Convergence to equilibrium for rough differential equations Samy Tindel Purdue University Barcelona GSE Summer Forum – 2017 Joint work with Aurélien Deya (Nancy) and Fabien Panloup (Angers) Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 1 / 23
Outline Setting and main result 1 Convergence to equilibrium for diffusion processes 2 Poincaré inequality Coupling method Elements of proof 3 Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 2 / 23
Outline Setting and main result 1 Convergence to equilibrium for diffusion processes 2 Poincaré inequality Coupling method Elements of proof 3 Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 3 / 23
Definition of fBm Definition 1. A 1-d fBm is a continuous process X = { X t ; t ∈ R } such that X 0 = 0 and for H ∈ (0 , 1): X is a centered Gaussian process 2 ( | s | 2 H + | t | 2 H − | t − s | 2 H ) E [ X t X s ] = 1 d -dimensional fBm: X = ( X 1 , . . . , X d ), with X i independent 1-d fBm Variance of increments: E [ | δ X j st | 2 ] ≡ E [ | X j t − X j s | 2 ] = | t − s | 2 H Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 4 / 23
Examples of fBm paths H = 0 . 35 H = 0 . 5 H = 0 . 7 Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 5 / 23
System under consideration Equation: dY t = b ( Y t ) dt + σ ( Y t ) dX t , t ≥ 0 (1) Coefficients: x ∈ R d �→ σ ( x ) ∈ R d × d smooth enough σ = ( σ 1 , . . . , σ d ) ∈ R d × d invertible σ − 1 ( x ) bounded uniformly in x X = ( X 1 , . . . , X d ) is a d -dimensional fBm, with H > 1 3 Resolution of the equation: Thanks to rough paths methods ֒ → Limit of Wong-Zakai approximations Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 6 / 23
Illustration of ergodic behavior Equation with damping: dY t = − λ Y t dt + dX t Simulation: For 2 values of the parameter λ 0.8 4 3 0.4 2 1 0.0 0 −0.4 −1 0 2 4 6 8 10 0 2 4 6 8 10 Figure: H = 0 . 7, d = 1, λ = 0 . 1 Figure: H = 0 . 7, d = 1, λ = 3 Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 7 / 23
Coercivity assumption for b Hypothesis: for every v ∈ R d , one has � v , b ( v ) � ≤ C 1 − C 2 � v � 2 v ) v ( b Interpretation of the hypothesis: Outside of a compact K ⊂ R d , K ⊂ R 2 b ( v ) ≃ − λ v with λ > 0 Arbitrary behavior Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 8 / 23
Ergodic results for equation (1) Brownian case: If X is a Brownian motion and b coercive Exponential convergence of L ( X t ) to invariant measure µ Markov methods are crucial See e.g Khashminskii, Bakry-Gentil-Ledoux Fractional Brownian case: If X is a fBm and b coercive Markov methods not available Existence and uniqueness of invariant measure µ , when H > 1 3 → Series of papers by Hairer et al. ֒ Rate of convergence to µ : ◮ When σ ≡ Id : Hairer ◮ When H > 1 2 and further restrictions on σ : Fontbona–Panloup Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 9 / 23
Main result (loose formulation) Theorem 2. Let H > 1 3 , equation dY t = b ( Y t ) dt + σ ( Y t ) dX t Y unique solution with initial condition µ 0 µ unique invariant measure Then for all ε > 0 we have: t ) − µ � tv ≤ c ε t − ( 1 �L ( Y µ 0 8 − ε ) Remark: Subexponential (non optimal) rate of convergence This might be due to the correlation of increments for X Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 10 / 23
Outline Setting and main result 1 Convergence to equilibrium for diffusion processes 2 Poincaré inequality Coupling method Elements of proof 3 Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 11 / 23
Outline Setting and main result 1 Convergence to equilibrium for diffusion processes 2 Poincaré inequality Coupling method Elements of proof 3 Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 12 / 23
Poincaré and convergence to equilibrium Theorem 3. Let X be a diffusion process. We assume: µ is a symmetrizing measure, with Dirichlet form E Poincaré inequality: Var µ ( f ) ≤ α E ( f ) Then the following inequality is satisfied: − 2 t � � Var µ ( P t f ) ≤ exp Var µ ( f ) α Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 13 / 23
Comments on the Poincaré approach Remarks: Theorem 3 asserts that 1 ( d ) − → µ, exponentially fast X t The proof relies on identity ∂ t P t = LP t 2 ֒ → Hard to generalize to a non Markovian context One proves Poincaré with Lyapunov type techniques 3 ֒ → Coercivity enters into the picture Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 14 / 23
Outline Setting and main result 1 Convergence to equilibrium for diffusion processes 2 Poincaré inequality Coupling method Elements of proof 3 Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 15 / 23
A general coupling result Proposition 4. Consider: Two processes { Z t ; t ≥ 0 } and { Z ′ t ; t ≥ 0 } A coupling (ˆ Z , ˆ Z ′ ) of ( Z , Z ′ ) We set � t ≥ 0; ˆ Z u = ˆ � τ = inf Z ′ u for all u ≥ t Then we have: �L ( Z t ) − L ( Z ′ t ) � tv ≤ 2 P ( τ > t ) Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 16 / 23
Comments on the coupling method Proposition 4 is general, does not assume a Markov setting 1 ֒ → can be generalized (unlike Poincaré) In a Markovian setting 2 ֒ → Merging of paths a soon as they touch a 0 Merged path Path starting from a 0 Path starting from a 1 a 1 τ In our case 3 → We have to merge both Y , Y ′ and the noise ֒ Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 17 / 23
Outline Setting and main result 1 Convergence to equilibrium for diffusion processes 2 Poincaré inequality Coupling method Elements of proof 3 Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 18 / 23
Algorithmic view of the coupling Merging positions of Y x and Y µ by coupling Success yes Stick solutions Y x and Y µ by a Girsanov shift of the noise no yes Success Estimate for the merging time τ no Wait in order to to have Y x and Y µ back in a compact Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 19 / 23
Merging positions (1) Simplified setting: We start at t = 0, and consider d = 1 Effective coupling: We wish to consider y 0 , y 1 and h such that We have dy 0 t = b ( y 0 t ) dt + σ ( y 0 t ) dX t dy 1 t = b ( y 1 t ) dt + σ ( y 1 t ) dX t + h t dt Merging condition: y 0 0 = a 0 , y 1 0 = a 1 and y 0 1 = y 1 1 Computation of the merging probability: Through Girsanov’s transform involving the shift h Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 20 / 23
Merging positions (2) Generalization of the problem: We wish to consider a family { y ξ , h ξ ; ξ ∈ [0 , 1] } such that We have dy ξ t = b ( y ξ t ) dt + σ ( y ξ t ) dX t + h ξ t dt Merging condition: h 0 ≡ 0 y ξ y 0 1 = y 1 0 = a 0 + ξ ( a 1 − a 0 ) , 1 , Remark: Here y has to be considered as a function of 2 variables t and ξ Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 21 / 23
Merging positions (3) Solution of the problem: Consider a system with tangent process � ξ dy ξ � b ( y ξ 0 d η η � dt + σ ( y ξ t = t ) − t ) dX t t d ξ t = b ′ ( y ξ t ) ξ t dt + σ ′ ( y ξ t ) ξ t dX t and initial condition y ξ 0 = a 0 + ξ ( a 1 − a 0 ), ξ 0 = a 1 − a 0 Heuristics: A simple integrating factor argument shows that ∂ ξ y ξ ∂ ξ y ξ t = ξ t (1 − t ) , and thus 1 = 0 Hence y ξ solves the merging problem Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 22 / 23
Merging positions (4) Rough system under consideration: for t , ξ ∈ [0 , 1] � ξ � � dy ξ b ( y ξ 0 d η η dt + σ ( y ξ t = t ) − t ) dX t t d ξ t = b ′ ( y ξ t ) ξ t dt + σ ′ ( y ξ t ) ξ t dX t Then y ξ 1 does not depend on ξ ! Difficulties related to the system: t �→ y t is function-valued 1 Unbounded coefficients, thus local solution only 2 Conditioning = ⇒ additional drift term with singularities 3 Evaluation of probability related to Girsanov’s transform 4 Samy T. (Purdue) Convergence to equilibrium Barcelona 2017 23 / 23
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