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

Random Dynamical Systems on Fractal Sets

Markus B¨

• hm

Institute of Mathematics Friedrich-Schiller-University Jena

3rd Bremen Winter School and Symposium Bremen, 24.03.2015

Markus B¨

• hm (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) = u0 , u0 ∈ R+ . Then uu0(t) =: ϕ(u0, t) = u0et is a continuous dynamical system ϕ : R+ × R → R+. How does the system evolves if we have a random influence ω ∈ Ω? It is possible to consider such RDS

• ver fractals?

SG

A DS here for a fixed time over the SG

Markus B¨

• hm (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

• {F1, F2, · · · , FN} 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 K =

N

• i=1

Fi(K) =: F(K) , where existence and uniqueness was shown in [4, Hut81] or [6, Kig01] Approximate K by a sequence of graphs

• let xi be the unique fixed point of each map Fi of the IFS defining K, then the

boundary of K is a finite set V0 ⊂ {x1, x2, · · · , xN} ⊂ K.

• G0 is a complete graph on V0, inductively define Gm := F(Gm−1) with vertices

Vm = F m(V0), and V∗ := ∞

m=0 Vm

• 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 Fi(K)

intersect in only finitely many points, no conditions for symmetry or the ”size” of the Fi(K) are required

Markus B¨

• hm (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¨

• hm (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 : Vm → R,

Em(u, v) = 1 2

• x∈Vm
• y∼

mx

(u(y) − u(x))(v(y) − v(x))

• y ∼

m x means it exists an edge in Gm with endpoints x and y and we denote Em(u) := Em(u, u)

• by taking the appropriately renormalized limit of Em(u) for u : V∗ → R (s.t. {Em(u)}m≥0 is non-decreasing) we
• btain the energy form

E(u) := lim

m→∞ Em(u)

• 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) < ∞}, F0 = {u ∈ C(K) : E(u) < ∞, u|V0 = 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¨

• hm (FSU Jena)

RDS on Fractal Sets Bremen, 24.03.2015 5 / 15

Analysis on Fractals Energy& Hilbert spaces

Definition (The space H = L2(K, µ)) We define the function space H = L2(K, µ) = {u : K → R,

• K

|u(x)|2 dµ(x) < ∞}, clearly functions u must also be measurable w.r.t. the same σ-algebra the measure µ is acting. We denote the related norm for u : K → R by u2

2 =

• K |u(x)|2 dµ(x).

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) = −

• K

fv dµ, v ∈ F0

• 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, v) := E(u, v) + (u, v)2 and norm uF =

• E(u) + u2

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¨

• hm (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)| ≤ MuFvF for all u, v ∈ F, and (iii) E∗(u, v) is coercive, meaning ∃δ > 0 s.t. E∗(u) ≥ δu2

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¨

• hm (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 + dWt, u(0) := u0 ∈ H , (1) 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 Wt : 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∈NQek , ek H < ∞) we consider

a H-valued Q Wiener process Wt : 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 ≤ t0 < t1 < ... < tn < ∞, n ∈ N0 : the increments Wt0 , Wt1 − Wt0 , . . . , Wtn − Wtn−1 are independent random variables and (Wtn − Wtn−1 ) ∼ N (0, (tn − tn−1)Q) for all 0 ≤ tn−1 < tn, n ∈ N. Markus B¨

• hm (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 u(t) = S(t)u0 + t S(t − r)F(u) dr + t S(t − r) dWr , 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: (1) the mapping (t, ω) → θtω is (B(R) ⊗ F, F)-measurable; (2) θ0 = IdΩ; (3) θt+s = θt ◦ θs for all t, s ∈ R. Then the quadrupel (Ω, F, P, (θt)t∈R) is called a metric dynamical system. Markus B¨

• hm (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), (1) ϕ is (B(R+) ⊗ F ⊗ B(H), B(H))-measurable; (2) ϕ(0, ω, ·) = IdH for all ω ∈ Ω; (3) cocylce property is valid, ϕ(t + s, ω, x) = ϕ(t, θsω, ϕ(s, ω, x)) for all ω ∈ Ω, x ∈ H and s, t ∈ R+ . First we change to the equivalent two-sided canonical process Wt(ω) = ω(t), t ∈ R, ω ∈ Ω. To avoid measurability problems and exceptional sets we redefine the probability space (Ω, F, P) := (C0, B(C0), P0) where C0 := C(R; H), ω(0) = 0 is the path space. We restrict the measure on C0, s.t. P0 describes the unique Wiener measure, which is in addition θ-invariant and P∗(C0) = 1. In the following we consider the Wiener shift θt : Ω → Ω, ω → θtω(·) := ω(t + ·) − ω(t) for all t ∈ R like [1, Arn] used. Markus B¨

• hm (FSU Jena)

RDS on Fractal Sets Bremen, 24.03.2015 10 / 15

SPDE on a fractal RDE& OU process Corresponding to step (b) we want to use this random differential equation (RDE) and transform it into the SPDE (1), dv(t) = Av(t)dt + F(v(t) + Z(θtω))dt, ∀t ∈ R+ v(0) = v0 . (2) Definition (Ornstein-Uhlenbeck process) We consider the following standard stochastic equation, dZ(t) = AZ(t)dt + dWt, Z(0) := 0 . (3) The unique solution is given P-a.s. by Zt = t

0 S(t − r) dWr see [5, DaPr-Zab]. Now by changing to the canonical process

ω, we define a random variable Z(ω) :=

−∞ S(−r) dω(r) and hence we get

Z(θtω) =

−∞

S(t − r)dθtω(r) = S(t)Z(ω) + ω(t) + A t S(t − r)ω(r) dr the stationary Ornstein-Uhlenbeck process . Theorem (Solution of the RDE) Under the assumptions before there exists a global unique solution v : R+ → H with initial value v0 ∈ H for ω ∈ Ω by v(t, ω, v0) = v(t) = S(t)v0 + t S(t − r)F(v(r, ω, v0) + Z(θr ω)) dr . (4) As a conclusion of [1, Theorem 2.2.2, Arn] the solution v generates a random dynamical system (RDS) ψ : R+ × Ω × H → H . Markus B¨

• hm (FSU Jena)

RDS on Fractal Sets Bremen, 24.03.2015 11 / 15

SPDE on a fractal RDS on a p.c.f. fractal We define a conjugation T : Ω × H → H and its (in the second component) inverse T −1 : Ω × H → H by (ω, v) → T(ω, v) = v + Z(ω), (ω, v) → T −1(ω, v) = v − Z(ω), with Z(ω) the random variable related to Ornstein-Uhlenbeck process. If we set u(t) := T(θtω, v(t)) = v(t) + Z(θtω), (5) we obtain by starting with the RDE (2) our initial SPDE (1)! We formulate our final result in terms of this conjugation T such that u ∈ H given by (5) generates a RDS on the fractal. Conclusion/Corollary If v is a solution to the RDE (2) and ψ its related RDS, then ϕ : R+ × Ω × H → H given by ϕ(t, ω, u0) = T(θtω, ψ(t, ω, T −1(ω, u0))), is a RDS generated by the solution u = T(θtω, v) from the SPDE (1) with initial condition u0 := u(0) ∈ H. Markus B¨

• hm (FSU Jena)

RDS on Fractal Sets Bremen, 24.03.2015 12 / 15

Outlook

Ongoing work and future aims determine the global random attractor of the RDS and its Hausdorff dimension consider more general fractal settings allowing similiar operators A, non-linearities F and multiplicative noise study all results over random fractals, ...

Markus B¨

• hm (FSU Jena)

RDS on Fractal Sets Bremen, 24.03.2015 13 / 15

Outlook

Thank you for your attention! Thank you for your attention!

Markus B¨

• hm (FSU Jena)

RDS on Fractal Sets Bremen, 24.03.2015 14 / 15

References

[1] L. Arnold Random Dynamical Systems, Springer, Berlin, 2010. [2] M. Fukushima,Dirichlet Forms and Symmetric Markov Processes, de Gruyter Studies in Mathematics 19, Berlin, 1994. [3] J. Hu, X.Wang, Domains of Dirichlet Forms and Effective Resistance Estimates on p.c.f. Fractals, Studia Math. 177(2), 153-172, 2006. [4] J. E. Hutchinson. Fractals and Self-similarity, Indiana Univ. Math. J., 30(5):713747, 1981. [5] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. [6] J. Kigami, Analysis on Fractals, Cambridge University Press, Cambridge, 2001. [7] A. Klenke, Wahrscheinlichkeitstheorie, Springer, Berlin, 2006. [8] R.Sell, Y.You, Dynamics of Evolutinary Equations, Springer, New York, 2002. [9] R.S. Strichartz, Differential Equations on Fractals, A Tutorial, Princeton University Press, Princeton, 2006.

Markus B¨

• hm (FSU Jena)

RDS on Fractal Sets Bremen, 24.03.2015 15 / 15