Random dynamical systems and rough paths
Sebastian Riedel
Technische Universit¨atBerlin
Workshop “Rough Paths in Toulouse” INSA Toulouse 19.10.2017
Random dynamical systems and rough paths S. Riedel 1/24 Outline
1 Random dynamical systems: Motivation
2 Random dynamical systems and rough paths
3 Invariant measures for RDEs
Random dynamical systems and rough paths S. Riedel 2/24 Outline
1 Random dynamical systems: Motivation
2 Random dynamical systems and rough paths
3 Invariant measures for RDEs
Random dynamical systems and rough paths S. Riedel 3/24 It proved to be useful in particular when studying the long time behaviour of stochastic systems. Main tool: Multiplicative Ergodic Theorem; yields existence of Lyapunovexponents for linear systems (Oseledets ’68). For non-linear systems, one may look at their linearization, and the MET can still provide information about the local behaviour of the non-linear system (stable manifold theorem, Ruelle ’79).
RDS and MET
The theory of Random dynamical systems (RDS) provides a formalism which can be used to describe a large class of stochastic systems which evolve in time (like dynamical systems do for deterministic time evolutions).
Random dynamical systems and rough paths S. Riedel 4/24 Main tool: Multiplicative Ergodic Theorem; yields existence of Lyapunovexponents for linear systems (Oseledets ’68). For non-linear systems, one may look at their linearization, and the MET can still provide information about the local behaviour of the non-linear system (stable manifold theorem, Ruelle ’79).
RDS and MET
The theory of Random dynamical systems (RDS) provides a formalism which can be used to describe a large class of stochastic systems which evolve in time (like dynamical systems do for deterministic time evolutions). It proved to be useful in particular when studying the long time behaviour of stochastic systems.
Random dynamical systems and rough paths S. Riedel 4/24 For non-linear systems, one may look at their linearization, and the MET can still provide information about the local behaviour of the non-linear system (stable manifold theorem, Ruelle ’79).
RDS and MET
The theory of Random dynamical systems (RDS) provides a formalism which can be used to describe a large class of stochastic systems which evolve in time (like dynamical systems do for deterministic time evolutions). It proved to be useful in particular when studying the long time behaviour of stochastic systems. Main tool: Multiplicative Ergodic Theorem; yields existence of Lyapunovexponents for linear systems (Oseledets ’68).
Random dynamical systems and rough paths S. Riedel 4/24 RDS and MET
The theory of Random dynamical systems (RDS) provides a formalism which can be used to describe a large class of stochastic systems which evolve in time (like dynamical systems do for deterministic time evolutions). It proved to be useful in particular when studying the long time behaviour of stochastic systems. Main tool: Multiplicative Ergodic Theorem; yields existence of Lyapunovexponents for linear systems (Oseledets ’68). For non-linear systems, one may look at their linearization, and the MET can still provide information about the local behaviour of the non-linear system (stable manifold theorem, Ruelle ’79).
Random dynamical systems and rough paths S. Riedel 4/24 Differential of the flow solves a linear equation; existence of Lyanpunovexponents follow from the MET (under certain assumptions):
1 ξ(ω) P − lim log DξYt v ∈ {λl < . . . < λ1}, t→∞ t m 1 ≤ l ≤ m, v ∈ R . Possible to conclude existence of stable (or unstable) manifolds around stationary solutions for the original equation (following Ruelle’s strategy).
Stable manifolds for SDE
Stable manifold theorem for SDE. (Mohammed-Scheutzow ’99) Assume
ξ ξ ξ ξ m dYt = b(Yt ) dt + σ(Yt ) ◦ dBt(ω),Y0 = ξ ∈ R induces stochastic (semi-)flow and solution admits an ergodic invariant measure.
Random dynamical systems and rough paths S. Riedel 5/24 Possible to conclude existence of stable (or unstable) manifolds around stationary solutions for the original equation (following Ruelle’s strategy).
Stable manifolds for SDE
Stable manifold theorem for SDE. (Mohammed-Scheutzow ’99) Assume
ξ ξ ξ ξ m dYt = b(Yt ) dt + σ(Yt ) ◦ dBt(ω),Y0 = ξ ∈ R induces stochastic (semi-)flow and solution admits an ergodic invariant measure. Differential of the flow solves a linear equation; existence of Lyanpunovexponents follow from the MET (under certain assumptions):
1 ξ(ω) P − lim log DξYt v ∈ {λl < . . . < λ1}, t→∞ t m 1 ≤ l ≤ m, v ∈ R .
Random dynamical systems and rough paths S. Riedel 5/24 Stable manifolds for SDE
Stable manifold theorem for SDE. (Mohammed-Scheutzow ’99) Assume
ξ ξ ξ ξ m dYt = b(Yt ) dt + σ(Yt ) ◦ dBt(ω),Y0 = ξ ∈ R induces stochastic (semi-)flow and solution admits an ergodic invariant measure. Differential of the flow solves a linear equation; existence of Lyanpunovexponents follow from the MET (under certain assumptions):
1 ξ(ω) P − lim log DξYt v ∈ {λl < . . . < λ1}, t→∞ t m 1 ≤ l ≤ m, v ∈ R . Possible to conclude existence of stable (or unstable) manifolds around stationary solutions for the original equation (following Ruelle’s strategy).
Random dynamical systems and rough paths S. Riedel 5/24 It¯o-theory not really necessary; only used to make sense of initial equation. Markov property used only to speak about invariant measures. In fact, not really necessary, too, more general concept of invariant measures available in the theory of RDS (cf. below). One of our goals: Build a bridge between the “two cultures” (L. Arnold) Dynamical systems and Stochastic analysis.
Remarks
Remark.
Random dynamical systems and rough paths S. Riedel 6/24 Markov property used only to speak about invariant measures. In fact, not really necessary, too, more general concept of invariant measures available in the theory of RDS (cf. below). One of our goals: Build a bridge between the “two cultures” (L. Arnold) Dynamical systems and Stochastic analysis.
Remarks
Remark. It¯o-theory not really necessary; only used to make sense of initial equation.
Random dynamical systems and rough paths S. Riedel 6/24 One of our goals: Build a bridge between the “two cultures” (L. Arnold) Dynamical systems and Stochastic analysis.
Remarks
Remark. It¯o-theory not really necessary; only used to make sense of initial equation. Markov property used only to speak about invariant measures. In fact, not really necessary, too, more general concept of invariant measures available in the theory of RDS (cf. below).
Random dynamical systems and rough paths S. Riedel 6/24 Remarks
Remark. It¯o-theory not really necessary; only used to make sense of initial equation. Markov property used only to speak about invariant measures. In fact, not really necessary, too, more general concept of invariant measures available in the theory of RDS (cf. below). One of our goals: Build a bridge between the “two cultures” (L. Arnold) Dynamical systems and Stochastic analysis.
Random dynamical systems and rough paths S. Riedel 6/24 Moreover, we will be interested in: equilibrium of the system, invariant measures long time behaviour (convergence towards equilibrium, attractors,...) qualitative difference between random and deterministic system (stabilization, synchronization by noise, ...)
Remark: In general, solution Y to (1) not Markovian.
Long term goal: Prove more general stable manifold theorem for
dYt = b(Yt) dt + σ(Yt) dXt(ω) (1)
with t 7→ Xt rough paths lift of a stochastic process (e.g. fBm).
Random dynamical systems and rough paths S. Riedel 7/24 equilibrium of the system, invariant measures long time behaviour (convergence towards equilibrium, attractors,...) qualitative difference between random and deterministic system (stabilization, synchronization by noise, ...)
Remark: In general, solution Y to (1) not Markovian.
Long term goal: Prove more general stable manifold theorem for
dYt = b(Yt) dt + σ(Yt) dXt(ω) (1)
with t 7→ Xt rough paths lift of a stochastic process (e.g. fBm).
Moreover, we will be interested in:
Random dynamical systems and rough paths S. Riedel 7/24 long time behaviour (convergence towards equilibrium, attractors,...) qualitative difference between random and deterministic system (stabilization, synchronization by noise, ...)
Remark: In general, solution Y to (1) not Markovian.
Long term goal: Prove more general stable manifold theorem for
dYt = b(Yt) dt + σ(Yt) dXt(ω) (1)
with t 7→ Xt rough paths lift of a stochastic process (e.g. fBm).
Moreover, we will be interested in: equilibrium of the system, invariant measures
Random dynamical systems and rough paths S. Riedel 7/24 qualitative difference between random and deterministic system (stabilization, synchronization by noise, ...)
Remark: In general, solution Y to (1) not Markovian.
Long term goal: Prove more general stable manifold theorem for
dYt = b(Yt) dt + σ(Yt) dXt(ω) (1)
with t 7→ Xt rough paths lift of a stochastic process (e.g. fBm).
Moreover, we will be interested in: equilibrium of the system, invariant measures long time behaviour (convergence towards equilibrium, attractors,...)
Random dynamical systems and rough paths S. Riedel 7/24 Remark: In general, solution Y to (1) not Markovian.
Long term goal: Prove more general stable manifold theorem for
dYt = b(Yt) dt + σ(Yt) dXt(ω) (1)
with t 7→ Xt rough paths lift of a stochastic process (e.g. fBm).
Moreover, we will be interested in: equilibrium of the system, invariant measures long time behaviour (convergence towards equilibrium, attractors,...) qualitative difference between random and deterministic system (stabilization, synchronization by noise, ...)
Random dynamical systems and rough paths S. Riedel 7/24 Long term goal: Prove more general stable manifold theorem for
dYt = b(Yt) dt + σ(Yt) dXt(ω) (1)
with t 7→ Xt rough paths lift of a stochastic process (e.g. fBm).
Moreover, we will be interested in: equilibrium of the system, invariant measures long time behaviour (convergence towards equilibrium, attractors,...) qualitative difference between random and deterministic system (stabilization, synchronization by noise, ...)
Remark: In general, solution Y to (1) not Markovian.
Random dynamical systems and rough paths S. Riedel 7/24 Outline
1 Random dynamical systems: Motivation
2 Random dynamical systems and rough paths
3 Invariant measures for RDEs
Random dynamical systems and rough paths S. Riedel 8/24 d Example: Ω = C0(R, R ), θ Wiener shift:
θtω := ω(t + ·) − ω(t).
If (iv) holds, coordinate process has stationary increments.
Metric dynamical systems
Definition.
(Ω, F, P, (θt)t∈R) called metric dynamical system if (i) (ω, t) 7→ θtω is measurable
(ii) θ0 = IdΩ
(iii) θt+s = θt ◦ θs −1 (iv) P = P ◦ θt for all t ∈ R.
Random dynamical systems and rough paths S. Riedel 9/24 Metric dynamical systems
Definition.
(Ω, F, P, (θt)t∈R) called metric dynamical system if (i) (ω, t) 7→ θtω is measurable
(ii) θ0 = IdΩ
(iii) θt+s = θt ◦ θs −1 (iv) P = P ◦ θt for all t ∈ R.
d Example: Ω = C0(R, R ), θ Wiener shift:
θtω := ω(t + ·) − ω(t).
If (iv) holds, coordinate process has stationary increments.
Random dynamical systems and rough paths S. Riedel 9/24 d Ω := C0(R, R ) F := Borel sigma-algebra P := Law of X¯ θ := Wiener shift
If X¯ has stationary increments, can build MDS as follows:
Coordinate process X on this space has same law as X¯ and satisfies the cocycle (or helix) property:
Xt+s(ω) − Xs(ω) = Xt(θsω).
Construction of MDS from processes with stationary increments
d Assume that X¯ is R -valued process defined on some probability ¯ ¯ ¯ ¯ d ¯ space (Ω, F, P) with X(¯ω) ∈ C0(R, R ) for all ω¯ ∈ Ω.
Random dynamical systems and rough paths S. Riedel 10/24 d Ω := C0(R, R ) F := Borel sigma-algebra P := Law of X¯ θ := Wiener shift Coordinate process X on this space has same law as X¯ and satisfies the cocycle (or helix) property:
Xt+s(ω) − Xs(ω) = Xt(θsω).
Construction of MDS from processes with stationary increments
d Assume that X¯ is R -valued process defined on some probability ¯ ¯ ¯ ¯ d ¯ space (Ω, F, P) with X(¯ω) ∈ C0(R, R ) for all ω¯ ∈ Ω. If X¯ has stationary increments, can build MDS as follows:
Random dynamical systems and rough paths S. Riedel 10/24 Coordinate process X on this space has same law as X¯ and satisfies the cocycle (or helix) property:
Xt+s(ω) − Xs(ω) = Xt(θsω).
Construction of MDS from processes with stationary increments
d Assume that X¯ is R -valued process defined on some probability ¯ ¯ ¯ ¯ d ¯ space (Ω, F, P) with X(¯ω) ∈ C0(R, R ) for all ω¯ ∈ Ω. If X¯ has stationary increments, can build MDS as follows: d Ω := C0(R, R ) F := Borel sigma-algebra P := Law of X¯ θ := Wiener shift
Random dynamical systems and rough paths S. Riedel 10/24 Construction of MDS from processes with stationary increments
d Assume that X¯ is R -valued process defined on some probability ¯ ¯ ¯ ¯ d ¯ space (Ω, F, P) with X(¯ω) ∈ C0(R, R ) for all ω¯ ∈ Ω. If X¯ has stationary increments, can build MDS as follows: d Ω := C0(R, R ) F := Borel sigma-algebra P := Law of X¯ θ := Wiener shift Coordinate process X on this space has same law as X¯ and satisfies the cocycle (or helix) property:
Xt+s(ω) − Xs(ω) = Xt(θsω).
Random dynamical systems and rough paths S. Riedel 10/24 (i) ϕ is measurable and (t, ξ) 7→ ϕ(t, ω, ξ) is continuous for all ω ∈ Ω.
(ii) ϕ(0, ω, ·) = IdX for all ω ∈ Ω and
ϕ(t + s, ω, ·) = ϕ(t, θsω, ·) ◦ ϕ(s, ω, ·) “cocycle property”
for all s, t ∈ [0, ∞) and all ω ∈ Ω.
Random dynamical system
Definition. A continuous random dynamical system (RDS) on a topological space X is a metric DS (Ω, F, P, (θt)) with mapping
ϕ: [0, ∞) × Ω × X → X
such that
Random dynamical systems and rough paths S. Riedel 11/24 (ii) ϕ(0, ω, ·) = IdX for all ω ∈ Ω and
ϕ(t + s, ω, ·) = ϕ(t, θsω, ·) ◦ ϕ(s, ω, ·) “cocycle property”
for all s, t ∈ [0, ∞) and all ω ∈ Ω.
Random dynamical system
Definition. A continuous random dynamical system (RDS) on a topological space X is a metric DS (Ω, F, P, (θt)) with mapping
ϕ: [0, ∞) × Ω × X → X
such that (i) ϕ is measurable and (t, ξ) 7→ ϕ(t, ω, ξ) is continuous for all ω ∈ Ω.
Random dynamical systems and rough paths S. Riedel 11/24 Random dynamical system
Definition. A continuous random dynamical system (RDS) on a topological space X is a metric DS (Ω, F, P, (θt)) with mapping
ϕ: [0, ∞) × Ω × X → X
such that (i) ϕ is measurable and (t, ξ) 7→ ϕ(t, ω, ξ) is continuous for all ω ∈ Ω.
(ii) ϕ(0, ω, ·) = IdX for all ω ∈ Ω and
ϕ(t + s, ω, ·) = ϕ(t, θsω, ·) ◦ ϕ(s, ω, ·) “cocycle property”
for all s, t ∈ [0, ∞) and all ω ∈ Ω.
Random dynamical systems and rough paths S. Riedel 11/24 Example: SDEs
Example: ϕ(t, ω, ξ) := φ(0, t, ω, ξ) where
m m φ: [0, ∞[×[0, ∞[×Ω × R → R
solution (semi-)flow to SDE
dY = b(Y ) dt + σ(Y ) ◦ dB
if b, σ sufficiently “nice”.
Random dynamical systems and rough paths S. Riedel 12/24 Therefore, can deduce cocycle property only on sets of full measure which depend on respective time points (crude cocycle property). In the literature, there are perfection theorems available which can be used to obtain perfect, indistinguishable versions of crude cocycles (Arnold-Scheutzow ’95, Scheutzow ’96). Typical assumption: continuity of crude cocycle. Not a problem for a pathwise calculus.
Cocycle property for It¯o-SDEs
Using It¯o’stheory creates nullsets which depend on all data of the equation.
Random dynamical systems and rough paths S. Riedel 13/24 In the literature, there are perfection theorems available which can be used to obtain perfect, indistinguishable versions of crude cocycles (Arnold-Scheutzow ’95, Scheutzow ’96). Typical assumption: continuity of crude cocycle. Not a problem for a pathwise calculus.
Cocycle property for It¯o-SDEs
Using It¯o’stheory creates nullsets which depend on all data of the equation. Therefore, can deduce cocycle property only on sets of full measure which depend on respective time points (crude cocycle property).
Random dynamical systems and rough paths S. Riedel 13/24 Not a problem for a pathwise calculus.
Cocycle property for It¯o-SDEs
Using It¯o’stheory creates nullsets which depend on all data of the equation. Therefore, can deduce cocycle property only on sets of full measure which depend on respective time points (crude cocycle property). In the literature, there are perfection theorems available which can be used to obtain perfect, indistinguishable versions of crude cocycles (Arnold-Scheutzow ’95, Scheutzow ’96). Typical assumption: continuity of crude cocycle.
Random dynamical systems and rough paths S. Riedel 13/24 Cocycle property for It¯o-SDEs
Using It¯o’stheory creates nullsets which depend on all data of the equation. Therefore, can deduce cocycle property only on sets of full measure which depend on respective time points (crude cocycle property). In the literature, there are perfection theorems available which can be used to obtain perfect, indistinguishable versions of crude cocycles (Arnold-Scheutzow ’95, Scheutzow ’96). Typical assumption: continuity of crude cocycle. Not a problem for a pathwise calculus.
Random dynamical systems and rough paths S. Riedel 13/24 Remark.
(2) coincides with L. Arnold’s notion of the cocycle property for group-valued processes. on the first level, property (2) reads
Xs+t(ω) = Xs(ω) + Xt(θsω).
Rough path cocycles
Definition.
An MDS (Ω, F, P, (θt)) together with a rough path valued process X defined on this probability space is called a rough paths cocycle if the cocycle relation
Xs+t(ω) = Xs(ω) ⊗ Xt(θsω) (2)
holds for every s, t and ω ∈ Ω.
Random dynamical systems and rough paths S. Riedel 14/24 (2) coincides with L. Arnold’s notion of the cocycle property for group-valued processes. on the first level, property (2) reads
Xs+t(ω) = Xs(ω) + Xt(θsω).
Rough path cocycles
Definition.
An MDS (Ω, F, P, (θt)) together with a rough path valued process X defined on this probability space is called a rough paths cocycle if the cocycle relation
Xs+t(ω) = Xs(ω) ⊗ Xt(θsω) (2)
holds for every s, t and ω ∈ Ω.
Remark.
Random dynamical systems and rough paths S. Riedel 14/24 on the first level, property (2) reads
Xs+t(ω) = Xs(ω) + Xt(θsω).
Rough path cocycles
Definition.
An MDS (Ω, F, P, (θt)) together with a rough path valued process X defined on this probability space is called a rough paths cocycle if the cocycle relation
Xs+t(ω) = Xs(ω) ⊗ Xt(θsω) (2)
holds for every s, t and ω ∈ Ω.
Remark.
(2) coincides with L. Arnold’s notion of the cocycle property for group-valued processes.
Random dynamical systems and rough paths S. Riedel 14/24 Rough path cocycles
Definition.
An MDS (Ω, F, P, (θt)) together with a rough path valued process X defined on this probability space is called a rough paths cocycle if the cocycle relation
Xs+t(ω) = Xs(ω) ⊗ Xt(θsω) (2)
holds for every s, t and ω ∈ Ω.
Remark.
(2) coincides with L. Arnold’s notion of the cocycle property for group-valued processes. on the first level, property (2) reads
Xs+t(ω) = Xs(ω) + Xt(θsω).
Random dynamical systems and rough paths S. Riedel 14/24 Proof. Straightforward!
RDEs induce RDS
Theorem (Bailleul, R., Scheutzow ’17). If X is a rough paths cocycle, the flow of the rough differential equation
dY = b(Y ) dt + σ(Y ) dX,
induces a continuous cocycle.
Random dynamical systems and rough paths S. Riedel 15/24 RDEs induce RDS
Theorem (Bailleul, R., Scheutzow ’17). If X is a rough paths cocycle, the flow of the rough differential equation
dY = b(Y ) dt + σ(Y ) dX,
induces a continuous cocycle.
Proof. Straightforward!
Random dynamical systems and rough paths S. Riedel 15/24 Theorem (Bailleul, R., Scheutzow). If a geometric rough paths valued process X¯ has stationary increments, there is a rough paths cocycle X with the same law as X¯ .
Proof. d As for R -valued processes.
Existence of rough cocycles
More interesting question: Existence of rough cocycles.
Random dynamical systems and rough paths S. Riedel 16/24 Proof. d As for R -valued processes.
Existence of rough cocycles
More interesting question: Existence of rough cocycles.
Theorem (Bailleul, R., Scheutzow). If a geometric rough paths valued process X¯ has stationary increments, there is a rough paths cocycle X with the same law as X¯ .
Random dynamical systems and rough paths S. Riedel 16/24 Existence of rough cocycles
More interesting question: Existence of rough cocycles.
Theorem (Bailleul, R., Scheutzow). If a geometric rough paths valued process X¯ has stationary increments, there is a rough paths cocycle X with the same law as X¯ .
Proof. d As for R -valued processes.
Random dynamical systems and rough paths S. Riedel 16/24 Proof. May assume w.l.o.g. that X is defined on MDS and satisfies cocycle property. Can check: Also Xε and canonical lift Xε satisfies cocycle property. From convergence, X¯ has stationary increments. Assertion follows by previous theorem. Example. fBm with Hurst parameter H ∈ (1/4, 1/2].
RDEs induce RDS III
Theorem (Bailleul, R., Scheutzow). If X has stationary increments and the iterated integrals of Xε converge towards a rough paths valued process X¯ in law, there is a rough paths cocycle X with the same law as X¯ .
Random dynamical systems and rough paths S. Riedel 17/24 May assume w.l.o.g. that X is defined on MDS and satisfies cocycle property. Can check: Also Xε and canonical lift Xε satisfies cocycle property. From convergence, X¯ has stationary increments. Assertion follows by previous theorem. Example. fBm with Hurst parameter H ∈ (1/4, 1/2].
RDEs induce RDS III
Theorem (Bailleul, R., Scheutzow). If X has stationary increments and the iterated integrals of Xε converge towards a rough paths valued process X¯ in law, there is a rough paths cocycle X with the same law as X¯ .
Proof.
Random dynamical systems and rough paths S. Riedel 17/24 Can check: Also Xε and canonical lift Xε satisfies cocycle property. From convergence, X¯ has stationary increments. Assertion follows by previous theorem. Example. fBm with Hurst parameter H ∈ (1/4, 1/2].
RDEs induce RDS III
Theorem (Bailleul, R., Scheutzow). If X has stationary increments and the iterated integrals of Xε converge towards a rough paths valued process X¯ in law, there is a rough paths cocycle X with the same law as X¯ .
Proof. May assume w.l.o.g. that X is defined on MDS and satisfies cocycle property.
Random dynamical systems and rough paths S. Riedel 17/24 From convergence, X¯ has stationary increments. Assertion follows by previous theorem. Example. fBm with Hurst parameter H ∈ (1/4, 1/2].
RDEs induce RDS III
Theorem (Bailleul, R., Scheutzow). If X has stationary increments and the iterated integrals of Xε converge towards a rough paths valued process X¯ in law, there is a rough paths cocycle X with the same law as X¯ .
Proof. May assume w.l.o.g. that X is defined on MDS and satisfies cocycle property. Can check: Also Xε and canonical lift Xε satisfies cocycle property.
Random dynamical systems and rough paths S. Riedel 17/24 Example. fBm with Hurst parameter H ∈ (1/4, 1/2].
RDEs induce RDS III
Theorem (Bailleul, R., Scheutzow). If X has stationary increments and the iterated integrals of Xε converge towards a rough paths valued process X¯ in law, there is a rough paths cocycle X with the same law as X¯ .
Proof. May assume w.l.o.g. that X is defined on MDS and satisfies cocycle property. Can check: Also Xε and canonical lift Xε satisfies cocycle property. From convergence, X¯ has stationary increments. Assertion follows by previous theorem.
Random dynamical systems and rough paths S. Riedel 17/24 RDEs induce RDS III
Theorem (Bailleul, R., Scheutzow). If X has stationary increments and the iterated integrals of Xε converge towards a rough paths valued process X¯ in law, there is a rough paths cocycle X with the same law as X¯ .
Proof. May assume w.l.o.g. that X is defined on MDS and satisfies cocycle property. Can check: Also Xε and canonical lift Xε satisfies cocycle property. From convergence, X¯ has stationary increments. Assertion follows by previous theorem. Example. fBm with Hurst parameter H ∈ (1/4, 1/2].
Random dynamical systems and rough paths S. Riedel 17/24 Outline
1 Random dynamical systems: Motivation
2 Random dynamical systems and rough paths
3 Invariant measures for RDEs
Random dynamical systems and rough paths S. Riedel 18/24 Invariant measures for RDS
Definition. Let (Ω, F, P, (θt), ϕ) be a RDS on a measurable space (X, B). Then the mapping
Θt :Ω × X → Ω × X
(ω, x) 7→ (θtω, ϕ(t, ω, x))
is called skew product. A probability measure µ on F ⊗ B is called invariant for ϕ if it has −1 first marginal P and if µ ◦ Θt = µ for all t ≥ 0.
Random dynamical systems and rough paths S. Riedel 19/24 (Ω × X, F ⊗ B, µ, (Θt)t≥0) itself MDS. Existence of invariant measures implies existence of a stationary state, i.e. a random variable Z :Ω → X for which ϕ(t, ω, Z(ω)) = Z(θtω). In particular, t 7→ ϕ(t, ·,Z) is stationary. One-to-one correspondence between “Markovian” invariant measures for RDS and (classical) invariant measures for Markov semigroups (Crauel ’91).
We will ask for existence of invariant measures for RDS induced by RDEs
dY = b(Y ) dt + σ(Y ) dX(ω).
Invariant measures II
Remark.
Random dynamical systems and rough paths S. Riedel 20/24 Existence of invariant measures implies existence of a stationary state, i.e. a random variable Z :Ω → X for which ϕ(t, ω, Z(ω)) = Z(θtω). In particular, t 7→ ϕ(t, ·,Z) is stationary. One-to-one correspondence between “Markovian” invariant measures for RDS and (classical) invariant measures for Markov semigroups (Crauel ’91).
We will ask for existence of invariant measures for RDS induced by RDEs
dY = b(Y ) dt + σ(Y ) dX(ω).
Invariant measures II
Remark.
(Ω × X, F ⊗ B, µ, (Θt)t≥0) itself MDS.
Random dynamical systems and rough paths S. Riedel 20/24 One-to-one correspondence between “Markovian” invariant measures for RDS and (classical) invariant measures for Markov semigroups (Crauel ’91).
We will ask for existence of invariant measures for RDS induced by RDEs
dY = b(Y ) dt + σ(Y ) dX(ω).
Invariant measures II
Remark.
(Ω × X, F ⊗ B, µ, (Θt)t≥0) itself MDS. Existence of invariant measures implies existence of a stationary state, i.e. a random variable Z :Ω → X for which ϕ(t, ω, Z(ω)) = Z(θtω). In particular, t 7→ ϕ(t, ·,Z) is stationary.
Random dynamical systems and rough paths S. Riedel 20/24 We will ask for existence of invariant measures for RDS induced by RDEs
dY = b(Y ) dt + σ(Y ) dX(ω).
Invariant measures II
Remark.
(Ω × X, F ⊗ B, µ, (Θt)t≥0) itself MDS. Existence of invariant measures implies existence of a stationary state, i.e. a random variable Z :Ω → X for which ϕ(t, ω, Z(ω)) = Z(θtω). In particular, t 7→ ϕ(t, ·,Z) is stationary. One-to-one correspondence between “Markovian” invariant measures for RDS and (classical) invariant measures for Markov semigroups (Crauel ’91).
Random dynamical systems and rough paths S. Riedel 20/24 Invariant measures II
Remark.
(Ω × X, F ⊗ B, µ, (Θt)t≥0) itself MDS. Existence of invariant measures implies existence of a stationary state, i.e. a random variable Z :Ω → X for which ϕ(t, ω, Z(ω)) = Z(θtω). In particular, t 7→ ϕ(t, ·,Z) is stationary. One-to-one correspondence between “Markovian” invariant measures for RDS and (classical) invariant measures for Markov semigroups (Crauel ’91).
We will ask for existence of invariant measures for RDS induced by RDEs
dY = b(Y ) dt + σ(Y ) dX(ω).
Random dynamical systems and rough paths S. Riedel 20/24 Then ( R 0 e−αs dX (ω) for α < 0 Z(ω) := −∞ s R ∞ −αs − 0 e dXs(ω) for α > 0 is a stationary state.
An explicit example
Example. Consider
dY = αY dt + dX(ω), α ∈ R,X fBm.
Random dynamical systems and rough paths S. Riedel 21/24 An explicit example
Example. Consider
dY = αY dt + dX(ω), α ∈ R,X fBm.
Then ( R 0 e−αs dX (ω) for α < 0 Z(ω) := −∞ s R ∞ −αs − 0 e dXs(ω) for α > 0 is a stationary state.
Random dynamical systems and rough paths S. Riedel 21/24 m Proof. Can find a compact subset of R on which cocycle can be defined; existence follows by standard fixed-point theorems (cf. [Crauel, 2002]).
Existence of invariant measures
Theorem. Let X be a rough paths cocycle. If b and σ have compact support, the cocycle map induced by the RDE
dY = b(Y ) dt + σ(Y ) dX(ω)
admits an invariant measure.
Random dynamical systems and rough paths S. Riedel 22/24 Existence of invariant measures
Theorem. Let X be a rough paths cocycle. If b and σ have compact support, the cocycle map induced by the RDE
dY = b(Y ) dt + σ(Y ) dX(ω)
admits an invariant measure.
m Proof. Can find a compact subset of R on which cocycle can be defined; existence follows by standard fixed-point theorems (cf. [Crauel, 2002]).
Random dynamical systems and rough paths S. Riedel 22/24 References
L. Arnold. Random Dynamical Systems. Springer, 1998. I. Bailleul, S. Riedel, M. Scheutzow. Random dynamical systems, rough paths and rough flows. Journal of Differential Equations, 2017. H. Crauel. Random Probability Measures on Polish Spaces. Taylor and Francis, 2002. S. Mohammed, M. Scheutzow. The stable manifold theorem for stochastic differential equations. The Annals of Probability, 1999.
Random dynamical systems and rough paths S. Riedel 23/24 Thank you.
Random dynamical systems and rough paths S. Riedel 24/24