T f L 2 t f n, n N f f n L 2 t, n. f = f t (ω L 2 T f m t (ω := f t (ω 1 m,m (f t (ω, m N f m t (ω F t - T ft m 2 L = E 2 T T E < ft 2 (ω 1 m,m (f t (ωdt ft 2 (ωdt f m L 2 T f f m 2 L 2 T T = E ft 2 1 { ft >m}dt (m f ft ϵ (ω := ϵ 1 f s (ωds, ϵ > f ϵ L 2 T (t ϵ f f ϵ L 2 T, (ϵ f t n 1 ft n := f tj (ω1 (tj,t j+1 (t, t j = T j/n j= f n f f n L 2 T, (n 11

13 f L 2 t 2.14 f n, n N f f n L 2 t, n M(f n = (M t (f n M T 2.13 {M(f n } M T Cauchy 2.7 M T {M(f n } M(f = (M t (f M T M t (f f s db s, t, T f(sdb s f L 2 T Brown B = (B t t t, T t, L 2 := {f = (f t t ; T > (f t t,t L 2 T } M := {M = (M t t ; T > (M t t M T } f = (f t L 2 M t (f = f s db s, t M(f = (M t (f t M ( f, g L 2, a, b R (i M = (M t (f t M M(f, M(g t = (ii M t (af + bg = am t (f + bm t (g, f s g s ds, t, T. t, a.s. (iii B t = (Bt i i i d (Ω, F, P d (F t -Brown Bt i (F t-brown {B i } 1 i d f L 2 M i t (f = f s db i s, t M i (f, M j (g t, i j (i E f s dbs i g s dbs j = δ ij E δ ij = { 1, i = j,, i j. f s g s ds. =, t 12

14 2.3 d X t = (Xt i 1 i d Xt i = X i + fk(sdb i s k + A i t, t. (2.7 B t = (B k t 1 k N N (F t -Brown 1 i d, 1 k N X i F - f i k = (f i k (t t L 2 A i t (F t- A i = ω A i t(ω t φ C 2 b (Rd φ(x t = φ(x i,j= (2.7 x i (X sda i s x i (X sf i k(sdb k s 2 φ x i x (X sfk(sf i j j k (sds, t, a.s. (2.8 dx i t = fk(tdb i t k + da i t dφ(x t = x i (X tdx i t Taylor 2 dφ(x t = x i (X tdx i t i,j=1 2 φ x i x j (X tdx i tdx j t (2.9 13

15 dbt k dbt k = δ kk dt, dbt k da i t =, da i tda j t =. ( N ( N dxtdx i j t = fk(tdb i t k + da i t f j k (tdb k t = fk(tf i j k (tdt + da j t (2.9 dφ(x t = i,j=1 x i (X tf i k(tdb k t + 2 φ x i x j (X tf i k(tf j k (tdt x i (X tda i t step1 t, T, T > fk i(t fk i (t { = t < t 1 < < t n = T } i, k φ(x t φ(x = {φ(x tm t φ(x tm 1 t} m=1 (2.8 s, t m (1 m n t m 1 s < t t m φ(x t φ(x s = x (X rfk(rdb i k i r s s i,j=1 x i (X rda i r s 2 φ x i x (X rfk(rf i j j k (rdr (2.1 14

16 s, t f i k (t t m 1, t m F s - f i k (:= f i k (t m 1 X i t X i t = X i + = X i + ( m 1 f i k(t l 1 (B k t i t B k t l 1 t + A i t f i k(t l 1 (B k t l B k t l 1 + f i k(t m 1 (B k t B k t m 1 + A i t X i t = X i s + f i k(b k t B k s + (A i t A i s, s < t T (2.11 (2.1 step2 step3 Einstein s, t n t l = t s n l + s, l n Taylor θ = θ l(ω (, 1 φ(x t φ(x s = {φ(x tl φ(x tl 1 } = x (X i t l 1 (Xt i l Xt i l φ 2 x i x (Y l(x i j t l Xt i l 1 (X j t l X j t l 1 Y l := X tl + θ(x tl 1 X tl (2.11 φ(x t φ(x s = I n (1 + I n (2 + I n (3 + I n (4 + I n (5 15

17 I (1 n := I (2 n := I (3 n := 1 2 I (4 n := x (X i t l 1 fk(b i t k l B k t l 1, x (X i t l 1 (A i t l A i t l 1, 2 φ x i x (Y lfk(b i k j t l Bt k l 1 f j k (B k t l Bt k l 1, I (5 n := φ x i x (Y lfk(b i k j t l Bt k l 1 (A j t l A j t l 1, 2 φ x i x (Y l(a i j t l A i t l 1 (A j t l A j t l 1. step3 I (1 n I (2 n f i k s I (3 n 1 2 s x (X rdb k i r in L 2, (2.12 x (X rda i i r ω Ω, (2.13 fkf i j 2 φ k x i x (X rdr in L 2, (2.14 j s I (4 n ω Ω, (2.15 I (5 n ω Ω. (2.16 {n} {n } (2.12-(2.16 a.s.- (2.1 (2.12 I n (1 I (1 n = f i k s Φ i,n (rdb k r Φ i,n (r Φ i,n (r = x i (X t l 1 1 (tl 1,t l (r 16

18 x i Φ i,n(r x (X r1 i (s,t (r (n. L 2 I n (1 x (X t L 2 - (2.12 i (2.13 A i t(ω ω t I n (2 Stieltjes (2.13 (2.15 I n (4 2 φ x i x (Y l j f k B i t k l Bt k l 1 A j t l A j t l 1 sup x R d sup x R d 2 φ x i x j (x 2 φ x i x j (x f k i max 1 l n Bk t l Bt k l 1 A j t l A j t l 1 f k i max 1 l n Bk t l Bt k l 1 sup A j r l A j r l 1. = {s = r < r 1 < < r n = t} s, t Bt k t s, t max 1 l n Bk t l Bt k l 1 (n (2.15 (2.16 max 1 l n Bk t l B k t l 1 max 1 l n Ai t l A i t l 1 (2.14 ψ C b (R Ĩ (3 n Ĩ(3,k,k n := ψ(y l (B k t l B k t l 1 (B k t l B k t l 1 Ĩ (3 n δ kk ψ(x r dr in L 2 (2.17 s 17

19 k, k Z l := (B k t l B k t l 1 (B k t l B k t l 1 (t l t l 1 δ kk { } 2 { E δ kk ψ(y l (t l t l 1 = E Ĩ (3 n J (1 n := J (2 n := 2 Eψ(Y l 2 Zl 2 1 l 1 <l 2 n Eψ(Y l1 Z l1 ψ(y l2 Z l2 EZl 2 = (1 + δkk (t l t l 1 2 J n (1 E sup ψ(x 2 Zl 2 x R d = sup ψ(x 2 (1 + δ kk x R d } 2 ψ(y l Z l = J n (1 + J n (2 (t l t l 1 (n ψ(y l1 Z l1 ψ(x l2 1 Z l2 EZ l2 = Eψ(Y l1 Z l1 ψ(x l2 1Z l2 = J n (2 J n (2 = 2 Eψ(Y l1 Z l1 {ψ(y l2 ψ(x l2 1}Z l2 1 l 1 <l 2 n Schwarz Z l1 Z l2 J n (2 2 E sup ψ(x Z l1 ψ(y l2 ψ(x l2 1 Z l2 1 l 1 <l 2 n x R d 2 sup ψ(x EZl 2 1 Zl 2 2 E{ψ(Y l2 ψ(x l2 1} 2 x R d 2 sup x R d ψ(x 1 l 1 <l 2 n E max {ψ(y l 2 ψ(x l2 1} 2 2 l n sup ψ(x (1 + δ kk (t s E 2 x R d 18 1 l 1 <l 2 n ( t s (1 + δ kk n max 2 l n {ψ(y l 2 ψ(x l2 1} 2 2

20 max 2 l n {ψ(y l 2 ψ(x l2 1} 2 n Ĩ (3 n δ kk ψ(y l (t l t l 1 in L 2 ψ(y l (t l t l 1 n ψ(y l (t l t l 1 Ĩ(3 n δ kk ψ(x r dr L 2 s + Ĩ(3 n δkk s s δ kk ψ(x r dr a.s.- ψ(x r in L 2. ψ(y l (t l t l 1 L 2 ψ(y l (t l t l 1 δ kk ψ(x r dr (2.17 fk i step4 fk i = (f k i(t L2 fk i f i,(n k f i,(n k X i,(n t = X i + X i,(n step3 φ(x (n t = φ(x (n x i (X(n s i,j=1 da i s f i,(n k (sdbs k + A i t x i (X(n s f i,(n k (sdbs k 2 φ x i x j (X(n s f i,(n k (sf j,(n k (sds (2.18 s L 2 19

21 n sup X (n t X t in L 2 t T (2.19 sup X (n t X t a.s. (2.19 t T x i (X(n s x (X s a.s. (2.2 i ( (a+b 2 2(a 2 +b 2, a, b R { t } 2 E x i (X(n s f i,(n k (sdbs k x (X sfk(sdb i k i s { } 2 2E x i (X(n s (f i,(n k (s fk(sdb i s k { ( + 2E x i (X(n s } 2 x (X s fk(sdb i k i s 2 ( 2 sup x R x (x d i f i,(n k fk i 2 L + 2 T x i (X(n 2 x (X f i i k (n ( φ x i x j C 2 φ x i x j (X(n s {f i,(n k L 2 T x (X sda i i s a.s. (sf j,(n k (s fk(sf i j k (s}ds f i,(n k (sf j,(n k (s fk(sf i j k (s ds a.s. (2.21 C ( φ x i x j (X(n s 2 φ x i x (X s j a.s. 2

22 2 φ x i x j (X(n s fk(sf i j k (sds 2 φ x i x (X sfk(sf i j j k (sds a.s. (2.22 (2.21 (2.22 ( a.s. i,j=1 2 φ x i x j (X sf i k(sf j k (sds 21

23 3 3.1, R d Borel α i k (t, x, bi (t, x, 1 i d, 1 k N d N α R d b α = α(t, x := (α i k(t, x 1 i d, 1 k N b = b(t, x := (b i (t, x 1 i d 3.1. (Ω, F, P (F t t N (F t -Brown B t = (B k t 1 k N X t = (X i t 1 i d R d dx t = α(t, X t db t + b(t, X t dt. (3.1 dxt i = αk(t, i X t dbt k + b i (t, X t dt, 1 i d α b 3.2. X = (X t t x R d (3.1 X (Ω, F, P (F t - R d - (i i, k (α i k(t, X t t L 2, (b i (t, X t t L 1 loc(,. ( 2 (ii X t = x + T b i (t, X t dt <, T a.s. α(s, X s db s + Xt i = x i + αk(s, i X s dbs k + 22 b(s, X s ds. (3.2 b i (s, X s ds, 1 i d

24 d N α α 2 := i,k (α i k T > K = K T > (i t, T, x, y R d α(t, x α(t, y + b(t, x b(t, y K x y. (3.3 (ii α(t, x + b(t, x K(1 + x. (3.1 X = (X i t X i L 2, i. T >, T step1 Picard d X (n = (X (n t t,t, n N n = 1 X (1 t = x X (n 1 t X (n 2 t, n 2 X (n t = x + α(s, X (n 1 s db s + b(s, X (n 1 s ds (3.4 (3.4 { α i k (t, X (n 1 t L 2 T, b i (t, X (n 1 t L 1 (, T a.s. n 2 X (n 1 = (X (n 1 t (F t -, (3.5 E sup X (n 1 t 2 < t T (3.6 n = 2 X (n 1 t = x (3.5 (3.6 (3.5 (3.6 X (n 1 t αk i 23

25 Borel (3.5 αk i (t, X(n 1 t F t (ii (3.6 T E α i k T αk(t, i X (n 1 t 2 dt E T E T K 2 E = K 2 E < T (t, X(n 1 t L 2 T T b i (t, X (n 1 t dt T K = KT α(t, X (n 1 t 2 dt {K(1 + X (n 1 t } 2 dt T ( sup s T ( sup s T K(1 + X (n 1 t dt ( 1 + sup X s (n 1 s T ( 1 + sup X s (n 1 s T X s (n 1 + sup X s (n 1 2 dt s T X s (n 1 + sup X s (n 1 2 s T dt < b i (t, X (n 1 t L 1 (, T a.s. (3.5 (3.6 (3.4 X (n t (F t - (3.5 X (n t (3.6 X (n t (a + b + c 3(a + b + c a, b, c R E sup X (n t 2 = E sup x + 2 α(s, X s (n 1 db s + b(s, X s (n 1 ds t T t T 2 3 x 2 + 3E α(s, X s (n 1 db s + 3E b(s, X s (n 1 ds sup t T sup t T 2 M i t = αk(s, i X s (n 1 dbs k Doob 24

26 E sup t T 2 α(s, X s (n 1 db s = E sup 2.16 E(M i t 2 = E (3.7 4E t T {(Mt (Mt d 2 } E sup (Mt sup (Mt d 2 t T t T 4(E(M 1 t E(M d t 2 N αk(s, i X s (n 1 2 ds 4E Schwarz 2 { E sup b(s, X s (n 1 ds E sup t T { T E T E sup X (n t 2 3 x E t T < αk(s, i X s (n 1 2 ds T T = 4E t T ( T E T = T E (3.7 αk(s, i X s (n 1 2 ds α(s, X s (n 1 2 ds } 2 b(s, X s (n 1 ds 1 b(s, X s (n 1 ds ( T 1 2 ds b(s, X s (n 1 2 ds } 2. b(s, X s (n 1 2 ds T α(s, X s (n 1 2 ds + 3T E d X (n, n N 25 b(s, X s (n 1 2 ds

27 step2 X (n n n 3, t, T (3.3 r 2 E sup X r (n X r (n 1 2 2E sup {α(s, X s (n 1 α(s, X s (n 2 }db s r t r t r 2 + 2E sup {b(s, X s (n 1 b(s, X s (n 2 }ds r t 8E α(s, X s (n 1 α(s, X s (n 2 2 ds + 2tE b(s, X s (n 1 b(s, X s (n 2 2 ds C 1 E X (n 1 s X (n 2 s 2 ds. (3.8 C 1 = (8 + 2T K 2 (3.8 C n 2 1 s s1 C 2 := E C n 2 1 C 2 t n 2 (n 2! sn 4 E sup X (2 t X (1 t 2 ds n 3 ds 1 ds. t T sup X (2 t X (1 t 2 C 2 < t T E sup X r (n X r (n 1 2 (C 1 t n 2 C 2 r t (n 2! (3.9 t = T Chebyshev ( P sup X (n t X (n 1 t (C 1T n/4 (C 1 T n 2 C t T (n! n/4 2 (n 2! n=2 (n!1/2 (C 1 T n/2 (3.9 (n! 1/2 (n 2! (C 1T (n 4/2 < Borel-Cantelli P n s.t. n n sup X (n t X (n 1 t < (C 1T n/4 = 1 t T (n! 1/4 26

28 n > m sup X (n t X (m t sup t T k=m+1 X (n t E k=m+1 t T X (k t + X (k 1 t 1 (n! 1/4 (C 1T n/4 (n, m a.s., T { sup 1/2 sup X (n t X (m t 2 E t T X (n t k=m+1 k=m+1 t T k=m+1 E X (k t X (k 1 t } 2 1/2 sup X (k t X (k 1 t 2 t T C 2 (C 1 T k 2 (k 2! 1/2 (n, m L 2 -Cauchy X t E sup X t 2 <, t T E sup X (n t X t 2 (n t T E sup t T T 4E T 4K 2 E 4K 2 T E α(s, X s (n db s α(s, X s (n α(s, X s 2 ds X (n s X s 2 ds α(s, X s db s 2 sup X s (n X s 2 (n s T αk(s, i X s (n dbs k 27 α i k(s, X s db k s in L 2.

29 b i (s, X s (n ds b i (s, X s ds (3.4 n (3.2 t, T a.s. t P ( t, T (3.2 = 1 X t step3 X t, X t (3.1 a.s. τ l := inf{t ; X t X t l} (inf =, l N τ Markov τ 2 E f s dbs k = E = E E τ fs 2 ds fs 2 1,τ ds f 2 s τds τl E X t τl X t τ l 2 C 1 E X s X s 2 ds C 1 E X s τl X s τ l 2 ds 3.4 C 1 E X s τl X s τ l 2 ds, t, T E X t τl X t τ l 2 =, t, T. P (X t τl = X t τ l = 1, t, T X t, X t lim l τ l = 3.4. φ t, t, T φ t C 1 + C 2 φ s ds, t, T, C 1, C 2. φ t C 1 e C2t, t, T 28

30 3.5. (Ω, F, P (F t t (F t -Brown B = (B t t (3.1 (3.2 X = (X t t X (W d, B(W d (3.1 φ = φ(x C 2 b (Rd φ(x t = φ(x i,j=1 x i bi (s, X s ds = φ(x + + ( x i (X sα i k(s, X s db k s 2 φ x i x j (X sα i k(s, X s α j k (s, X sds b i (s, X s x i (X s α i k(s, X s x i (X sdb k s i,j=1 α i k(s, X s α j k (s, X s 2 φ x i x j (X s ds. dφ(x t = α i k(t, X t x i (X tdb k t + A t φ(x t dt. A t φ(x := 1 2 i,j=1 a ij (t, x 2 φ x i x j (x + b i (t, x x i (x a(t, x = (a ij (t, x 1 i,j d := α(t, x t α(t, x a ij := αk(t, i xα j k (t, x 29

31 (F t - φ(x t φ(x A s φ(x s ds (F t - α, b t A := A t t Aφ(x := 1 2 i,j=1 a ij (x 2 φ x i x j (x + b i (x x i (x 3.6. (W d, B(W d P x R A- (i P (w = x = 1. (ii φ C 2 b (Rd M t (φ = M t (w, φ := φ(w t φ(w Aφ(w s ds P (B(W d t - w t, t w W d t ( B(W d t := s>t σ(w r ; r s x R d A- 3.2 Feynman-Kac α, b x R d (3.1 x R d X = (X t t P x E x f C b (R d V, g C b (R d α, b 3.3 (ii 3

32 3.7. u = u(t, x C 1,2 ((, R d C(, R d Cauchy u t = Au + V u + g, t >, x Rd (3.1 u(, x = f(x, x R d (3.11 T > C = C T, p = p T > u(t, x C(1 + x p, t, T, x R d (3.12 { u(t, x = E x f(x t exp } V (X s ds + Feynman-Kac 3.8. X t p > 1, T > E x sup X t p <, x R d t T. T > { } M t := u(t t, X t exp V (X s ds + { s } g(x s exp V (X r dr ds { s } g(x s exp V (X r ds, t, T Xt := T t, Xt d+1 := V (X s ds, ˆX t := (Xt, Xt 1,, Xt d, X d+1 t 31

33 φ(x, x 1,, x d, x d+1 = u(x, x 1,, x d exp(x d+1 dφ( ˆX t = d+1 i= x i ( ˆX t d ˆX i t d+1 i,j= 2 φ x i x j ( ˆX t d ˆX i td ˆX j t d+1 = x ( ˆX i t d ˆX t i φ 2 x i x ( ˆX j t dxtdx i j t i= i,j=1 = u { } t (T t, X t exp V (X s ds ( dt { u t } ( N + x (T t, X t exp V (X i s ds αk(x i t dbt k + b i (X t dt { } + u(t t, X t exp V (X s ds V (X t dt + 1 { 2 u t } ( N 2 x i x (T t, X t exp V (X j s ds αk(x i t α j k (X tdt i,j=1 { u t } ( N = x (T t, X t exp V (X i s ds αk(x i t dbt k + ( u t (T t, X t + Au(T t, X t + V (X t u(t t, X t { } (3.1 dt g(x t exp V (X s ds dm t = { u t } x (T t, X tαk(x i i t exp V (X s ds dbt k σ n := inf{t > ; X t > n} (M t t,t E x M = E x M T σn (3.13 ( =E x u(t, x = u(t, x (3.12 K { } exp V (X s ds dt. 32

34 { M T σn u(t T σ T σn n, X T σn exp V (X s ds} T σn { s } + g(x s exp V (X r dr ds { T C(1 + X T σn p Kds} exp T + ( C 1 + sup X t p exp(kt + KT exp(kt t T { T } K exp Kdr ds 3.8 M T σn (M T σn n (3.13 n (

35 4 Grushin Grushin R 2 { ( L = 1 2 ( } 2 + x 2. (4.1 2 x y R 2 (x, y u t = Lu (4.2 ( p(t, (x, y, (z, v ((x, y, (z, v R 2 f C b (R 2 u(t, (x, y = f(z, vp(t, (x, y, (z, vdzdv R 2 u(t, (x, y u(, = f (4.2 Grushin 2,3 (Ω, F, P (F t t 2 (F t - Brown B t = (Bt k 1 k 2 X (x,y t α((x, y = ( 1 x = (X (x,y,k t 1 k 2 dx t = α(x t db t, X = (x, y α((x, y t α((x, y = Feynman-Kac ( 1 x 2 u(t, (x, y = Ef(X (x,y t (4.2 u(, = f 34

36 4.1. (x, y, (z, v R 2 p(t, (x, y, (z, v = 1 ( ξ 1ξ(v 3 2πt sinh ξ exp 1 y 2 R p(t, (x, y, (z, v u t ξ sinh ξ {(x2 + z 2 cosh ξ 2xz} (4.3 = Lu 4.2. ϕ :, R ϕ(sdb2 s N(, ϕ(s2 ds. ϕ n (t = ϕ(2 n t/2 n 2 n t ϕ n (sdbs 2 = ϕ(i/2 n {Bt (i+1/2 2 n B2 t i/2 n} i= ϕ n(sdb 2 s N(, ϕ n(s 2 ds n /t dξ. ( X (x,y t = x + Bt 1, y + (x + B 1 sdb 2 s Ef(X (x,y t = Ef(X (x.y t x + Bt 1 1 = z e (x z2 /2t dz (4.4 2πt R E x + B 1 t = z x + B 1 t = z 35

37 F 1 Bt 1 (t σ- F 1 2 Bt 2 (t 4.2 ( Ef(X (x,y t F 1 = E f a, y + ϕ(sdbs 2 (a=x+bt 1,ϕ=B2 = f(x + Bt 1 1 (v y2, v exp ( dv R 2πh x t 2h x t h x t = (4.4 Ef(X (x,y t 1 = f(z, ve exp ( R 2 2πh x t R (x + B 1 s 2 ds (v y2 x + B 1 2h x t = z t e 1λu 1 2πa e u2 /2a du = e aλ2 /2, 1 e u2 /2a = 1 e 1λu e aλ2 /2 dλ 2πa 2π R (4.5 Ef(X (x,y 1 exp ( 2πt (4.5 t = f(z, v 1 R 2π e 1λ(v y Ee λ2hx/2 t x + B 1 1 t = z e (x z2 /2t dλdvdz 3 2πt 4,Theorem 5.8.2,p268 Ee λ2 h x t /2 x + B 1 t = z 1 2πt e (x z2 /2t = 1 ( λt 2πt sinh(λt exp λ 2 coth(λt{x2 2xz sech(λt + z 2 } = 1 ( λt 2πt sinh(λt exp 1 λt 2t sinh(λt {(x2 + z 2 cosh(λt 2xz} (4.6 (x z2 dvdz 2t 36

38 ( 1 R 2π e 1λ(v y 1 λt 2πt sinh(λt exp 1 λt 2t sinh(λt {(x2 + z 2 cosh(λt 2xz} dλ = 1 ( 3 e 1ξ(v y/t ξ 2πt R sinh ξ exp 1 ξ/t 2 sinh ξ {(x2 + z 2 cosh ξ 2xz} dξ (ξ = λt = 1 ( ξ 1ξ(v 3 2πt sinh ξ exp 1 ξ y 2 sinh ξ {(x2 + z 2 cosh ξ 2xz} /t dξ R (4.6 p(t, (x, y, (z, v X (x,y t L V 1 = x, V 1, V 2 V 2 = x y V 1, V 2 = V 1 V 2 V 2 V 1 = y V(x, y = {av 1 (x, y + bv 2 (x, y a, b R}, W(x, y = {av 1 (x, y + bv 2 (x, y + cv 1, V 2 (x, y a, b, c, R} { 1, (x = dim V(x, y =, dim W(x, y = 2 2, (x x = p(t, (x, y, (x, y t cosh ξ 1 = 4 sinh 2 (ξ/2, sinh ξ = 2 sinh(ξ/2 cosh(ξ/2, p(t, (x, y, (x, y = 1 3 2πt R ξ sinh ξ exp 37 ( (ξ/2 tanh(ξ/2 x 2 dξ 2t

39 (, y p(t, (, y, (, y = 1 ξ 3 2πt R sinh ξ dξ t p(t, (, y, (, y 1/ t 3 x g(ξ = ξ tanh ξ, ξ R g (ξ = 4ξ + e2ξ e 2ξ (e ξ + e ξ 2 g(ξ ξ = g(ξ = ξ 2 + O(ξ 4, (ξ ( ξ R sinh ξ exp (ξ/2 tanh(ξ/2 x 2 dξ 2t ξ sinh ξ exp ( x2 ξ 2 dξ = 1 2π 4t (t ξ= 8t x 2 R g(t h(t (t lim t g(t/h(t = 1 x (x, y p(t, (x, y, (x, y 1 π x t (t t p(t, (x, y, (x, y 1/t 38

40 1,,, C-H. Chang, D-C. Chang, B. Gaveau, P. Greiner, and H-P. Lee, Geometric analysis on a step 2 Grushin operator, Bull. Inst. Math. Acad. Sinica(New Series 4 (29, O. Calin, D-C Chang, K. Furutani, C. Iwasaki, Heat kernels for elliptic and sub-elliptic operators, Birkhäuser, New York, H. Matsumoto and S. Taniguchi, Stochastic Analysis Itô and Malliavin Calculus in tandem, Cambridge Univ. Press, Cambridge,

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