aa r X i v : . [ m a t h . P R ] O c t Stochastic flows with reflection
Andrey PILIPENKO
Institute of Mathematics of Ukrainian Academy of Sciences, Dept. of Stochastic Processes, 3Tereschenkovskaya Str., 01601 Kiev-4, UKRAINE. ( [email protected])
Summary:
Some topological properties of stochastic flow ϕ t ( x ) generated by stochastic dif-ferential equation in a R d + with normal reflection at the boundary are investigated. Sobolev differ-entiability in initial condition is received. The absolute continuity of the measure-valued process µ ◦ ϕ − t , where µ ≪ λ d , is studied. Flows generated by SDEs in Euclidean space is a well-studied topic nowa-days. It is well known for example (cf. [1] and ref. therein) that if thecoefficients of SDE are Liphitzian then the SDE generates a flow of homeo-morphisms, if coefficients are of the class C n + ε then SDE generates C n -flowof diffeomorphisms, equations for derivatives are obtained by formal differ-entiation of the SDE etc.Note that the similar questions for SDEs with reflection is much harderto answer. Even the problems about coalescence of two reflecting Brownianmotions [2, 3, 4, 5] or differentiability of the Brownian reflecting flow ( σ ( x ) = const ) [6, 7] need accurate and non-trivial considerations.The article below was published in Reports of Ukrainian Nat.Acad. of Sci.[8] (2005). Only some new references or minor remarks are added.Assume that functions a k : R d + → R d satisfy the Lipschitz condition. Here R d + = R d − × [0 , ∞ ). Consider an SDE in R d + with normal reflection from theboundary: dϕ t ( x ) = a ( ϕ t ( x )) dt + P mk =1 a k ( ϕ t ( x )) dw k ( t )++ nξ ( dt, x ) , t ∈ [0 , T ] ,ϕ ( x ) = x, ξ (0 , x ) = 0 , x ∈ R d + , (1)where { w k ( t ) , k = 1 , . . . , m } are independent Wiener processes, n = (0 , . . . , , R d − × { } , for each fixed x ∈ R d + a process ξ ( t, x )is non-decreasing in t , and ξ ( t, x ) = Z t { ϕ s ( x ) ∈ R d − ×{ }} ξ ( ds, x ) , i.e. ξ ( t, x ) is increasing only on those instants of time when ϕ t ( x ) ∈ R d − × } . Lipschitz property of the coefficients ensures the existence and theuniqueness to the solution of (1), cf. [9].
1. Existence of continuous modification.Theorem 1 [10] . There exists a modification of the processes ϕ t ( x ) , ξ ( t, x ) (it will be denoted in the same way) such that1) for any x ∈ R d + , the pair ( ϕ t ( x ) , ξ ( t, x )) , t ≥ , is a solution of (1);2) for any ω ∈ Ω processes ϕ t ( x ) , ξ ( t, x ) are continuous in a pair of argu-ments ( t, x ) , t ≥ , x ∈ R d + . The Theorem 1 is proved in a way similar to the corresponding proofused for the solution of SDE without reflection, cf. [1], with the use ofKolmogorov’s theorem on existence of continuous modification.It will be assumed further that ϕ t ( x ) , ξ ( t, x ) are already continuous.
2. The joint motion of solutions started from different initialpoints.
It is well known [1] that a solution of an SDE (without reflection)generates a flow of diffeomorphisms. However, the injectivity for reflectingflow can be failed as the following example shows.
Example 1.
Let d = 1 , m = 1 , a = 0 , a = 1 , i.e. ϕ t ( x ) is the reflectedBrownian motion in R started from x ≥ ϕ t ( x ) = x + w ( t ) + ξ ( t, x ) , x ≥ . It is easy to see that ϕ t ( x ) , ξ ( t, x ) is of the form ϕ t ( x ) = w ( t ) − min ≤ s ≤ t w ( s ) , x = 0 ,w ( t ) + x, x > τ ( x ) ≥ t,ϕ t (0) , x > τ ( x ) < t,ξ ( t, x ) = ( − min τ ( x ) ≤ s ≤ t w ( s ) , τ ( x ) < t, , τ ( x ) ≥ t, where τ ( x ) is a moment, when the process x + w ( t ) gets zero for the firsttime.In other words, ϕ t ( x ) is moving as x + w ( t ) before hitting 0, and then amotion of ϕ t ( x ) coincides with the reflected Brownian motion ϕ t (0) startedfrom zero.The similar situation takes place in multi-dimensional space. heorem 2. [10] Denote by τ ( x ) = inf (cid:8) t ≥ ϕ t ( x ) ∈ R d − × { } (cid:9) themoment of the first hitting the hyperplane R d − × { } by a solution startedfrom x ∈ R d + . Then there exists a set Ω of probability 1 such that for all ω ∈ Ω thefollowing statements hold true:1) for all x, y ∈ R d + , x = y and t < max { τ ( x ) , τ ( y ) } the inequality ϕ t ( x ) = ϕ t ( y ) is satisfied;2) for any x ∈ R d + there exists y = y ( x, ω ) ∈ R d − × { } , such that ϕ τ ( x ) ( x ) = ϕ τ ( x ) ( y ) if τ ( x ) < ∞ . Moreover, ϕ t ( x ) = ϕ t ( y ) for t ≥ τ ( x ) . Remark.
Informally this theorem can be formulated in the following way.A particle started from a point x ∈ R d − × (0 , ∞ ) does not hit any otherparticle before getting the hyperplane R d − × { } . At the instant τ ( x ) itcoalesces with some other particle, which started from R d − × { } . After thisboth particles moves together.
3. Characterization of inner and boundary points of random set ϕ t ( R d + ) . Theorem 3.[11]
For almost all ω and all t ∈ [0 , T ] the following equalityof random sets takes place ∂ϕ t ( R d + ) = ϕ t ( ∂ R d + ) = ϕ t { x ∈ R d + : τ ( x ) ≤ t } , where τ ( x ) = inf { s ≥ ϕ s ( x ) ∈ R d − × { }} is the moment of the firsthitting the hyperplane R d − × { } by the solution started from x .Moreover, for all R > Hausdorff measure H d − of the set ∂ϕ t ( R d + ) ∩ { x ∈ R d + : k x k ≤ R } is finite.
4. Differentiability with respect to initial condition.
As in Example 1, there is no reasons to expect that a solution of (1) iscontinuously differentiable in x even when coefficients of the SDE are infinitedifferentiable. However, it can be proved that for any t the mapping x → ϕ t ( x ) belongs to a Sobolev space ∩ p> W p, loc ( R d + , R d ) . The equations for ∇ ϕ t are not classical equations of stochastic analysis.We need the following definition. efinition 1. Let w ( t ) , . . . , w m ( t ) be independent Wiener processes, F t = σ ( w k ( s ) , k = 1 , m, s ≤ t ) , a k : R l × R p → R l , b k : R l × R p → R p , k =0 , . . . , m, and x t be a continuous F t -measurable stochastic process. Considera random measure-valued process ν ( t ) = δ { x ( t )=0 } , where δ is a probabilitymeasure on R , assigned unit mass to a point zero.A pair ( y t , z t ) of F t -adapted processes satisfies the equation ( dy t = a ( y t , z t ) dt + P mk =1 a k ( y t , z t ) dw k ( t ) − y t − dν ( t ) ,dz t = b ( y t , z t ) dt + P mk =1 b k ( y t , z t ) dw k ( t ) , t ≥ , (2)if: 1) y t , t ≥ z t , t ≥ z t = z + R t b ( y s , z s ) ds + P mk =1 R t b k ( y s , z s ) dw k ( s ) , t ≥ ω the set { t ≥ x t = 0 } is contained in { t ≥ y t = 0 } ;5) for any stopping time τ such that x τ = 0 a.s., the following equalityholds true y t = y τ + Z tτ a ( y s , z s ) ds + m X k =1 Z tτ a k ( y s , z s ) dw k ( s )for all t ∈ [ τ, ◦ τ ) , ◦ τ = inf { t ≥ τ : x t = 0 } . Theorem 4. [12]
Assume that functions a k , b k , k = 0 , m satisfy Lipschitzcondition. Then there exists a unique solution of (2) for any non-randominitial condition ( y , z ) . Theorem 5. I. [11] If functions a k : R d + → R d , k = 0 , m satisfyLipschitz condition then for a.a. ω a mapping R d + ∋ u → ϕ t ( u ) ∈ R d belongsto the space ∩ p> W p, loc ( R d + , R d ) for a.a. t ≥ . II. [12]
Assume that functions a k : R d + → R d , k = 0 , m , are contin-uously differentiable and their derivatives are bounded. Suppose also that P mk =1 ( a k,d ( x )) > for all x ∈ R d − × { } , where a k,d is the d -th coordinateof a function a k = ( a k, , . . . , a k,d ) T . hen the Sobolev derivative ∇ ϕ t ( x ) satisfies the SDE d ∇ ϕ t ( x ) = ∇ a ( ϕ t ( x )) ∇ ϕ t ( x ) dt + m P k =1 ∇ a k ( ϕ t ( x )) ∇ ϕ t ( x ) dw k ( t ) −− P ∇ ϕ t − ( x ) n ( dt, x ) , ∇ ϕ ( x ) = 1I , (3) where is an identity matrix, P is a matrix corresponding to the orthoprojec-tion on the d -th coordinate of the space R d , n ( dt, x ) is a point random measuresuch that n ( { t } , x ) = 1 iff ϕ t ( x ) belongs to the hyperplane R d − × { } . Remark.
Equation (3) is understood in the sense of Definition 1. In thiscase we take the d -th coordinate of ϕ t ( x ) as x t , the d -th row of ∇ ϕ t ( x ) as y t ,the first ( d −
1) rows of ∇ ϕ t ( x ) and ϕ t ( x ) as the process z t . Remark.
The process ∇ ϕ t ( x ) can be chosen measurable in t, x, ω. Remark.
The similar result for constant diffusion coefficient was obtainedin [7]. Moreover, it was proved that for all x and a.a. ω the mapping ϕ t iscontinuously differentiable in some neighborhood of x. Let us compare Sobolev differentiability and usual differentiability of themapping ϕ t ( · , ω ) . It is well known that if the diffusion matrix is a constantthen for a.a. ω and all t the mapping x → ϕ t ( x, ω ) satisfies Lipschitz con-dition. Therefore ϕ t ( x ) is differentiable for λ d -a.a. x ∈ G by Rademacher’stheorem [14]. Since the usual and Sobolev derivatives are equal (if they ex-ist), so equation for usual derivative coincides with that for Sobolev. It isnot difficult to prove that the usual derivatives exist not only for a.a. x , a.a. ω, but for all x and a.a. ω . However almost everywhere local continuousdifferentiability is not evident.It should be noted that [7] does not imply that for a.a. ω the mapping x → ϕ t ( x, ω ) is continuously differentiable. Really, for the process fromExample 1: P ( x → ϕ t ( x ) is continuously differentiable) = 0but for each x > P ( x → ϕ t ( x ) is continuously differentiable in some neighborhood of x ) = 1!The fact that R d + ∋ x → ϕ t ( x ) is not continuously differentiable seems tobe typical, because rank ∇ ϕ t ( x ) ≤ d − τ ( x ) ≤ t and ϕ t ( x ) , τ ( x ) > t , oincides with the flow without reflection, so det ∇ ϕ t ( x ) , τ ( x ) > t, k x k ≤ r, is separated from zero for any r > .
5. Absolute continuity of image-measures driven by ϕ t . Let µ be afinite measure in R d + which is absolute continuous w.r.t. Lebesgue measure.Consider a measure-valued process µ t = µ ◦ ϕ − t , t ≥ . Let us introduce a random set O t ( ω ) = { x ∈ R d + : t < τ ( x ) } , where τ ( x ) = inf { s ≥ ϕ s ( x ) ∈ R d − × { }} is the moment of the first hitting thehyperplane R d − × { } by the process ϕ s ( x ). Theorem 6. [13]
For a.a. ω and every t ≥ a measure µ t is representedas a sum of orthogonal measures µ t = µ (cid:12)(cid:12) O t ◦ ϕ − t + µ (cid:12)(cid:12) R d + \ O t ◦ ϕ − t , such thata) the first measure is absolute continuous w.r.t. d -dimensional Lebesguemeasure and the second one is singular;b) the support of measure µ (cid:12)(cid:12) R d + \ O t ◦ ϕ − t is contained in the set ϕ t ( R d − ×{ } ) of the σ -finite ( d − -dimensional Hausdorff measure H d − . The proof of the first part of the theorem follows from [16], and the secondpart follows from Theorem 3.The next theorem gives a sufficient condition that ensures the absolutecontinuity of µ (cid:12)(cid:12) R d + \ O t ◦ ϕ − t with respect to H d − (cid:12)(cid:12) ∂ϕ t ( R d + ) , which is the restrictionof H d − to the set ∂ϕ t ( R d + ). Theorem 7. [13]
Assume that for µ -a.a. x ∈ R d + : P (rank ∇ ϕ t ( x ) ≥ d − , t ≥
0) = 1 . (4) Then with probability 1 the absolute continuity µ (cid:12)(cid:12) R d + \ O t ◦ ϕ − t ≪ H d − (cid:12)(cid:12) ∂ϕ t ( R d + ) (5) holds for all t ≥ . Here µ (cid:12)(cid:12) R d + \ O t , H d − (cid:12)(cid:12) ∂ϕ t ( R d + ) are restrictions of measures µ, H d − to the sets R d + \ O t , ∂ϕ t ( R d + ) , respectively. Remark.
In contrast to Hausdorff measure H d − in R d , its restriction tothe set ∂ϕ t ( R d + ) = ϕ t ( R d + \ O t ) is a σ -finite measure (Theorem 3). So thenotion of absolute continuity in (5) does not require any specification.The proof is provided by using the co-area formula [14] similarly to thecase m = n , cf. [16, 15] he verification of Theorem 7 conditions is quite difficult. Let us givemore simple sufficient conditions that ensure (4). Assume that functions a k , ≤ k ≤ m , are continuously differentiable.Denote by U st ( x ) , s ≤ t , a solution of the following linear SDE: ( dU st ( x ) = ∇ a ( ϕ t ( x )) U st ( x ) dt + P mk =1 ∇ a k ( ϕ t ( x )) U st ( x ) dw k ( t ) , t ≥ s,U ss ( x ) = 1I . Let us represent a random set A = A t ( x ) = { s ∈ [0 , t ] : ϕ ds ( x ) > } asa disjoint union of random intervals A = [ α ( x ) , β ( x )) ∪ ( α ( x ) , β ( x )] ∪ ∞ S k =2 ( α k ( x ) , β k ( x )) , where α ( x ) = 0 , β ( x ) = t. Let P be the same as in The-orem 5. Theorem 8. [13]
Let the conditions of the second part of Theorem 5 besatisfied, t ≥ . Assume that for all k ≥ and a.a. x ∈ U P (rank((1 − P ) U α k β k ( x )(1 − P )) = d −
1) = 1 . (6) Then relation (4) holds true.
Corollary.
Assume that for all x ∈ R d + , s ≤ t : P (cid:16) ϕ dt ( x ) = 0 , det k e U st ( x ) k = 0 (cid:17) = 0 , where e U st ( x ) is the matrix getting outfrom U st ( x ) by deleting last raw and last column. Then (6) fulfills.Observe that the assumption of the Corollary is the requirement of non-hitting zero by two-dimensional Ito process, and this is usually easier to checkthan (6). Example 2.
Let d = 2, then assumptions of the Corollary to the Theorem8 are satisfied if for all x ∈ R , y ∈ R , y = 0 vectors ( a k, ( x )) ≤ k ≤ m and( ∇ y a k, ( x )) ≤ k ≤ m , are linear independent.
6. Flows generated by SDE with reflection in arbitrary set.
Let G ⊂ R d be a closed set with smooth boundary such that the following SDEwith normal reflection on the boundary of G has a unique strong solutiondefined for all x ∈ G, t ≥ dϕ t ( x ) = a ( ϕ t ( x )) dt + P mk =1 a k ( ϕ t ( x )) dw k ( t )++ n ( ϕ t ( x )) ξ ( dt, x ) , t ≥ , x ∈ G,ϕ ( x ) = x, ξ (0 , x ) = 0 , ξ ( t, x ) = R t { ϕ s ( x ) ∈ ∂G } ξ ( ds, x ) , here n ( x ) is the inward normal at a boundary point x ∈ ∂G , ξ ( t, x ) iscontinuous and non-decreasing in t process for every fixed x ∈ G. Assume that for any x ∈ ∂G there exists a neighborhood O ( x ) and C -diffeomorphism α x which transforms the set O ( x ) ∩ G into { x ∈ R d : k x k ≤ , x d ≥ } in such a manner that ∇ α x ( y ) = n ( y ) = (0 , . . . , , , y ∈ ∂G ∩ O ( x ) . Applying a localization of solutions, all statements on flows in half-spacecan be easily generalized (of course with natural changes) to the case of SDEin G. For example, relation (6) will be of the form: P (rank( P ( ϕ β k ) U α k β k ( x ) P ( ϕ α k )) = d −
1) = 1 , (7)where P ( x ) is orthoprojection on the orthogonal complement to n ( x ) where x ∈ ∂G. Example 3.
Let G = { x ∈ R : k x k ≤ } be a unit disk, ϕ t ( x ) , t ≥ , x ∈ G be a Brownian motion in G with reflection on the boundary. I.e. ϕ t ( x ) isa solution of the SDE ( dϕ t ( x ) = dw ( t ) + n ( ϕ t ( x )) ξ ( dt, x ) , t ≥ ,ϕ ( x ) = x, ξ (0 , x ) = 0 , x ∈ G, (8)where w ( t ) is a two-dimensional Wiener process.Let us describe inner and boundary points of the set ϕ t ( G ). Now, thestopping time τ ( x ) from Theorem 2 is of the form τ ( x ) = inf { t ≥ x + w ( t ) ∈ ∂G } . So, the set of inner points of ϕ t ( G ) is equal to { x + w ( t ) : x ∈ G, t < τ ( x ) } . Introduce a stopping time σ = inf { t ≥ k w ( t ) k = 2 } . Observe thatsup x τ ( x ) ≤ σ . Therefore the analogues of Theorems 2,3 imply that for every t ≥ σ the random set ϕ t ( G ) coincides with the nowhere dense set ϕ t ( ∂G ) offinite Hausdorff measure H . Moreover, P ( ∀ t ≥ σ ∀ x, k x k < ∃ y ∈ ∂G, x = y : ϕ t ( x ) = ϕ t ( y )) = 1 . It is interesting to compare this result with [2], where it is proved that anytwo solutions of (8) started from different initial points of G never meet eachother with probability 1, that is ∀ x, y ∈ G, x = y : P ( ∃ t ≥ ϕ t ( x ) = ϕ t ( y )) = 0 . ote that in this example U st ( x ) is an identity matrix, so condition (7)is obviously satisfied. Thus for any absolute continuous measure µ on G wehave the absolute continuity µ (cid:12)(cid:12) G \ O t ◦ ϕ − t ≪ H (cid:12)(cid:12) ∂ϕ t ( G ) (9)with probability 1. In particular, for a.a. ω and all t ≥ σ : µ ◦ ϕ − t ≪ H (cid:12)(cid:12) ∂ϕ t ( G ) . Observe that if G is not a unit disk but any domain with ”nice” boundary,and the boundary does not contain any two perpendicular segments, then (7)and so (4),(9) are also satisfied. Exactly the same condition on the boundaryappears in [6]. Moreover, it can be easily shown that if the boundary containstwo perpendicular segments then (4),(7), and (9) are false. References [1] Kunita H. Stochastic flows and stochastic differential equations.
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