Quasi-invariant flows under low exponential integrability of divergence on the Wiener space Shizan Fang Universit´ e de Bourgogne City University of HongKong June 29th, 2009 1
Introduction In 1983, A.B. Cruzeiro proved in Equations diff´ erentielles sur l’espace de Wiener et formules de cameron-Martin non lin´ eaires , J. Funct. Anal. 54 (1983), 206–227 that on the Wiener space ( W , H , µ ) for a vector field A : W → H in the Sobolev space D 2 ∞ ( W , H ) if for all λ > 0, � e λ ( | A | H + | div µ ( A ) | ) d µ < + ∞ , ( i ) W � e λ |∇ A | H ⊗ H < + ∞ ( ii ) W then there exists a unique flow of measurable maps U t : W → W such that ( U t ) ∗ µ = K t µ and 2
� t U t ( x ) = x + A ( U s ( x )) ds . 0 This result has been generalized later by several authors. V. Bogachev and E.M. Wolf, Absolutely continuous flows generated by Sobolev class vector fields in finite and infinite dimensions , J. Funct. Anal. 167 (1999), 1–68. G. Peters, Anticipating flows on the Wiener space generated by vector fields of low regularity , J. Funct. Anal. 142 (1996), 129–192. But at the same time, she proved that on R d if A ∈ C 3 and � R d e λ 0 ( | A | + | div γ ( A ) | ) d γ < + ∞ for some λ 0 > 0, then the similar results on R d hold. Notice no condition on the gradient. In this case, the ordinary differential equation with coefficient A admits the unique solution up to the explosion time; so the second condition insures the non-explosion and the existence of density. 3
For the Wiener space case, even though A is smooth in the sense of Malliavin calculus, it is not in general continuous with respect to the Banach norm of W . We have to use a procedure of smoothing and the estimate on e |∇ A | H × H is usually needed. In 1989, R.J. Diperna and P.L. Lions discussed vector fields in W 1 , 1 loc ( R d ) in the paper R.J. DiPerna and P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511–547. No conditions were needed on the e λ |∇ A | . Let’s explain a bit their method. 4
Let A be a C 1 vector field on R d , having bounded derivative. Then the differential equation dX t dt = A ( X t ) , X 0 = x (1) defines a flow of global diffeomorphisms x → X t ( x ) of R d ; the inverse map x → X − 1 ( x ) solves t dX − 1 t = − A ( X − 1 ) , X − 1 = x . (2) t 0 dt Let θ ∈ C 1 ( R d ) and set u t = θ ( X − 1 ). Then u t solves the t transport equation du t u 0 = θ ∈ C 1 . dt + A · ∇ u t = 0 , (3) 5
Now for A belonging to a Sobolev space W 1 , q ( R d ) with q ≥ 1, We say that u ∈ L ∞ ([0 , T ] , L p ( R d )) solves (3) in distribution sense if � � � � − α ′ F u t − α div( FA ) u t dxdt = R d α (0) F u 0 dx (4) [0 , T ] × R d where α ∈ C ∞ c ([0 , T )) and F ∈ C ∞ c ( R d ). An useful concept in this respect is the notion of renormalized solutions : for any β ∈ C 1 b ( R ), β ( u t ) solves again (35). A basic result in DiPerna-Lions’s paper is 6
Theorem Let u n t = u t ∗ χ n . Then u n t satisfies du n t dt + A · ∇ u n t = c n ( u t , Z ) , (5) here c n ( f , A ) = ( D A f ) ∗ χ n − D A ( f ∗ χ n ) and || c n ( f , A ) || L 1 ≤ C || f || L p ( ||∇ A || L q + || div( A ) || L q ) . (6) 7
Gaussian case: Return to the case where A is good and ( X t ) t ∈ R is the flow of diffeomorphisms associated to A : dX t dt = A ( X t ) , X 0 = x . Then for the standard Gaussian measure γ on R d , ( X t ) ∗ γ = K t γ with K t satisfying �� t � p 2 t � � � || K t || p K t = exp div γ ( A )( X − s ) ds , L p ≤ R d exp p − 1 | div γ ( A ) | d γ 0 (7) where div γ is the divergence with respect to the Gaussian measure γ . Suppose that � R d e λ 0 | div γ ( A ) | d γ < + ∞ for some small λ 0 . (8) Then for p > 1 fixed, K t ∈ L p ( γ ) only for t ≤ ( p − 1) λ 0 . p 2 8
Let T > 0 be given. Then for t ∈ [0 , T ], � � R d e λ 0 | div γ ( A ) | d γ, R d K t | log K t | d γ ≤ T || div γ ( A ) || L 2 N (9) where N λ 0 ≥ T . In fact, using the explicit expression given in (7), 4 we have � t log K t ( X t ) = div γ ( A )( X t − s ) ds . 0 Then � � R d K t | log K t | d γ = R d | log K t ( X t ) | d γ � t � ≤ R d | div γ ( A )( X t − s ) | d γ ds . 0 Note by (7) for p = 2 and t ≤ T 0 = λ 0 4 , � R d e 4 T 0 | div γ ( A ) | d γ. || K t || 2 L 2 ≤ (10) 9
Using the property of flow, for 0 ≤ t ≤ T 0 : � � R d | div γ ( A )( X t + T 0 ) | d γ = R d | div γ ( A )( X t ) | K T 0 d γ, which is less, by Cauchy-Schwarz inequality than �� � 1 / 2 R d | div γ ( A )( X t ) | 2 d γ || K T 0 || L 2 , again by Cauchy-Schwarz inequality, less than || div γ ( A ) || L 4 || K t || 1 / 2 L 2 || K T 0 || L 2 . Set B = sup 0 ≤ t ≤ T 0 || K t || 2 L 2 . By induction, we get for t ∈ [0 , T ], � R d | div γ ( A )( X t ) | d γ ≤ || div γ ( A ) || L 2 N B 1 / 2 N · · · B 1 / 2 ≤ || div γ ( A ) || L 2 N B , where N is such that NT 0 ≥ T . Now the estimate (10) leads to (9). 10
Now let Z ∈ D p 1 ( R d , γ ) such that � R d e λ 0 ( | Z | + | div γ ( Z ) | ) d γ < + ∞ for some small λ 0 > 0 . (11) We regularize Z by Ornstein-Uhlenbeck semi-group P ε � R d f ( e ε x + � 1 − e − 2 ε y ) d γ ( y ) . P ε f ( x ) = Take a subsequence ε m → 0 and set B m = P ε m Z . Then for λ < λ 0 small enough, � R m e λ | div γ ( B m ) | d γ < + ∞ . sup (12) m ≥ 1 Let ( X m t ) t ∈ R solve dX m t = B m ( X m X m t ) , 0 = x , dt and K m be the density of ( X m t ) γ with respect to γ . t 11
By second term in (7), for any q > 1, there is a small T 0 such that t || q || K m sup sup L q < + ∞ , (13) m ≥ 1 t ∈ [0 , T 0 ] and for all fixed T > 0, � R d K m t | log K m sup sup t | d γ < + ∞ . (14) m ≥ 1 t ∈ [0 , T ] Let { ℓ 1 , · · · , ℓ d } be the dual basis of R d . Consider u m i ( t , x ) = ℓ i ( X m t ( x )). Then we saw that { u m i ; t ∈ [0 , T ] } solves the transport equations du m i − B m · ∇ u m u m = 0 , i (0) = ℓ i . (15) i dt Using (13), we have 12
� i ( t , x ) | q d γ ( x ) < + ∞ , R d | u m sup sup t ∈ [0 , T ] . (16) m ≥ 1 t ∈ [0 , T ] Let q ′ = p − 1 . Taking q = q ′ and q = 2 q ′ in (16), we see that p { u m i ; m ≥ 1 } and { ( u m i ) 2 ; m ≥ 1 } are bounded in L q ′ ([0 , T ] × R d ). We can choose a subsequence such that i ) 2 → w i weakly in L q ′ as m → + ∞ . u m → v i and ( u m i It is easy to check that v i ( t ) and w i ( t ) solve the transport equation du dt − Z · ∇ u = 0 , (17) with respectively initial condition ℓ i and ℓ 2 i . If the equation (17) has the unique renormalized solution , then v 2 i solves (17) with the initial value ℓ 2 i ; therefore v 2 i = w i . Therefore as m → + ∞ , i ) 2 → v 2 u m → v i and ( u m i weakly, from which we deduce that i � � 2 d γ dt = 0 . � u m � � lim i ( t , x ) − v i ( t , x ) (18) m → + ∞ [0 , T ] × R d 13
Now we define d � X t ( x ) = v i ( t , x ) e i , (19) i =1 where { e 1 , · · · , e d } is the canonical basis of R d . By (18), � t ( x ) − X t ( x ) | 2 d γ dt = 0 . [0 , T ] × R d | X m lim (20) m → + ∞ By (14) and (20), we obtain that for almost all t ∈ [0 , T ], the density K t of ( X t ) ∗ γ with respect to γ exists and � R d K t | log K t | d γ ≤ C T < + ∞ , (21) where C T > 0 is a constant independent of t ∈ [0 , T ]; there exists a small T 0 > 0 such that for almost all t ∈ [0 , T 0 ], � q 2 t � � || K t || q L q ≤ R d exp q − 1 | div γ ( Z ) | d γ. (22) 14
In what follows, we will consider the subsequence such that X m converges to X almost everywhere. Then by (20) and (22), uniformly with respect to [0 , T 0 ], as m → + ∞ � t � t B m ( X m in all L q ( R d , γ ) . s ) ds → Z ( X s ) ds 0 0 Now letting m → + ∞ in � t X m B m ( X m t ( x ) = x + s ( x )) ds ; 0 it holds in L 1 ([0 , T 0 ] × R d ), � t X t ( x ) = x + Z ( X s ( x )) ds . (23) 0 Note that the right hand side of (23) is continuous with respect to t ∈ [0 , T 0 ] for γ -a.e x ∈ R d . Now we redefine � t ˜ X t ( x ) = x + Z ( X s ( x )) ds , for t ∈ [0 , T 0 ] . 0 15
Obviously for t ∈ [0 , T 0 ]: � t ˜ Z ( ˜ X t ( x ) = x + X s ( x )) ds . (24) 0 Now for each t ∈ [0 , T 0 ], the density K t of (˜ X t ) ∗ γ with respect to γ admits the explicit expression �� t � div γ ( Z )(˜ K t ( x ) = exp X − s ) ds , (25) 0 where ˜ X − s solves (24) replacing Z by − Z . Definition For t ∈ [0 , T 0 ], we define X t + T 0 ( x ) = ˜ ˜ X t ( ˜ X T 0 ( x )) , and so on, we obtain { X t ; t ∈ [0 , T ] } . 16
Theorem Let Z ∈ D p 1 ( R d , γ ) with p > 1 and suppose that for a small λ 0 > 0 , � R d e λ 0 ( | Z | + | div γ ( Z ) | ) d γ < + ∞ . Then there exists a unique flow of maps { ˜ X t ; t ∈ [ − T , T ] } such that ( ˜ X t ) ∗ γ = K t γ with K t given in (25) and � t ˜ Z ( ˜ X t ( x ) = x + X s ) ds . 0 See F. Cipriano and A. B. Cruzeiro , J. Diff. Equ. (2005), 183-201. 17
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