Random Dynamical Systems on Fractal Sets Markus B ohm Institute of - PowerPoint PPT Presentation

Random Dynamical Systems on Fractal Sets Markus B¨ ohm Institute of Mathematics Friedrich-Schiller-University Jena 3 rd Bremen Winter School and Symposium Bremen, 24.03.2015 Markus B¨ ohm (FSU Jena) RDS on Fractal Sets Bremen, 24.03.2015 1 / 15

Outline & Foundations Outline Outline 1 Foundations and analysis on fractals 2 Stochastic partial differential equations in Hilbert spaces 3 A random dynamical system on a p.c.f. fractal Motivation/Example: Consider a function u : R → R + which fulfills the standard ODE u ( t ) = u ( t ) ˙ , t ∈ R , u (0) = u 0 , u 0 ∈ R + . Then u u 0 ( t ) =: ϕ ( u 0 , t ) = u 0 e t is a continuous dynamical system SG ϕ : R + × R → R + . How does the system evolves if we have a random influence ω ∈ Ω? It is possible to consider such RDS A DS here for a fixed time over the SG over fractals? Markus B¨ ohm (FSU Jena) RDS on Fractal Sets Bremen, 24.03.2015 2 / 15

Outline & Foundations Foundations Class of self-similiar fractals - post critcally finite ( p.c.f. ) fractals - { F 1 , F 2 , · · · , F N } collection of contractive similarities on a complete metric space ( X , d ) - there exists a unique nonempty compact subset K of X ( self-similiar set ) that satisfies N � K = F i ( K ) =: F ( K ) , i =1 where existence and uniqueness was shown in [4, Hut81] or [6, Kig01] Approximate K by a sequence of graphs - let x i be the unique fixed point of each map F i of the IFS defining K , then the boundary of K is a finite set V 0 ⊂ { x 1 , x 2 , · · · , x N } ⊂ K . - G 0 is a complete graph on V 0 , inductively define G m := F ( G m − 1 ) with vertices V m = F m ( V 0 ), and V ∗ := � ∞ m =0 V m - the fractal K is the closure of V ∗ w.r.t. d - p.c.f means, there exists a finite subset P in the word space, such that F i ( K ) intersect in only finitely many points, no conditions for symmetry or the ”size” of the F i ( K ) are required Markus B¨ ohm (FSU Jena) RDS on Fractal Sets Bremen, 24.03.2015 3 / 15

Outline & Foundations Foundations Example SG is a p.c.f. fractal which can be approximated by a sequence of graphs via an iterated function system Markus B¨ ohm (FSU Jena) RDS on Fractal Sets Bremen, 24.03.2015 4 / 15

Analysis on Fractals Energy& Hilbert spaces Analysis on a p.c.f. fractal (where the classical derivative isn’t possible) - define an energy form for u , v : V m → R , 1 � � E m ( u , v ) = ( u ( y ) − u ( x ))( v ( y ) − v ( x )) 2 y ∼ mx x ∈ Vm - y ∼ m x means it exists an edge in G m with endpoints x and y and we denote E m ( u ) := E m ( u , u ) - by taking the appropriately renormalized limit of E m ( u ) for u : V ∗ → R (s.t. {E m ( u ) } m ≥ 0 is non-decreasing) we obtain the energy form E ( u ) := m →∞ E m ( u ) lim - if u ∈ C ( K ) := { u : K → R continuous } we write E ( u ) := E ( u | V ∗ ) - domain of the energy F = { u ∈ C ( K ) : E ( u ) < ∞} , F 0 = { u ∈ C ( K ) : E ( u ) < ∞ , u | V 0 = 0 } Borel regular probability measure - let ( K , B ( K ) , µ ) be a probability space with Borel σ -algebra B ( K ) and related Borel regular measure µ - further we assume that: µ ( V ∗ ) = 0 and µ ( A ) > 0 for all non-empty open A ⊂ K - all self-similiar measures are examples for such measures Markus B¨ ohm (FSU Jena) RDS on Fractal Sets Bremen, 24.03.2015 5 / 15

Analysis on Fractals Energy& Hilbert spaces Definition (The space H = L 2 ( K , µ )) We define the function space � | u ( x ) | 2 d µ ( x ) < ∞} , H = L 2 ( K , µ ) = { u : K → R , K clearly functions u must also be measurable w.r.t. the same σ -algebra the measure µ is acting. K | u ( x ) | 2 d µ ( x ). We denote the related norm for u : K → R by � u � 2 2 = � Laplacian on a fractal - defined in terms of energy, for u ∈ F and f ∈ C ( K ) we say ∆ u = f if and only if, � E ( u , v ) = − fv d µ, v ∈ F 0 K - in general: if the equation above holds for f ∈ H , we say u ∈ D (∆ H ) Remarks - H is a separable Hilbert space, F is a Hilbert space with inner product � E ( u ) + � u � 2 E ∗ ( u , v ) := E ( u , v ) + ( u , v ) 2 and norm � u � F = 2 for functions u , v ∈ F - ( E , F ) is a local regular Dirichlet form on H ⇒ F is dense in H , [2, Fuku94] Markus B¨ ohm (FSU Jena) RDS on Fractal Sets Bremen, 24.03.2015 6 / 15

Analysis on Fractals Analytic semigroups Lax-Milgram property The inner product E ∗ ( u , v ) : F × F → R as a bilinear form fulfills the Lax-Milgram property. A bilinear form is said to satisfy the Lax-Milgram property, if the following conditions hold: (i) F and H are Hilbert spaces and F ֒ → H , (ii) E ∗ ( u , v ) is bounded in the F -norm, i.e. ∃ M ≥ 0 s.t. |E ∗ ( u , v ) | ≤ M � u � F � v � F for all u , v ∈ F , and (iii) E ∗ ( u , v ) is coercive, meaning ∃ δ > 0 s.t. E ∗ ( u ) ≥ δ � u � 2 F for all u ∈ F . Theorem If E ∗ ( u , v ) satisfies the Lax-Milgram property and F is dense in H, then the operator A : F → H (e.g. the negative Laplacian) generates an analytic semigroup ( S ( t )) t ≥ 0 . Moreover the operator − A is a sectorial operator. See [8, S.-Y.02] for more information. Markus B¨ ohm (FSU Jena) RDS on Fractal Sets Bremen, 24.03.2015 7 / 15

SPDE on a fractal SPDE Let (Ω , F , P ) be an arbitrary probability space and consider the stochastic partial differential equation (SPDE) for t ∈ R + , du ( t ) = ( Au ( t ) + F ( u )) dt + dW t , (1) u (0) := u 0 ∈ H , where A is the negative Laplacian and the infinitesimal generator of the analytic semigroup S ( t ) t ≥ 0 . We need to define an appropriate nonlinear perturbation term F : H → H and a related H -valued process W t : Ω → H . Definition (Nemytskii operator) Let f : K × R → R , then F : H → H is the Nemytskii operator generated by f , u �→ f ( · , u ( · )). We assume that f is Lipschitz continuous in its second component, therefore F is Lipschitz continuous. For example choose F ( u )[ x ] = f ( x , u ( x )) = g ( x ) sin u ( x ), for all x ∈ K and u , g ∈ C ( K ). Definition ( Q -Wiener process) For a given nonnegative, symmetric operator Q ∈ L ( H ) with finite trace (i.e. tr Q := � k ∈ N � Qe k , e k � H < ∞ ) we consider a H -valued Q Wiener process W t : Ω → H for t ∈ R + on (Ω , F , P ), that is W (0) = 0 P -a.s., W has P -a.s. continuous paths, for all 0 ≤ t 0 < t 1 < ... < t n < ∞ , n ∈ N 0 : the increments W t 0 , W t 1 − W t 0 , . . . , W tn − W tn − 1 are independent random variables and ( W tn − W tn − 1 ) ∼ N (0 , ( t n − t n − 1 ) Q ) for all 0 ≤ t n − 1 < t n , n ∈ N . Markus B¨ ohm (FSU Jena) RDS on Fractal Sets Bremen, 24.03.2015 8 / 15

SPDE on a fractal SPDE Theorem (Mild solution) Under the assumptions above a mild solution of the SPDE (1) is given P -a.s. by � t � t u ( t ) = S ( t ) u 0 + S ( t − r ) F ( u ) dr + S ( t − r ) dW r , 0 0 for all t ∈ R + , like in [5, DaPr-Zab] formulated. How does a version of this solution generates a random dynamcial system? What is a random dynamical system? Heuristic procedure (a) Define MDS and RDS and choose a proper setting, (b) Use a given RDE endowed with an auxiliary process which generates a RDS ψ , (c) Define the Ornstein-Uhlenbeck process Z ( t , ω ), (d) Use the conjugation T between the solutions of the differential equations to generate the RDS to the solution of the SPDE. Definition (Metric Dynamcial System) Let θ : R × Ω → Ω be a family of P -preserving transformations having the following properties: the mapping ( t , ω ) �→ θ t ω is ( B ( R ) ⊗ F , F )-measurable; (1) (2) θ 0 = Id Ω ; θ t + s = θ t ◦ θ s for all t , s ∈ R . (3) Then the quadrupel (Ω , F , P , ( θ t ) t ∈ R ) is called a metric dynamical system. Markus B¨ ohm (FSU Jena) RDS on Fractal Sets Bremen, 24.03.2015 9 / 15

SPDE on a fractal RDS Definition (Random Dynamical System) A random dynamical system is a mapping ϕ : R + × Ω × H → H , ( t , ω, x ) �→ ϕ ( t , ω, x ) , ϕ is ( B ( R + ) ⊗ F ⊗ B ( H ) , B ( H ))-measurable; (1) ϕ (0 , ω, · ) = Id H for all ω ∈ Ω; (2) (3) cocylce property is valid, ϕ ( t + s , ω, x ) = ϕ ( t , θ s ω, ϕ ( s , ω, x )) for all ω ∈ Ω, x ∈ H and s , t ∈ R + . First we change to the equivalent two-sided canonical process W t ( ω ) = ω ( t ), t ∈ R , ω ∈ Ω. To avoid measurability problems and exceptional sets we redefine the probability space (Ω , F , P ) := ( C 0 , B ( C 0 ) , P 0 ) where C 0 := C ( R ; H ), ω (0) = 0 is the path space. We restrict the measure on C 0 , s.t. P 0 describes the unique Wiener measure, which is in addition θ -invariant and P ∗ ( C 0 ) = 1. In the following we consider the Wiener shift θ t : Ω → Ω , ω �→ θ t ω ( · ) := ω ( t + · ) − ω ( t ) for all t ∈ R like [1, Arn] used. Markus B¨ ohm (FSU Jena) RDS on Fractal Sets Bremen, 24.03.2015 10 / 15