Stochastic Stochastic parabolic and wave equations with geometric constraints

Transcription

1 Stochastic Stochastic parabolic and wave equations with geometric constraints Zdzisław Brzeźniak, University of York, UK based on joint research with M. Ondreját (Prague), B. Goldys & T. Jegaraj (Sydney), A. de Bouard (Paris) and A. Prohl (Tübingen) Iasi, 6th July, 212

2 Outline Prerequisites from Differential Geometry Prerequisites from Stochastic Analysis Stochastic geometric wave equations Stochastic geometric heat equations Stochastic Landau-Lifshitz equations with applications to ferromagnetism

3 Prerequisites from Differential Geometry Prerequisites from Differential Geometry Let M be compact Riemanian manifold of dimension m without boundary which we assume is isometrically embedded into an euclidean space R d. In particular, the scalar product in the tangent space T m, where m M, is equal to the restriction of the scalar product in R d. m M u, v T m M n T m M, i.e. n R d T m M

4 Prerequisites from Differential Geometry Prerequisites from Differential Geometry Assume that M R n : y : R R n Newton s second law ÿ = f (y, ẏ)

5 Prerequisites from Differential Geometry Prerequisites from Differential Geometry Assume that M R n : y : R R n Newton s second law ÿ = f (y, ẏ) P Ty M[ÿ] = P Ty M[f (y, ẏ)] D t ẏ = P Ty M[f (y, ẏ)] Covariant derivative

6 Prerequisites from Differential Geometry Prerequisites from Differential Geometry The second fundamental form γ : (a, b) M X(t) T γ(t) M Ẋ(t) R n D t X = P Tγ(t) MẊ A γ (X, γ) = P (TγM) Ẋ A p : T p M T p M (T p M), Ẋ = D t X + A γ (X, γ) p M

7 Prerequisites from Differential Geometry Prerequisites from Differential Geometry The second fundamental form γ : (a, b) M X(t) T γ(t) M Ẋ(t) R n D t X = P Tγ(t) MẊ A γ (X, γ) = P (TγM) Ẋ A p : T p M T p M (T p M), Ẋ = D t X + A γ (X, γ) p M

8 Prerequisites from Differential Geometry Prerequisites from Differential Geometry The second fundamental form γ : (a, b) M X(t) T γ(t) M Ẋ(t) R n D t X = P Tγ(t) MẊ A γ (X, γ) = P (TγM) Ẋ A p : T p M T p M (T p M), Ẋ = D t X + A γ (X, γ) p M

9 Prerequisites from Differential Geometry Prerequisites from Differential Geometry The second fundamental form γ : (a, b) M X(t) T γ(t) M Ẋ(t) R n D t X = P Tγ(t) MẊ A γ (X, γ) = P (TγM) Ẋ A p : T p M T p M (T p M), Ẋ = D t X + A γ (X, γ) p M

10 Prerequisites from Differential Geometry Prerequisites from Differential Geometry The second fundamental form γ : (a, b) M X(t) T γ(t) M Ẋ(t) R n D t X = P Tγ(t) MẊ A γ (X, γ) = P (TγM) Ẋ A p : T p M T p M (T p M), Ẋ = D t X + A γ (X, γ) p M

11 Prerequisites from Differential Geometry Prerequisites from Differential Geometry The second fundamental form γ : (a, b) M X(t) T γ(t) M Ẋ(t) R n D t X = P Tγ(t) MẊ A γ (X, γ) = P (TγM) Ẋ A p : T p M T p M (T p M), p M Ẋ = D t X + A γ (X, γ)

12 Prerequisites from Differential Geometry Prerequisites from Differential Geometry The second fundamental form γ : (a, b) M X(t) T γ(t) M Ẋ(t) R n D t X = P Tγ(t) MẊ A γ (X, γ) = P (TγM) Ẋ A p : T p M T p M (T p M), Ẋ = D t X + A γ (X, γ) p M Special case: if X(t) = γ(t), then γ(t) = D t γ(t) + A γ(t) ( γ(t), γ(t) )

13 Prerequisites from Stochastic Analysis Wiener process Assume that (Ω, F, P) is a probability space with complete filtration F = (F t ) t. An R d -valued standard F-Wiener process is a family W = (W t ) t of R d -valued process such that W =, W t W s is F s independent for t > s, W t is an N(, ti) random variable, i.e. gaussian with mean and variance t, for t >. In other words, the density of W t is the function p(t, ), where p(t, x) is the fundamental solution of the heat equation t u = 1 2 u = 1 2 d xi x i u i=1

14 Prerequisites from Stochastic Analysis Itô integral T ξ(s)dw (s) := N ξ(s i 1 ) ( W (s i ) W (s i 1 ) ) i=1 for an adapted L(R d, R k )-valued step process ξ = N i=1 ξ(s i 1)1 [si 1,s i ), = s < s 1 < < s N = T.

15 Prerequisites from Stochastic Analysis Itô integral T ξ(s)dw (s) := N ξ(s i 1 ) ( W (s i ) W (s i 1 ) ) i=1 for an adapted L(R d, R k )-valued step process ξ = N i=1 ξ(s i 1)1 [si 1,s i ), = s < s 1 < < s N = T. Itô isometry (with T2 being the Hilbert-Schmidt norm) T T E ξ(s)dw (s) 2 = E ξ(s) 2 T 2 ds

16 Prerequisites from Stochastic Analysis Itô integral T ξ(s)dw (s) := N ξ(s i 1 ) ( W (s i ) W (s i 1 ) ) i=1 for an adapted L(R d, R k )-valued step process ξ = N i=1 ξ(s i 1)1 [si 1,s i ), = s < s 1 < < s N = T. Itô isometry (with T2 being the Hilbert-Schmidt norm) E T ξ(s)dw (s) 2 = E T ξ(s) 2 T 2 ds This can be generalized two fold: (i) Replace R d by a Hilbert space K with ONB (e j ) j=1, so we get a cylindrical Wiener process W (t) = i=1 w j(t)e j, where (w j ) j=1 are independent standard R-valued Wiener process, (ii) Replace R k by a Hilbert space (or even a Banach space of sufficient geometrical smoothness, so called 2-smooth) E.

17 Prerequisites from Stochastic Analysis Itô integral If K, resp. E, is a Hilbert, resp. 2-smooth Banach space, then T ξ(s)dw (s) := N ξ(s i 1 ) ( W (s i ) W (s i 1 ) ) i=1

18 Prerequisites from Stochastic Analysis Itô integral If K, resp. E, is a Hilbert, resp. 2-smooth Banach space, then T ξ(s)dw (s) := N ξ(s i 1 ) ( W (s i ) W (s i 1 ) ) i=1 Burkholder inequality (with T2 being the γ-radonifying norm), if 1 < p <, t E sup ξ(s)dw (s) p C p (T )E [ T ξ(s) 2 T 2 ds ] p/2 t [,T ]

19 Prerequisites from Stochastic Analysis Itô integral If K, resp. E, is a Hilbert, resp. 2-smooth Banach space, then T ξ(s)dw (s) := N ξ(s i 1 ) ( W (s i ) W (s i 1 ) ) i=1 Burkholder inequality (with T2 being the γ-radonifying norm), if 1 < p <, t E sup ξ(s)dw (s) p C p (T )E [ T ξ(s) 2 T 2 ds ] p/2 t [,T ] Naturally extended to progressively measurable γ(k, E)-valued process.

20 Prerequisites from Stochastic Analysis Itô and Stratonovich differential equations Itô integral allows one to define a solution to a SDEs dx = g(x) dw + f (x) dt (1) as a solution to the integral equation (in which the 1 st integral is the Itô one) t t x(t) = x() + g(x(s)) dw(s) + f (x(s)) ds, t. (2) A solution to (2) is not stable with respect to the piecewise linear approximations but a solution to Stratonovich problem, where t g(x(s)) dw(s) := t g(x(s)) dw(s)+ 1 2 has such a stability property (Wong-Zakai phenomenon). t tr ( g (x(s))g(x(s)) ) ds,

21 Prerequisites from Stochastic Analysis Brownian Motion on a riemannian manifold Let M be compact Riemanian manifold of dimension m without boundary which we assume is isometrically embedded into an euclidean space R d. Let a C function π : Rd L(R d, R d ) satisfy π(m) : R d T m M orthogonal projection,m M If a M, a solution x s.t. x() = a, to Stratonovich equation dx = π(x) dw(t) = π(x) dw (3) is an M-valued Brownian Motion. In particular, the density (w.r.t. the riemannian volume measure) of x(t) is equal to u(t, ; a) where u(t, ; a) solves the heat equation (with the Laplace-Beltrami operator) on M such that u(, ; a) = δ a.

22 Wave Equations - why they important? The fundamental equation of wave mechanics vibrating strings, membranes, 3D elastic bodies seismic waves ultrasonic waves (detection of flaws in materials) sound waves, water waves electromagnetic waves Quantum mechanics General relativity Yang-Mills theory Optics Geometric wave equations

23 Wave Equation - critical points of action functional Lagrange - Euclidean L(u) = Du, Du Eucl + V (u) R d Du, Du Eucl := u 2 + u = 1 2 V (u) Lagrange - Minkowski L(u) = Du, Du Mink + V (u) R 1+d Du, Du Mink := u t 2 + u 2 u tt u = 1 2 V (u)

24 Deterministic Wave Equation u tt u + f (u, Du) = in R R d u(, x) = u (x), u t (, x) = v (x) t u(t) = K (t) u + K (t) v K (t s) f (u(s), Du(s)) ds { } sin(t y ) K (t) = F 1 y

25 Stochastic Wave Equation u tt u + f (u, Du) = g(u, Du)Ẇ in R + R d u(, x) = u (x), u t (, x) = v (x) t u(t) = K (t) u + K (t) v t + K (t s) g(u(s), Du(s)) dw { } sin(t y ) K (t) = F 1 y K (t s) f (u(s), Du(s)) ds

26 Stochastic Wave Equation u tt u + f (u, Du) = g(u, Du)Ẇ in R + R d u(, x) = u (x), u t (, x) = v (x) t u(t) = K (t) u + K (t) v t + K (t s) g(u(s), Du(s)) dw K (t s) f (u(s), Du(s)) ds 1 E W (t, x)w (s, y) = min {t, s} Γ(x, y) 2 E W (t, x)w (s, y) = min {t, s} Γ(x y)

27 Stochastic Wave Equation u tt u + f (u, Du) = g(u, Du)Ẇ in R + R d u(, x) = u (x), u t (, x) = v (x) t u(t) = K (t) u + K (t) v t + K (t s) g(u(s), Du(s)) dw K (t s) f (u(s), Du(s)) ds Finite speed of propagation (Huygen s principle)

28 spatially homogenous Wiener process Consider (Ω, F, P), F = (F t ), Γ S µ = ˆΓ spectral measure W is S (R d )-valued F-Wiener process: E W (t), ϕ 2 = t Γ, ϕ ϕ( ), ϕ S (R d ). Then the RKHS K can be explicitly found and Law of W (t) is translation invariant. If Γ C(R d ) then EW (t, x)w (s, y) = t s Γ(x y). Examples: Euclidean free field dµ dx = (2π) d 2 ( x 2 + m 2 ) 1. Space-time white noise Γ = δ.

29 Geometric Wave Equations M a compact Riemannian manifold u (x) M for x R d v (x) T u (x)m for x R d Definition A map u : R R d M is a solution of a GWE iff D t u t d i=1 D x i u xi = in R R d u(, x) = u (x) u t (, x) = v (x)

30 Wave Equation - critical points again Lagrange - Minkowski u : R R d M L(u) = Du, Du dt dx R 1+d Mink Du, Du = Mink ut 2 T + um u 2 T um d D t u t = D xi u xi i=1

31 Geometric Wave Equations M is a submanifold in R n A is the second fundamental form of M γ : R M is a curve Lemma The following properties holds: 1 D t t γ(t) = tt γ(t) A γ(t) ( t γ(t), t γ(t)) 2 A γ(t) ( t γ(t), t γ(t)) tt γ(t) A γ(t) ( t γ(t), t γ(t))

32 Geometric Wave Equations Lemma M is a submanifold in R n A is the second fundamental form of M γ : R M is a curve The following properties holds: 1 D t t γ(t) = tt γ(t) A γ(t) ( t γ(t), t γ(t)) 2 A γ(t) ( t γ(t), t γ(t)) tt γ(t) A γ(t) ( t γ(t), t γ(t)) Corollary and, denote the norm and the inner product in R n tt γ(t) A γ(t) ( t γ(t), t γ(t)), tt γ(t) = tt γ(t) A γ(t) ( t γ(t)) 2,

33 Geometric Wave Equations cont. Example If M is a sphere then u is a solution of the SGWE if and only if u tt u + ( u t 2 u 2 )u = u = 1

34 Blow-up and the global existence: deterministic case Global existence strong solutions d = 1 (Gu 8 ) weak solutions d = 1 (Zhou 99 ) weak solutions d = 2 (Müller, Struwe 96 ) M is a sphere M is a homogeneous riemannian manifold (Freire 96 ) Blow-up d 3 (Cazenave, Shatah, Tahvildar-Zadeh 98 ) Non-uniqueness d 3 (Cazenave, Shatah, Tahvildar-Zadeh 98 )

35 Stochastic wave equations with values in R This is a very well developed subject. Let me just mention few names. [Carmona and Nualart 88, Mueller 97 ] [Mueller 97, Dalang and Frangos 98, Millet and Sanz-Solé 99, Peszat and Zabczyk ] [Peszat and Zabczyk, Peszat 2, Dalang and Nualart 4] [Ondrejat 7, 11, Kim 6 and many others]

36 The existence and uniqueness of global strong solutions to SGWE in R 1+1 We will always assume that E W (t, x)w (s, y) = min {t, s}γ(x y) M a compact Riemannian manifold g p : T p M T p M T p M, p M is Cb 1 D t u t D x u x = g(u, u t, u x )Ẇ (4) Theorem (1: ZB and M. Ondreját (JFA 27)) If Γ Cb 2 then An intrinsic solution an extrinsic solution.

37 The existence and uniqueness of global strong solutions to SGWE in R 1+1 We will always assume that E W (t, x)w (s, y) = min {t, s}γ(x y) M a compact Riemannian manifold g p : T p M T p M T p M, p M is C 1 b D t u t D x u x = g(u, u t, u x )Ẇ (4) Theorem (1: ZB and M. Ondreját (JFA 27)) If Γ Cb 2 then and u Hloc 2 (R, M) and v Hloc 1 (R, TM) are such that v (x) T u (x)m, x R, then a unique global strong solution to (4). This solution has continuous Hloc 2 H1 loc trajectories.

38 The existence of a global weak solution for SGWE in R 1+1 M a compact Riemannian manifold g, g,..., g d continuous D t u t D x u x = [g(u) + g (u)u t + d g i (u)u xi ]Ẇ i=1 Theorem (2: ZB & M.Ondrejat (Comm PDEs (211)) If Γ C b then there exists a global weak solution to SGWE provided u Hloc 1 (R, M) and v L 2 loc (R, TM) are such that v (x) T u (x)m, for a.a. x R. The solution has weakly continuous H 1 loc L2 loc trajectories.

39 Global Existence in R 1+d M a compact Riemannian homogeneous space g, g,..., g d continuous D t u t D x u x = [g(u) + g (u)u t + d g i (u)u xi ]Ẇ i=1 Theorem (3: ZB and M. Ondreját (Ann Prob to appear)) If Γ C b then there exists a global weak solution to SGWE provided u Hloc 1 (R, M) and v L 2 loc (R, TM) are such that v (x) T u (x)m, for a.a. x R. The solution has weakly continuous H 1 loc L2 loc trajectories.

40 Physical background We consider a ferromagnetic material filling a domain D R d, d 3, u(t, x) the magnetic moment at x D at time t, For temperatures not too high (below Curie point) u(t, x) = 1, x D

41 Energy functional Landau-Lifshitz 1935, Gilbert 1955 Every configuration φ : D S 2 R 3, φ H 1 of magnetic moments minimizes the energy functional E(φ) = a 1 φ 2 dx + 1 v 2 dx H φdx 2 D 2 R d D exchange energy magnetostatic energy, H- given external field. v = (1 D φ ), on R d

42 Landau-Lifshitz-Gilbert equation H(u) = D u E(u) = a 1 u v + H u t = λ 1 u H(u) λ 2 u (u H(u)) on D u n = on D u (x) = 1 on D where λ 2 > and from now on λ 1 = λ 2 = 1.

43 Connection with harmonic maps problem but u 2 = 1 on D then E(φ) = 1 2 u t D φ 2 dx = u (u u) u (u u) = (u u)u u 2 u, u u =, u u = u 2 We obtain heat flow of harmonic maps: u t = u + u 2 u

44 Previous works A. Visintin 1985: weak existence, d 3, Chen and Guo 1996, Ding and Guo 1998, Chen 2, Harpes 24: existence and uniqueness of partially regular solutions, d = 2 C. Melcher 25: existence of partially regular solutions, d = 3, R. V. Kohn, M. G. Reznikoff, E. Vanden-Eijnden 27, large deviations A. Desimone, R. V. Kohn, S. Müller, F. Otto 22, thin film approximations R. Moser 24, thin film approximations, magnetic vortices

45 Thermal noise Néel 1946: H = noise. E(φ) = H φ D H = hdw h : D R 3, W Brownian Motion important problem: to study noise-induced transition between minima of E

46 Stochastic Landau-Lifshitz-Gilbert-Equation I H(u) = D u E(u) = u v + hdw u t = u H(u) u (u H(u)) on D u n = on D u (x) = 1 on D F dw is a Stratonovitch integral: F(u) dw = 1 2 F (u) F(u)dt + FdW

47 Stochastic Landau-Lifshitz-Gilbert-Equation II H(u) = u Pu + hdw u t = u H(u) u (u H(u)) on D u n = on D u (x) = 1 on D (5)

48 Introduction Stochastic geometric wave equations Stochastic Landau-Lifshitz equations Proof of some of the results on Numerical experiments (joint works with L. Banas and A. Prohl) Figure: Switching mechanism: u(t, x) for space-time white noise withn).

49 Integration by parts N Neumann Laplacian { D ( N ) = u H 2 : u n. Lemma D D } =, on D If v H 1 and u D ( N ) then u N, v dx = u, ( v) u dx.

50 Weak martingale solution A system (Ω, F, F, P, W, u), where F = (F t ) t is a solution to (5) iff for every T > and φ C ( D, R 3 ), ( u( ) C [, T ]; H 1,2), P a.s. E sup u(t) 2 L <, u(t, x) 2 R 3 = 1, t T Leb P a.e. u(t), ϕ u, ϕ = t t u, ( ϕ) u ds + t u, (u ϕ) u ds + G(u)Pu, ϕ ds t G(u)f = u f + u (u f ), Pu, ϕ = u, ϕ G(u)h, ϕ dw

51 Notation Given u H 1,2 we define u u as a measurable L 2 -valued function such that u u, ϕ = u, u ( ϕ)

52 Weak existence for d = 3 Theorem (4: ZB, T Jegaraj and B Goldys (AMReX, 212)) Let u H 1, h L W 1,3 and u (x) = 1. Then there exists a solution (Ω, F, F, P, W, u) to the SLLGEs such that for all T > T E u u 2 dt <, t t u(t) = u + u uds u (u u)ds t + G(u)Pu ds + t G(u)h dw (s), u C α ( [, T ], L 2), α < 1 2.

53 Non-uniqueness Non-uniqueness for d = 3 Theorem (6: ZB and Anne de Bouard (in preparation)) There exists u H 1 such that u (x) = 1 and a non-trivial h L W 1,3 such that there exist infinity many solution (Ω, F, F, P, W, u) to the SLLGEs such that for all T > T E u u 2 dt <, t u(t) = u + u uds t + G(u)h dw (s), t u C α ( [, T ], L 2), α < 1 2. u (u u)ds

54 Comments on the proof of Theorem 1 To avoid unnecessary difficulties stemming from the language of differential geometry we assume that M = S 2 R 3. We take a tubular neighbourhood O of M as O := {x R 3 : 1 2 < x < 2} M = S 2 O R 3.

55 Comments on the proof of Theorem 1 Define an involution map h : O x x x 2 O We extend h to the whole R 3. Note that h(x) = x, x R 3 iff x M, x R 3. Define a map S q (x, y) = 1 2 (d 2 q h) ( (d q h)(x), (d q h)(y) ), q, x, y R 3.

56 Comments on the proof of Theorem 1 Define an involution map h : O x x x 2 O We extend h to the whole R 3. Note that h(x) = x, x R 3 iff x M, x R 3. Define a map S q (x, y) = 1 2 (d 2 q h) ( (d q h)(x), (d q h)(y) ), q, x, y R 3.

57 Comments on the proof of Theorem 1 For M = S 2 R 3 the SGWE takes the form ( u = u xx ) u tt u + ( u t 2 u 2 )u = g(u, u t, u x )Ẇ, ; u = 1 (1) u() = u, u t () = v (2) Instead of SGWE (1-2) we consider SPDE with values in R 3 : u tt u = S u (u t, u t ) S u (u x, u x ) + G(u, u t, u x )Ẇ (3) u() = u, u t () = v (2) where G is a suitable extension of g to R 3 (which is in some sense h-invariant on O) and with initial data as earlier.

58 Comments on the proof of Theorem 1 Problem (3-2) has a unique continuous Hloc 2 -valued solution u(t), t [, τ), where τ is the exit time of u(t) from O. By the construction of S and of G, a process ũ(t) := h(u(t)), t [, τ) is also a solution of (3). Because u (x) S 2 and v (x) T u(x) S 2, by the construction of the involution map h, ũ satisfies the same initial condition as u, i.e. (2). By the uniqueness of solutions to (3-2), u = ũ, i.e. h(u(t, x)) = u(t, x), x, t < τ. Since the fixed point set of h in O is S 2 we infer that u(t, x) S 2, x, t < τ. (4)

59 Comments on the proof of Theorem 1 Then using (4) we can employ some energy estimates to conclude that τ = and that u is a solution to (1-2). { utt u + ( u t 2 u 2 )u = g(u, u t, u x )Ẇ u = 1 (1) u() = u, u t () = v (2)

60 Proof of Theorem 3: Intro The assumptions of the Theorem imply that (Moore-Schlafly Theorem) M2 There exists a C -class function F : R n [, ) such that M = {x : F(x) = } and F is constant outside some large ball in R n. M3 There exist a finite sequence (A i ) N i=1 of skew symmetric linear operators on R n such that for each i {1,, N}, F(x), A i x =, for every x R n, (6) A i p T p M, for every p M. (7) M4 There exist a family ( h ij )1 i,j N of C -class R-valued functions on M such that N N ξ = h ij (p) ξ, A i p R na j p, p M, ξ T p M. (8) i=1 j=1

61 Proof of Theorem 3: Main idea Method: penalization and approximation tt U m = U m m F(U m )+f m (U m, (t,x) U m )+g m (U m, (t,x) U m ) dw m We get tightness of sequences the sequences U m, V m = t U m and of, for every i {1,, N}, Mi n := V m, A i U m R d. We prove that these three sequences have limits U, V, M i, which satisfy certain integral equations. Finally, using the previous page, we show that the process U is a solution.

62 Proof of Theorem 4 Uniform estimates for the Galerkin approximations u n, Tightness of the family of probability laws {L (u n ) : n 1}, Identification of the limit

63 Proof of Theorem 4: Galerkin approximations {e n } n=1 eigenbasis of N in L 2 and π n orthogonal projection onto H n = lin {e 1,..., e n }. { dun = (G n (u n ) u n (u n ) + G n (u n ) Pu n ) dt + G n (u n ) h dw, u n () = π n u G n (u)f = π n (u n f ) π n (u n (u n f )) For every n 1 there exists a unique strong solution in H n.

64 Proof of Theorem 4: uniform estimates Lemma (Let h L W 1,3 and u H 1.) Then for p 1, β > 1 2 and T > u n (t) L 2 = u n () L 2, P a.s. [ ] sup E n sup u n (t) 2p L 2 t [,T ] <, T sup E u n (t) u n (t) L 2 dt <, n ( T sup E u n (t) ( u n (t) u n (t) ) p/2 2 L dt) <. n 3/2 T ( sup E π n un (t) ( u n (t) u n (t) )) 2 H β dt <. n

65 Proof of Theorem 4: tightness Lemma (For any p 2, q [2, 6) and β > 1 2 ) the set of laws {L (u n ) : n 1} is tight on L p (, T ; L q ) C (, T ; H β).

66 Proof of tightness For β > 1 2, α < 1 2 and p > 2 Then for β < γ < 1 sup E u n 2 W α,p (,T ;H β ) <. n L p (, T ; H 1) W α,p (, T ; H β) L p (, T ; H γ ), with compact embedding by Flandoli&Gatarek 1995 and tightness on L p (, T ; H γ ) L p (, T ; L q ) follows. Again by Flandoli&Gatarek 1995 W α,p (, T ; H β 1) C (, T ; H β ), β > β 1, αp > 1, with compact embedding.

67 Doss-Sussman method Simplified stochastic Landau-Lifshitz-Gilbert equation: du = [u u u (u u)]dt + (u h) dw, t >, x D, u n =, t, x D, u(, x) = u (x), x D.

68 Doss-Sussman method: auxiliary facts Bx = x a, x R 3 Then e tb is a group of isometries and ( ) ( ) e tb (x y) = e tb x e tb y, x, y R 3. For h H 2 put Gφ = φ h, φ L 2 Then ( e tg) is again a group of isometries in L 2 and e tg φ = φ + (sint)gφ + (1 costt)g 2 φ

69 Doss-Sussman method III: transformation Let Then where dv dt v(t) = e W (t)g u(t). = v R(t)v v (v R(t)v) (9) R(t)v = e W (t)g e W (t)g v

70 Doss-Sussman method: transformation continued. Lemma For φ H 2 t e tg e tg φ = φ + e sg Ce sg φ ds, with Cφ = φ h + 2 i ( φ x i ) ( h x i ). If v R 3 = 1 then we obtain { dv dt = R(t)v + v R(t)v + e tb v 2 v v() = u. (1)

71 Doss-Sussman: regularity Theorem (5: ZB and B Goldys (in preparation)) Let h H 2 and u W 1,4. Then for every ω there exists T = T (ω) > such that equation (1) has a unique solution u on [, T ) with the property u C (, T ; W 1,4) and v(t, x) R 3 = 1, t < T, x D.

72 Proof of Theorem 5 Equation (1) is a strongly elliptic quasi-linear system Show that there exists a mild solution v C (, T ; W 1,4) Use maximal regularity and ultracontractivity of the heat semigroup to "bootstrap" the regularity of solutions. Show that v(t, x) = 1. Note that (9) can be written in the form dv dt = v + v v + v 2 v + v L(t, v) + v (v L(t, v)) with L linear and L(t, v L 2 C v H 1 where C is a finite random variable.

73 Theorem (6: ZB and B Goldys (in preparation)) The process u(t) = e W (t)g v(t) is a unique solution of the stochastic Landau-Lifshitz-Gilbert equation on [, T ) satisfying for every n 1 conditions Proof: take E T n N v(s) 2 2 < E sup v(t) 2 <, t T n u(t) = e W (t)g v(t). Use the Itô formula to obtain the estimates.

74 Stochastic LLG in 1D D = [, 1] pathwise uniqueness maximal regularity large deviations Pathwise uniqueness Theorem (7: ZB, T Jegaraj and B Goldys (in preparation)) Let u 1, u 2 : [, T ] L 2 be two progressively measurable solutions, such that u i C ( [, T ], L 2) L 8 ( [, T ], H 1). Then u 1 ( ) = u 2 ( ) P-a.s.