Spatial asymptotic behavior of homeomorphic global flows for non-lipschitz SDEs

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1 Bull. Sci. math Satial asymtotic behavior of homeomorhic global flows for non-lischitz SDEs Zongxia Liang Deartment of Mathematical Sciences, Tsinghua University, Beijing 84, China Received 3 October 6 Available online 8 December 6 Abstract Let x φ s,t x be a R d -valued stochastic homeomorhic flow roduced by non-lischitz stochastic differential equation φ s,t x = x + t s σφ s,u x dw u + t s bφ s,u x du, wherew = W,W,... is an infinite sequence of indeendent standard Brownian motions. We first give some estimates of modulus of continuity of {φ s,t }, then rove that the flow φ s,t x, whenx nears infinity, grows slower than Z ex{c ln ln x } for some constant c> and integrable random variable Z via lemma of Garsia Rodemich Rumsey Lemma abbreviated as GRR Lemma imroved by Arnold and Imkeller [L. Arnold, P. Imkeller, Stratonovich calculus with satial arameters and anticiative roblems in multilicative ergodic theory, Stochastic Process. Al ] and moment estimates for one- and two-oint motions. 6 Elsevier Masson SAS. All rights reserved. MSC: rimary 6H, 34F5; secondary 6G7, 37C, 34K5 Keywords: Stochastic homeomorhic flow; Non-Lischitz conditions; GRR Lemma; Satial growth rate. Introduction and main results Let l denote the usual Hilbert sace of R-valued sequences with inner roduct, l, σ : R d R d l and b : R d R d be measurable functions. W = W,W,... is an infinite sequence of indeendent standard Brownian motions on a comlete filtered robability sace This work is suorted by NSFC and SRF for ROCS, SEM. address: zliang@math.tsinghua.edu.cn /$ see front matter 6 Elsevier Masson SAS. All rights reserved. doi:.6/j.bulsci.6..

2 Z. Liang / Bull. Sci. math Ω, F, {F t } t [,], P satisfying the usual conditions. Consider the following stochastic differential equations abbreviated as SDEs dx t x = σ X t x dw t + b X t x dt, X x = x R d.. In this aer we make the following assumtions on coefficients of SDEs.: There exist constants C >, C > and η, e such that H σx σy := d i= σ i x σ i y l C x y ρ η x y, x,y R d ; H bx by C ρ η x y, x,y R d, where the concave function ρ η x on [, is defined by { x log ρ η x = x, x η, η log η + log η x η, x > η.. Recently research on stochastic homeomorhic flow associated to non-lischitz SDE has arisen a great interest. Malliavin, Airault, Ren, Fang and other authors firstly established bounded homeomorhic flow on the grou of diffeomorhisms of the circle and d-dimensional shere S d cf. [,5 7,,,4,7] and references therein, then Liang [3], Zhang [8], Fang, Imkeller and Zhang [8] studied global flows on R d by non-lischitz SDE. They roved that, under hyotheses H and H, SDEs. can roduce a stochastic global homeomorhic flow, that is, a ma φ : [, R d Ω R d such that. φ st x,, t s, solves SDEs. with initial condition X s = x for each s, x R d ;. φ st,ωis a homeomorhism for each s t, ω Ω; 3. φ st,ω= φts,ωfor each s,t, ω Ω; 4. φ su,ω= φ tu,ω φ st,ω s,t,u, ω Ω; 5. s, t φ st,ω is continuous from [, to the homeomorhisms on R d. The flows can be unbounded at infinity. The main urose of the aer is to study satial asymtotic behavior of the homeomorhic flow φ st when the satial variable nears infinity. The first result of this aer rovides some estimates of modulus of continuity for the flow {φ s,t }. We now state the result as follows. Theorem.. Assume that H and H hold. Let T>be fixed. Then there exist constants β>, C>and Φ δ -integrable nonnegative random variable Z such that for all x, y R d with x > y and ω Ω su φ,t x φ,t y t Z max {, x + x α} ex {8δβ ln + ln + x + x α} x y max{, x + x α } ex{ 8dδln + z } + αz α dz..3

3 48 Z. Liang / Bull. Sci. math In articular, for all x R d, ω Ω, we have φ,t x φ,t Z max {, x + x α} { ex 8δβ ln + ln + x + x α}.4 su t where Φ δ x = ex{ δt }x t dt is a Young function on [, and δ is such that Z satisfies EΦ δ Z < +. The function ln + x := max{ln x,} on [, denotes the ositive art of the logarithm, α = ex{ C T C T }. Our second result states that the growth rate of the flow φ st ex{γ ln ln x } for some γ>. The result is the following. at infinity can be at most Theorem.. Assume that H and H hold. Let T>be fixed. Then there exist constants δ> and β>such that the random variables + su s,t T Y = su φ stx { ex } 8δβ ln + ln + x.5 x R d + x and + x { Y = su su x R d s,t T + φ st x ex } 8δβ ln + ln + x.6 are Φ δ -integrable, that is, EΦ δ Y i <+, i =,. Recall that Imkeller and Scheutzow [9] roved similar results under strong Lischitz and regularity conditions imosed on coefficients of SDEs.. The main aim of this aer is to remove the regularity conditions and weaken Lischitz condition. Examle 5. in Section 5 will state that this work is interesting. We will use the line of [9] to rove Theorems.., but the rice to ay is that we have to overcome obstacles in doing moment estimates for one- and two-oint motions, choosing metric dx,y for using GRR lemma etc. because of non-lischitz conditions. We will use Gronwall Bellman Bihari inequalities, recise B D G inequalities for semimartingales imroved by Barlow and Yor [3] and aroaches used in [8,9] to arrive at our conclusions. Our results also seem to show that the globally bounded condition on coefficients in [5] can be droed in our setting. We exect our results would be of interest for theory of stochastic homeomorhic flow associated to non-lischitz SDEs. This aer is organized as follows. Section is to resent some fundamental tools which will be used later. In Section 3 we will rovide some moments estimates for one- and two-oint x motions of {φ,t } and { + φ,t x }. Sections 4 and 5 devote to roving Theorems. and., resectively. In Section 6 we will establish a stochastic flow roduced by non-lischitz SDE and the flow will tend to infinity when the satial variable nears infinity.. Preliminaries Let X,d and M,ρ be two searable metric saces, m a locally finite Borel measure on X, BX.LetΦ : R + R + be increasing, right continuous and Φ =. For any measurable function g : X M we define measurable function g by { ρgx,gy gx,y = dx,y if x y, ifx = y

4 Z. Liang / Bull. Sci. math for x,y X. Firstly, we have the following extension of GRR Lemma see [, Theorem ]: Lemma.. Let f : X M be continuous. If f satisfies V := Φ fx,y mdxmdy < +, X X then for any x,y X, ρ fx,fy 8 su z {x,y} dx,y Φ 4V mk ε z dz, where K ε z denotes the ε-ball centered at z X. Secondly, we resent recise B D G inequalities for semimartingales imroved by Barlow and Yor see [3, Proosition 4.] as follows: Lemma.. There exists a universal constant c such that, for any and any continuous martingale {M t } with M =, we have Mt c M t,. where Mt = su s t M t, M is the quadratic variation of M, denotes the norm in L Ω, F, P. Finally, we give the following three lemmas. Lemma.3. Gronwall Bellman Bihari inequalities, see [6, Theorem.3.] Let u and f be nonnegative continuous functions defined on R + =[, +. Let Wu be a continuous nondecreasing function defined on R + and Wu > on, +.If t ut k + fsw us ds. for t R +, where k is a nonnegative constant, then for t t ut G Gk t + fsds,.3 where Gr = r r dr Ws, r>, r >, G is the inverse function of G and t R + is chosen such that Gk + t fsds DomG, for all t R +, lying in the interval t t. Lemma.4. Assume that f is a concave and increasing function on [, with ρ =, X is a random variable and belongs to L Ω, F, P. Then f X f X.4 for any.

5 5 Z. Liang / Bull. Sci. math Proof. Let gx = f x / for any. Obviously g = and g is an increasing continuous function on [, +. The right derivative of g defined by g + x = f x/ /x / f + x/ is also a nonnegative increasing right continuous function on, +. Hence gx is a concave increasing function on [, with g = for. Alying Jensen s inequality to gx := [fx / ],wehave E [ f X ] = Eg X g E X = [ f X ]..5 So.5 imlies.4. Let ρ η be the concave function defined by.. It is easy to see that the function has the following roerties. Lemma.5. i ρ η x ρ η x for η η and x ; ii x ρ η x + ρ η +x+ for and x. In the end of this section we define an exonential function on [, + by Φ δ x = ex{ δt }x t dt.6 for δ>. Then we have the following estimates on Φ δ and its inverse. Lemma.6. See [9] for Lemma. Let δ> and denote {[ { } ]} K = ex 4δ ln ex c d. Then for t we have a b Φ δ t K ex 4δ ln + t,.7 { π ln t } Φ δ t δ ex..8 4δ 3. Moment estimates for one- and two-oint motions In this section we will give moments estimates for one- and two-oint motions of {φ,t } and x { + φ,t x } for roving Theorems... We assume that flow {φ s,t } satisfies the flowing SDEs, { dφs,tx = σφ s,tx dw t + bφ s,tx dt, φ s,s x = x R d. The first result of this section deals with the flow {φ,t }. 3. Theorem 3.. Assume that H and H hold. Then for any T>, x,y R d and there exists constant C > such that

6 Z. Liang / Bull. Sci. math E su { φ,t x } { ex cc + C T + } + x, 3. t T E su t T { φ,t x φ,t y } ex { 3 hc T + + 3hc C T } x y + x y α 3.3 where α = ex{ 3C T 3c C T } and h = log η. Proof. It suffices to rove Theorem 3. for because of Hölder s inequality. By H and H there exists a nonnegative constant C = C η, σ, b such that for all x R d { di= σ i x C l + x, 3.4 bx C + x. Let Y t x = su s t φ,s x and ψ t x = Y t x for all x R d. Using Lemma. in [3], {M t := t σφ,sx dw s,t } is a continuous martingale and its stochastic contraction can be defined by M t = t σφ,sx ds. Since t φ,t x = x + σ φ,s x t dw s + by 3.4, the following inequality holds for all x R d ψ T x x + su M t + C t T T b φ,s x ds, 3.5 [ + ψs x ] ds. 3.6 By Lemmas.,.4 and 3.4, there exists c> such that for any su t T Consequently, M t c T cc cc cc ψ T x x +cc T σ φ,s x ds T T T + Y s x ds + Y s x ds + ψ s x ds + ψ s x ds. 3.7 T [ + C + ψs x ] ds. 3.8

7 5 Z. Liang / Bull. Sci. math Squaring both sides of 3.8 and using Hölder s inequality, we have ψt x 3 x + 3 [ cc + C T ] T + ψ t x dt. 3.9 It follows immediately from 3.9 that + ψ T x + 3 x + 3[ cc + C T ] T + ψ t x + 3 x dt. 3. Using Gronwall s lemma, ψ T x ex { cc + C T + } + x. 3. Therefore E su t T { φ,t x } { ex cc + C T + } + x. 3. The 3. yields the assertion 3.. We now turn to roving the inequality 3.3. Let Z t x, y = su s t φ,s x φ,s y, ϕ t x, y = Z t x, y for any and x,y R d. Using H, H, Lemmas.,.4 and.5, we have t su [ σ φ,s x σ φ,s y ] dw s t T cc cc cc cc cc Similarly, su t T t T T Z t x, yρ η Zt x, y dt ρ η Z t x, y dt T ρ η Z t x, y dt T T ρ η Z t x, y dt ρ η ϕ t x, y dt. 3.3 [ b φ,s x b φ,s y ] ds C T ρ η ϕt x, y dt. 3.4

8 Z. Liang / Bull. Sci. math The inequalities 3.3, 3.4 and entail that φ,t x φ,t y = x y + + t ϕ T x, y x y +cc + C T t [ σ φ,s x σ φ,s y ] dw s [ b φ,s x b φ,s y ] ds 3.5 T ρ η ϕ t x, y dt ρ η ϕt x, y dt. 3.6 We define ρx = ρ η x + ρ η x for x. ρx is also an increasing concave function on R + by Lemma. in []. Squaring both sides of 3.6, we have ϕt x, y 3 x y + 3 T C T + c C By Lemma.3, ρ ϕ t x, y dt. 3.7 ϕ T x, y G G 3 x y + 3T C T + c C, 3.8 where Gx = x x dy ρy for some x. dy ρ η y for some x. Since the inverse functions G and G of G and Let Gx = x x G are strictly increasing, it is easy to see from 3.8 that ϕ T x, y G G 3 x y + 3T C T + c C. 3.9 If 3 x y η, then since for sufficiently small η and t η log η Gt = log, <t η 3. log t and G t = ex { log η ex{ t} }, t <, 3. we deduce from 3.9 to 3. that ϕ T x, y 3 x y α 3. where α = ex{ 3C T 3c C T }.

9 54 Z. Liang / Bull. Sci. math If 3 x y >η, then since for t>η t Gt = t dy a + hy = a + ht h log, a + ht where a = η and h = log η, G G 3 x y + 3T C T + c C a a + 3h x y h + t ex { 3h C a + ht T + c C T } a η + 3 x y ex { 3h C T + c C T } a 6 x y ex { 3h C T + c C T }. 3.3 The inequalities 3.9 and 3.3 yield ϕ T x, y 6ex { 3h C T + c C T } x y 3.4 for 3 x y >η. Therefore from 3. and 3.4 it follows that for any x,y R d ϕ T x, y ex { 3h C T + c C T + 3 } x y + x y α, 3.5 which imlies { E su φ,t x φ,t y } t T ex { 3 hc T + + 3hc C T } x y + x y α. 3.6 Thus we comlete the roof of Theorem 3.. For t and x R d we define F t x ={ following. x + φ,t x }. The second result of this section is the Theorem 3.. Assume that H and H hold. Then for any T>, x,y R d and we have { } E su t T + φ,t x ex { 3 + c C T + 38C T } + x, 3.7 { E su F t x F t y } t T ex { hc T + 8C T + 6 c T C + } hc x y + x y α 3.8 where α = ex{ 3C T 3c C T } and h = log η.

10 Z. Liang / Bull. Sci. math Proof. It suffices to rove Theorem 3. for. Define Bt = φ,t x and D t = Then t Bt = x + σ Bs t dw s + and the contraction of B is the following b Bs ds db i t db i t = σ i Bs,σj Bs l ds. Let fx= for x R d. Then + x f x i x = x i + x, f x = x j x i + x + 8x ix j + x 3. Alying Itô s formula to stochastic rocess X and function f,wehave D t = t + x t + 4 d Bs, σ Bs dw s + Bs Bs, bbs + Bs ds d t i= j= t σbs + Bs ds B i sb j s σ i Bs, σ j Bs l + Bs 3 ds + Bt. + x Mt A t A t + A 3 t. 3.9 Let J t = su s t D s. It follows from 3.9 that J T + x + su M s + s t By the same way as in 3.7: 3 i= su s t A i s. 3.3 su M s T cc Jt dt, 3.3 s t su s t su s t A s 3C A s C T T J s ds, 3.3 J s ds, 3.33

11 56 Z. Liang / Bull. Sci. math su s t A3 s 4C T We deduce from 3.3 to 3.34 that J T T + x + cc J s ds J t dt T + 8C J s ds. By the same arguments as in 3.: JT ex{ 3c C T + 38C T + 3 } + x. Therefore E su t T { + φ,t x } ex { 3 + c C T + 38C T } + x The roof of 3.7 is comlete. Next we rove 3.8. Since for x,y R d with y > x and t T su t T F t x F t y su t T su t T. x y su + φ,t x t T + φ,t y φ,t x φ,t y + x y su t T y + φ,t y +, 3.35 by using Theorem 3., 3.7 and an inequality: E XYZ 4 E X 4 E Y 4 E Z 4,we have { 4 x 4 4 } { E su t T + φ,t x y 4 4 } E su t T + φ,t y { E su φ,t x φ,t y 4 } t T ex { 3 + hc T + 8C T + 4 c T C + } hc x y + x y α 4. Consequently, ex { hc T + 8C T + 6c T C + } hc x y + x y α Similarly, ex { 3 + c C T + 38C T } x y + x y α. 3.37

12 Z. Liang / Bull. Sci. math The inequalities 3.35, 3.36 and 3.37 imly that su Ft x F t y ex { hc T + 8C T t T Thus + 6c T C + hc } x y + x y α { E su F t x F t y } { ex hc T + 8C T t T The roof of Theorem 3. has been done. 4. Proof of Theorem. + 6 c T C + hc } x y + x y α The urose of this section is to rove Theorem.. Through this and the next section we choose the measure m in GRR Lemma as follows: mdx = g x λdx 4. where λ denotes Lebesgue measure on R d and gx = x d ln + x β, x ; β>issuch that for some constant C 3 > mr d = lim R C 3 [ r d dr R r d ln + r β + ] r d dr r d ln + r β β + C 3 < d Proof of Theorem.. Fix T>wedefine ρ φ x φ y = su φ,t x φ,t y, t T d α x, y = x y + x y α { d T x, y = max d α x, y }, U = Φ δ [ [, R d R d Φ δ ρφ x φ y d T x, y ] mdxmdy, where α = ex{ 3C T 3c C }, [, ; δ will be secified later. Using Theorem 3. and Fubini Theorem, we have EΦ δ U = R d R d [ ] ρφ x ex{ δ φ y }E d mdxmdy d T x, y

13 58 Z. Liang / Bull. Sci. math R d R d [ ex {[ δ + 3 hc T + + 3hc C T ] ] } d mdxmdy π [ mr d δ [3hC T + + 3hc C T ] ] < + rovided that δ [3hC T + + 3hc C T ] >. Therefore V Φ δu < +. Letting fx= + x + x α for x [, +, by Lemma. and.7, we have for x,y R d with x > y ρ φ x, φ y 8 su z {x,y} d T x,y Φ δ 8K ex { 4δ ln + V } su z {x,y} f x y 4V mk ε z dε { ex 4δ ln + m K ε z } dε 4.3 because {, x = y, d T x, y = + x y, < x y, x y + x y α, x y > and d T x, y f x y. Since K ε z ={x R d : d T x, z ε} K ε z := {x R d : f x z ε}={x R d : x z f ε} and λ K ε z = C 4 [f ε] d for some constant C 4 >, we have for z {x,y} m K ε z = g t λdt t K ε z g d T x, + d T x, y λ K ε z g d T x, + d T x, y λ K ε z = C 4 g d T x, + d T x, y [ f ε ] d C 4 g d α x, + d α x, y [ f ε ] d 4.4 where f denotes the inverse of f. Noting that there exists constant C 5 > such that 3 C 4 g d α x, + d α x, y [ f ε ] d C 5 + x + x α d ln + f x + x α β, 4.5 ε we have 4δ ln + 3 4δ ln + C 5 + 8dδln + + x + x α f ε + 8δβ ln + ln + x + x α. 4.6

14 Z. Liang / Bull. Sci. math It follows from that d T x,y { ex 4δ ln + m K ε z } dε } ex {4δ ln + C 5 ex {8δβ ln + ln + x + x α } f x y { ex 8dδln + + x + x α f ε = ex { 4δ ln + C 5 } ex { 8δβ ln + ln + x + x α } x y { ex 8dδln + + x + x α t } dε } d + t + t α C 6 max {, x + x α} ex {8δβ ln + ln + x + x α } x y max{, x + x α } ex{ 8dδln + } + αz α dz 4.7 z where C 6 = ex{ 4δ ln + C 5 }.LetY = ex{ 4δ ln + V }. We deduce from.8 that EΦ δ Y E πδ ex{ln + V } E πδ ex{ln + V}= πδ E + V<+. LetZ = 8KC 6 Y. By the definition.6 of Φ δ, Z is Φ δ -integrable. By 4.3 and 4.7, ρ φ x φ y Z max {, x + x α} ex {8δβ ln + ln + x + x α } x y max{, x + x α } ex{ 8dδln + } + αz α dz. 4.8 z Letting y = in 4.8, since C 7 = 8dδln ex{ + z } + αzα dz<+, ρ φ x φ C 7 Z max {, x + x α} { ex 8δβ ln + ln + x + x α }. 4.9 Therefore the roof of Theorem. follows from 4.8 and Proof of Theorem. The main urose of this section is to rove Theorem.. Since the roof is similar to that of Theorem. in [9], we only give a sketch of the roof here for reader s convenience.

15 6 Z. Liang / Bull. Sci. math Proof of Theorem.. Let Y = +Z +su t T φ,t and choose δ such that δ cc + C T + >. Using 3., Fubini Theorem and Theorem., we have EΦ δy < + and + su φ,t x { Y max, x + x α } { ex 8δβ ln + ln + x + x α } 5. t T for all x R d. By 3.8 and doing the same way as in Theorem., we can find α > and Φ δ -integrable nonnegative random variable Z such that for all x, + φ,t x C 8 + Z ex{ 8δβ ln + ln + x + x α } + x + x α 5. su t T for some constant C 8 >. Let Z = Y + C 8 + Z, then Z is Φ b,δ -integrable nonnegative random variable. If we define y s = φ,s x for x Rd, it is easy to see from 5. and 5. that + φ,t y s Z + ys + y s α { ex 8δβ ln + ln + y s + y s α } 5.3 and + φ,t y s Z + ys + y s α ex{ 8δβ ln + ln + y s + y s α } 5.4 where α = min{α, α}. The inequality 5.4 yields that + y s + y s α + x { Z ex 8δβ ln + ln + y s + y s α }. 5.5 By 5.3 and 5.5, + φ s,t x = + φ,t y s Z { + x ex 8δβ ln + ln + y s + y s α }. 5.6 Using 5.5 and inequality: ln + ln + x ln + ln + e + ln + ln + x for all x, there exists an α> such that ln + ln + + y s + y s α ln + ln + + ln + ln α Z + ln + ln + x. 5.7 It follows from 5.6 and 5.7 that there exists a constant C 9 such that + su s,t T φ s,t x C 9 Z { + x ex } 8δβ ln + ln α + Z { ex 8δβ ln + ln + + x }. 5.8 Choosing large δ> such that Z ex{ 8δβ ln + ln + α Z } is Φ δ-integrable, the roof of.5 follows from 5.8. Noting that φ,s y s = x, by 5.3 and 5.4, we have

16 Z. Liang / Bull. Sci. math φs,t x = + φ,t y s Z + x ex { 8δβ ln + ln + y s + y s α }. 5.9 Using 5.7, there exists a constant C > such that + φ s,t x { } C Z + x ex 8δβ ln + ln α + Z { } ex 8δβ ln + ln + x. 5. It immediately follows from 5. that + x { } su su x R d s,t T + φ s,t x ex 8δβ ln + ln + x } C Z { ex 8δβ ln + ln α + Z. 5. Since Z ex{ 8δβ ln + ln + α Z } is Φ δ-integrable for sufficiently large δ>, the roof of.6 follows from 5.. Thus we comlete the roof of Theorem.. 6. Examle In this section we will use Theorem. in [] or Theorem 5. in [4] to establish an examle satisfying Theorem. to state that our results are otimal. Here C denotes a universal ositive constants and may change from lace to lace. Examle 6.. Let α k = hk + c k3 k for some constants h> and c>, σ k x = sinkx α k and σ k x = coskx α k for any x Rand k {,,...}. Define σx= x ex { x } σ,σ,... x ex { x } σx. Then we have σx σy [ { } { } ] l = x ex x sinkx y ex y sinky α k= k + α k= k Since for sufficient small η> and x y η [ x ex { x } sinkx y ex { y } sinky ] [ { } { } ] x ex x coskx y ex y cosky sinkx sinky, + + ex{η} sinky and k= sinkx sinky C x y log αk x y by Theorem. in [], we have 4 C x y log x y.

17 6 Z. Liang / Bull. Sci. math Similarly, 5 C x y log x y. Therefore σx σy l C x y ρ η x y 6. because of k= <. Thus there exists stochastic flow {φ αk t } satisfying the following equality t φ t x = x + φ s x ex { φs x } σ φ s x dw s. It follows immediately from the last equality that φ t x = x ex { t t ex { φ s x } σ φ s x dw s ex { φ s x } σ φ s x l ds xf t. Since {ft} is an exonential martingale, E{ φ t x } = x +,as x +. By Theorem. + su t [,T ] φ t x Y = su x R + x ex{c ln + ln + x } is Φ δ -integrable random variable for large δ>. Acknowledgements This work is suorted by NSFC and SRF for ROCS, SEM. The author would like to thank both for their generous financial suort. References [] H. Airault, J.R. Ren, Modulus of continuity of the canonical Brownian motion on the grou of diffeomorhisms of the circle, J. Funct. Anal [] L. Arnold, P. Imkeller, Stratonovich calculus with satial arameters and anticiative roblems in multilicative ergodic theory, Stochastic Process. Al [3] M.T. Barlow, M. Yor, Semimartingale inequalities via the Garsia Rodemich Rumsey lemma, and alications to local times, J. Funct. Anal [4] G. Cao, K. He, On a tye of stochastic differential equations driven by countably many Brownian motions, J. Funct. Anal [5] S. Fang, Canonical Brownian motion on the diffeomorhism grou of the circle, J. Funct. Anal [6] S. Fang, T. Zhang, Isotroic stochastic flows of homeomorhisms on S d for the critical Sobolev exonent, J. Math. Pures Al. 6, in ress. }

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