We formulate and prove a {\it Local Stable Manifold Theorem\/} for stochastic
differential equations (sde's) that are driven by spatial Kunita-type
semimartingales with stationary ergodic increments. Both Stratonovich and
It\^o-type equations are treated. Starting with the existence of a stochastic
flow for a sde, we introduce the notion of a hyperbolic stationary trajectory.
We prove the existence of invariant random stable and unstable manifolds in the
neighborhood of the hyperbolic stationary solution. For Stratonovich sde's, the
stable and unstable manifolds are dynamically characterized using forward and
backward solutions of the anticipating sde. The proof of the stable manifold
theorem is based on Ruelle-Oseledec multiplicative ergodic theory

Michael K. R. Scheutzow

On R

Salah-Eldin A. Mohammed

English

arXiv

We formulate and prove a local stable manifold theorem for stochastic differential equations (SDEs) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itô-type equations are treated. Starting with the existence of a stochastic flow for a SDE, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich SDEs, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating SDE. The proof of the stable manifold theorem is based on Ruelle–Oseledec multiplicative ergodic theory

Mohammed, Salah-Eldin A.

Scheutzow, Michael K. R.

OpenSIUC

THE STABLE MANIFOLD THEOREM FORSTOCHASTIC DIFFERENTIAL EQUATIONS∗Salah-Eldin A. Mohammed‡ and Michael K. R. Scheutzow†We formulate a Local Stable Manifold Theorem for stochastic (ordinary) differentialequations in Euclidean space that are driven by multidimensional Brownian motion. Theresult is stated for Stratonovich stochastic differential equations, but it is also true forstochastic differential equations of Itoˆ-type under somehwat weaker conditions. In fact,the theorem holds for general stochastic differential equations driven by Kunita-type spatialsemimartingales with stationary ergodic increments. Details of the proof of the theoremwill appear eslewhere. The proof uses Ruelle-Oseledec multiplicative ergodic theory. ForStratonovich equations, the stable and unstable manifolds are dynamically characterizedusing forward and backward solutions of the anticipating stochastic differential equation.Consider the stochastic differential equationdx(t) = h(x(t)) dt+m∑i=1gi(x(t)) ◦ dWi(t), (I)where h, gi : Rd → Rd, 1 ≤ i ≤ m, are vector fields on Rd. Suppose that for somek ≥ 1, δ ∈ (0, 1), h is Ck,δb , viz. h has all derivatives Djh, 1 ≤ j ≤ k, continuous andglobally bounded, and Dkh is Ho¨lder continuous with exponent δ. Suppose also thatgi, 1 ≤ i ≤ m, are globally bounded and are in Ck+1,δb . Let W := (W1, · · · ,Wm) be∗ October 27, 1999‡The research of this author is supported in part by NSF Grants DMS-9503702, DMS-9703852 and byMSRI, Berkeley.†The research of this author is supported in part by MSRI, Berkeley.1m-dimensional Brownian motion on Wiener space (Ω,F , P ). Denote by θ : R × Ω → Ωthe canonical Brownian shiftθ(t, ω)(s) := ω(t+ s)− ω(t), t, s ∈ R, ω ∈ Ω.Let φ : R×Rd ×Ω→ Rd be the stochastic flow generated by (I). It is known thatφ satisfies the perfect cocycle propertyφ(t+ s, ·, ω) = φ(t, ·, θ(s, ω)) ◦ φ(s, ·, ω),for all s, t ∈ R and all ω ∈ Ω.Definition.Say that the stochastic differential equation (I) has a stationary trajectory if thereexists an F-measurable random variable Y : Ω→ Rd such thatφ(t, Y (ω), ω) = Y (θ(t, ω)) (1)for all t ∈ R and every ω ∈ Ω. In the sequel, we will always refer to the stationarytrajectory (1) by φ(t, Y ).By the Oseledec Theorem, the linearized cocycle (D2φ(t, Y (ω), ω), θ(t, ω)) admits anon-random finite Lyapunov spectrum defined bylimt→∞1tlog |D2φ(t, Y (ω), ω)(v)|, v ∈ Rd,and taking values in {λi}pi=1 with non-random (finite) multiplicities qi, 1 ≤ i ≤ p, andp∑i=1qi = d.2Definition.A stationary trajectory Y (θ(t, ω)) of (I) is said to be hyperbolic if E log+ |Y (·)| <∞,and if the linearized cocycle (D2φ(t, Y (ω), ω), θ(t, ω)) has a Lyapunov spectrum {λp <· · · < λi+1 < λi < · · · < λ2 < λ1} which does not contain 0.Let {U(ω), S(ω) : ω ∈ Ω} denote the unstable and stable subspaces for the linearizedcocycle (D2φ(t, Y (·), ·), θ(t, ·)).It can be shown that the (possibly anticipating) stationary trajectory φ(t, Y (ω), ω)is a solution of the anticipating Stratonovich stochastic differential equationdφ(t, Y ) = h(φ(t, Y )) dt+m∑i=1gi(φ(t, Y )) ◦ dWi(t), t > 0φ(0, Y ) = Y. (II)Furthermore, we can linearize the stochastic differential equation (I) along thestationary trajectory and then match the solution of the linearized equation with thelinearized cocycle D2φ(t, Y (ω), ω). That is to say, the (possibly non-adapted) processD2φ(t, Y (ω), ω), t ≥ 0, satisfies the associated Stratonovich linearized stochastic differen-tial equationdD2φ(t, Y ) = Dh(φ(t, Y ))D2φ(t, Y ) dt+m∑i=1Dgi(φ(t, Y ))D2φ(t, Y ) ◦ dWi(t), t > 0D2φ(0, Y ) = I.(III)In (III), the symbols D2, D denotes spatial (Fre´chet) derivatives.Similarly, the backward trajectories φ(t, Y ), D2φ(t, Y, ·), t < 0, satisfy the back-ward Stratonovich stochastic differential equations corresponding to (II) and (III) above.Equation (II) and its backward counterpart provide dynamic characterizations of the stableand unstable manifolds in the theorem below.3For any ρ ∈ R+ and x ∈ Rd denote by B(x, ρ) the open ball in Rd, center x andradius ρ. Denote by B¯(x, ρ) the corresponding closed ball. Let C(Rd) be the class of allnon-empty compact subsets of Rd. Give C(Rd) the Hausdorff metric d∗:d∗(A1, A2) := sup{d(x,A1) : x ∈ A2} ∨ sup{d(y,A2) : y ∈ A1}where A1, A2 ∈ C(Rd), and d(x,Ai) := inf{|x − y| : y ∈ Ai}, x ∈ Rd, i = 1, 2. LetB(C(Rd)) be the Borel σ-algebra on C(Rd) with respect to the metric d∗.We now state the local stable manifold theorem for the stochastic differential equa-tion (I) around a hyperbolic stationary solution. Details of the proof are given in [M-S].Theorem. (The Stable Manifold Theorem)Assume that the SDE (I) satisfies the given hypotheses. Suppose φ(t, Y ) is a hy-perbolic stationary trajectory of (I) with E log+ |Y | < ∞. Suppose the linearized cocycle(D2φ(t, Y (ω), ω), θ(t, ω), t ≥ 0) has a Lyapunov spectrum {λp < · · · < λi+1 < λi < · · · <λ2 < λ1}. Define λi0 := max{λi : λi < 0} if at least one λi < 0. If all λi > 0, setλi0 = −∞. (This implies that λi0−1 is the smallest positive Lyapunov exponent of thelinearized flow, if at least one λi > 0; in case all λi are negative, set λi0−1 =∞.)Fix ²1 ∈ (0,−λi0) and ²2 ∈ (0, λi0−1). Then there exist(i) a sure event Ω∗ ∈ F with θ(t, ·)(Ω∗) = Ω∗ for all t ∈ R,(ii) F-measurable random variables ρi, βi : Ω∗ → [0,∞), βi > ρi > 0, i = 1, 2, such thatfor each ω ∈ Ω∗, the following is true:There are Ck,² (² ∈ (0, δ)) submanifolds S˜(ω), U˜(ω) of B¯(Y (ω), ρ1(ω)) andB¯(Y (ω), ρ2(ω)) (resp.) with the following properties:(a) S˜(ω) is the set of all x ∈ B¯(Y (ω), ρ1(ω)) such that|φ(n, x, ω)− Y (θ(n, ω))| ≤ β1(ω) e(λi0+²1)n4for all integers n ≥ 0. Furthermore,lim supt→∞1tlog |φ(t, x, ω)− Y (θ(t, ω))| ≤ λi0for all x ∈ S˜(ω). Each stable subspace S(ω) of the linearized flow D2φ is tangent atY (ω) to the submanifold S˜(ω), viz. TY (ω)S˜(ω) = S(ω). In particular, dim S˜(ω) =dim S(ω) and is non-random.(b) lim supt→∞1tlog[sup{ |φ(t, x1, ω)− φ(t, x2, ω)||x1 − x2| : x1 6= x2, x1, x2 ∈ S˜(ω)}]≤ λi0 .(c) (Cocycle-invariance of the stable manifolds):There exists τ1(ω) ≥ 0 such thatφ(t, ·, ω)(S˜(ω)) ⊆ S˜(θ(t, ω)), t ≥ τ1(ω).AlsoD2φ(t, Y (ω), ω)(S(ω)) = S(θ(t, ω)), t ≥ 0.(d) U˜(ω) is the set of all x ∈ B¯(Y (ω), ρ2(ω)) with the property that|φ(−n, x, ω)− Y (θ(−n, ω))| ≤ β2(ω) e(−λi0−1+²2)nfor all integers n ≥ 0. Alsolim supt→∞1tlog |φ(−t, x, ω)− Y (θ(−t, ω))| ≤ −λi0−1.for all x ∈ U˜(ω). Furthermore, U(ω) is the tangent space to U˜(ω) at Y (ω). Inparticular, dim U˜(ω) = dim U(ω) and is non-random.(e) lim supt→∞1tlog[sup{ |φ(−t, x1, ω)− φ(−t, x2, ω)||x1 − x2| : x1 6= x2, x1, x2 ∈ U˜(ω)}]≤ −λi0−1.5(f) (Cocycle-invariance of the unstable manifolds):There exists τ2(ω) ≥ 0 such thatφ(−t, ·, ω)(U˜(ω)) ⊆ U˜(θ(−t, ω)), t ≥ τ2(ω).AlsoD2φ(−t, Y (ω), ω)(U(ω)) = U(θ(−t, ω)), t ≥ 0.(g) The submanifolds U˜(ω) and S˜(ω) are transversal, viz.Rd = TY (ω)U˜(ω)⊕ TY (ω)S˜(ω).(h) The mappingsΩ→ C(Rd), Ω→ C(Rd),ω 7→ S˜(ω) ω 7→ U˜(ω)are (F ,B(C(Rd)))-measurable.Assume, in addition, that h, gi, 1 ≤ i ≤ m have bounded derivatives of all orders.Then the local stable and unstable manifolds S˜(ω), U˜(ω) are C∞.Remarks.(i) The stable manifold theorem still holds for the Stratonovich SDE (I) if the boundedcondition on the g′is is replaced by the somewhat weaker requirement that thefunctionsRd 3 x 7→m∑l=1∂gil(x)∂xjgjl (x) ∈ Rare in Ck,δb for each 1 ≤ i, j ≤ d and some k ≥ 1, δ ∈ (0, 1).(ii) The stable manifold theorem holds for the Itoˆ SDEdx(t) = h(x(t)) dt+m∑i=1gi(x(t)) dWi(t), (II)6where h, gi : Rd → Rd, 1 ≤ i ≤ m, are in Ck,δb .Reference[M-S] Mohammed, S.-E. A., and Scheutzow, M. K. R., The Stable Manifold Theorem forStochastic Differential Equations, The Annals of Probability, Vol. 27, No. 2, (1999),615-652.Mathematical Sciences Research Institute (Berkeley) and Technical University of BerlinEmail: salah@math.siu.edu, ms@math.tu-berlin.deWeb page: http://salah.math.siu.edu7

The Stable Manifold Theorem for Stochastic Differential Equations

. We formulate and prove a Local Stable Manifold Theorem for stochastic differential equations (sde&apos;s) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Ito-type equations are treated. Starting with the existence of a stochastic flow for a sde, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich sde&apos;s, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde. The proof of the stable manifold theorem is based on Ruelle-Oseledec multiplicative ergodic theory. 1. Introduction.  Consider the following Stratonovich and Ito stochastic differential equations (sde&apos;s) on R  d  :  dOE(t) =  ffi  F (ffidt; OE(t)); t ? s OE(s) = x  9 ? = ? ;  (S) dOE(t) = F (dt; OE(t)); t ? s OE(s) = x  9 = ;  (I) defined on a f..

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The Stable Manifold Theorem For Stochastic Differential Equations

. We formulate and prove a Local Stable Manifold Theorem for stochastic differential equations (sde&apos;s) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Ito-type equations are treated. Starting with the existence of a stochastic flow for a sde, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich sde&apos;s, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde. The proof of the stable manifold theorem is based on Ruelle-Oseledec multiplicative ergodic theory. 1. Introduction.  Consider the following Stratonovich and Ito stochastic differential equations (sde&apos;s) on R  d  : dOE(t) =  ffi  F (ffidt; OE(t)); t ? s OE(s) = x  )  (S) dOE(t) = F (dt; OE(t)); t ? s OE(s) = x  )  (I) defined on a filtered proba..

The Stable Manifold Theorem for Stochastic Differential Equations

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