WEAK MARTINGALE SOLUTIONS TO THE STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION WITH MULTI-DIMENSIONAL NOISE VIA A CONVERGENT FINITE-ELEMENT SCHEME

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1 WEAK MARTINGALE SOLUTIONS TO THE STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION WITH MULTI-DIMENSIONAL NOISE VIA A CONVERGENT FINITE-ELEMENT SCHEME BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE Abstract. We propose an unconditionally convergent linear finite element sceme for te stocastic Landau Lifsitz Gilbert LLG) equation wit multi-dimensional noise. By using te Doss-Sussmann tecnique, we first transform te stocastic LLG equation into a partial differential equation tat depends on te solution of te auxiliary equation for te diffusion part. Te resulting equation as solutions absolutely continuous wit respect to time. We ten propose a convergent θ-linear sceme for te numerical solution of te reformulated equation. As a consequence, we are able to sow te existence of weak martingale solutions to te stocastic LLG equation. Contents 1. Introduction 1. Definition of a weak solution and te main result 3 3. Te auxiliary equation for te diffusion part 4 4. Equivalence of weak solutions Te finite element sceme Te main result Appendix 8 Acknowledgements 8 References 8 1. Introduction Te deterministic Landau-Lifscitz-Gilbert LLG) equation provides a basis for te teory and applications of ferromagnetic materials and fabrication of magnetic memories in particular, see for example [15, 9, 1, 17]. Te setting is described as follows. Let D be a bounded domain in R 3 wit a smoot boundary D, occupied by a ferromagnetic material. A configuration of magnetic moments across te domain D can be represented by a function on D taking values in S, te unit spere in R 3. According to te Landau and Lifscitz Date: April 1, 17. Matematics Subject Classification. Primary 35Q4, 35K55, 35R6, 6H15, 65L6, 65L, 65C3; Secondary 8D45. Key words and prases. stocastic partial differential equation, Landau Lifsitz Gilbert equation, finite element, ferromagnetism. 1

2 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE teory of ferrormagnetism [17], modified later by Gilbert [1], te time evolution of magnetic moments Mt, x) is described, in te simplest case, by te Landau-Lifscitz-Gilbert LLG) equation 1.1) M t M n = λ 1 M M λ M M M) in, T ) D, = in, T ) D, M, ) = M ) in D, were λ 1 and λ > are constants, and n stands for te outward normal vector on D; see e.g. [9]. Assuming tat M H 1, D, S ), one can sow tat 1.) Mt, x) = 1, t [, T ], x D In tis paper we are concerned wit a stocastic version of te LLG equation. Randomly fluctuating fields were originally introduced in pysics by Néel in [?] as formal quantities responsible for magnetization fluctuations. Te necessity of being able to describe deviations from te average magnetization trajectory in an ensemble of noninteracting nanoparticles was later empasised by Brown in [6, 7]. According to a non-rigorous argument of Brown, te magnetisation M evolves randomly according to a stocastic version of 1.1) tat takes te form, see [8] for more details about te pysical background and derivation of tis equation) dm = λ 1 M M λ M M M) ) dt + q M g i) dw i t), 1.3) M n = on, T ) D, M, ) = M in D, were g i W, D), i = 1,, q, satisfy te omogeneous Neumann boundary conditions and W i ) q is a q-dimensional Wiener process. In view of te property 1.) for te deterministic system, we require tat M also satisfies 1.). To tis end, we are forced to use te Stratonovic differential dw i t) in equation 1.3). Te matematical teory of equation 1.3) as been initiated only recently, in [8], were te existence of weak martingale solutions to 1.3) was proved for te case q = 1 using te Galerkin-Faedo approximations. We note tat te Galerkin-Faedo approximations do not usually provide a useful computational tool for solving an equation exactly. Te aim of tis paper is two-fold. We will prove te existence of solutions to te stocastic LLG equation 1.3) and at te same time will provide an efficient and flexible algoritm for solving numerically tis equation. To tis end we will use te finite element metod and a new transformation of te Stratonovic type equation 1.3) to a deterministic PDE 4.) wit coefficients determined by a stocastic ODE 3.6) tat can be solved separately. Te deterministic PDE we obtain as solutions tat are absolutely continuous wit respect to time, and ence is suitable for te construction of a convergent finite element sceme. Our approac is based on te Doss-Sussmann tecnique [11, 18]. Tis transformation was introduced in [13] to study te stocastic LLG equation wit a single Wiener process

3 FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 3 q = 1), in wic case te auxiliary ODE is deterministic. Since te vector fields u g i are non-commuting, te case of q > 1 is more difficult and requires new arguments. We apply te finite element metod to te PDE resulting from tis transformation and prove te convergence of linear finite element sceme to a weak martingale solution to 1.3) after taking an inverse transformation). Our proof is simpler tan te proof in [8] and covers te case of q > 1. We note ere tat under appropriate assumptions even te case of infinite-dimensional noise q = ) can be andled in exactly te same way. Let us recall tat te first convergent finite element sceme for te stocastic LLG equation was studied in [5] and is based on a Crank Nicolson type time-marcing evolution, relying on a nonlinear iteration solved by a fixed point metod. On te oter and, tere as been an intensive development of a new class of numerical metods for te LLG equation 1.1) based on a linear iterations, yielding unconditional convergence and stability [1, 3]. Te ideas developed tere are extended and generalized in [13, ] in order to take into account te stocastic term. A fully linear discrete sceme for 1.3) is studied in [13] but wit one-dimensional noise. Te metod is based on te so called Doss-Sussmann tecnique [11, 18], wic allows one to replace te stocastic partial differential equation PDE) by an equivalent PDE wit random coefficients. In contrast, [] considers, for a more general noise, a projection sceme applied directly to te original stocastic equation 1.3). However, tis approac requires a quite specific and complicated treatment of te stocastic term. In te current paper, we propose a convergent θ-linear sceme for te numerical solution of te tranformed equation and prove unconditional stability and convergence for te sceme wen θ > 1/. To te best of our knowledge tis is a new result for tis problem. Te paper is organised as follows. In Section we define te notion of weak martingale solutions to 1.3) and state our main result. In Section 3, we introduce an auxiliary stocastic ODE and prove some properties of solution necessary for te transformation of equation 1.3) to a deterministic PDE wit random coefficients. Details of tis transformation are presented in Section 4. We also sow in tis section ow a weak solution to 1.3) can be obtained from a weak solution of te reformulated form. In Section 5 we introduce our finite element sceme and present a proof for te stability of approximate solutions. Section 6 is devoted to te proof of te main teorem, namely te convergence of finite element solutions to a weak solution of te reformulated equation. Finally, in te Appendix we collect, for te reader s convenience, a number of facts tat are used in te course of te proof. Trougout tis paper, c denotes a generic constant tat may take different values at different occurrences. In wat follows we will also use te notation D T =, T ) D.. Definition of a weak solution and te main result In tis section we state te definition of a weak solution to 1.3) and present our main result. Before doing so, we introduce some suitable Sobolev spaces, and fix some notation. Te standing assumption for te rest of te paper is tat D is a bounded open domain in R 3 wit a smoot boundary. For any U R d, d 1, we denote by L U) te space of Lebesgue square-integrable functions defined on U and taking values in R 3. Te function space H 1 U) is defined as: { H 1 U) = u L U) : u } L U) for i d.. x i

4 4 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE Remark.1. For u, v H 1 D) we denote u v := u v, u v, u v ) x 1 x x 3 3 u u v := v x i x i 3 w v, u L D) := w v, u w L D). x i x i L D) Definition.. Given T, ) and a family of functions {g i : i = 1,..., q} L D), a weak martingale solution Ω, F, F t ) t [,T ], P, W, M) to 1.3), for te time interval [, T ], consists of a) a filtered probability space Ω, F, F t ) t [,T ], P) wit te filtration satisfying te usual conditions, b) a q-dimensional F t )-adapted Wiener process W = W t ) t [,T ], c) a progressively measurable process M : [, T ] Ω L D) suc tat.1) 1) M, ω) C[, T ]; H 1 D)) for ) P-a.e. ω Ω; ) E ess sup t [,T ] Mt) L D) < ; 3) Mt, x) = 1 for eac t [, T ], a.e. x D, and P-a.s.; 4) for every t [, T ], for all ψ C D), P-a.s.: Mt), ψ L D) M, ψ L D) = λ 1 M M, ψ L D) ds λ M M, M ψ) L D) ds + M g i, ψ L D) dw is). As te main result of tis paper, we will establis a finite element sceme defined via a sequence of functions wic are piecewise linear in bot te space and time variables. We also prove tat tis sequence contains a subsequence converging to a weak martingale solution in te sense of Definition.. A precise statement will be given in Teorem Te auxiliary equation for te diffusion part In tis section we introduce te auxiliary equation 3.1) tat will be used in te next section to define a new variable from M, and establis some properties of its solution. Let g 1,..., g q C D, R 3), be fixed. For i = 1,..., q, and x D we define linear operators G i x) : R 3 R 3 by G i x)u = u g i x). In wat follows we suppress te argument x. It is easy to ceck tat 3.1) 3.) and G i = G i, ) G i = G i.

5 FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 5 We will consider a stocastic Stratonovitc equation on te algebra L R 3) of linear operators in R 3 : 3.3) Z t = I + G i Z s dw i s), t. Lemma 3.1. Let g 1,..., g q C D, R 3). Ten te following olds. a) For every x D equation 3.3) as a unique strong solution, wic as a t-continuous version in L R 3). b) For every t and x D 3.4) Z t u = u P a.s for every u R 3. In particular, for every t te operator Z t is invertible and Z 1 t = Z t. c) If moreover g 1,..., g q C α D, R 3) for a certain α, 1) ten te mapping t, x) Z t x) as a continuous version in L R 3). Proof. Equation 3.3) can be equivalently written as an Itô equation 3.5) Z t = I + 1 G i Z s ds + G i Z s dw i s), t. Since te coefficients of equation 3.5) are Lipscitz, te existence and uniqueness of strong solutions to equation 3.5), and te existence of its continuous version is standard, see for example Teorem 18.3 in [16]. Hence, te same result olds for 3.3). To prove b) we fix x D, t and u R 3 and put Z u = Zu. Ten equation 3.5) yields 3.6) Z u t = u + 1 G i Z u s ds + G i Z u s dw i s). Applying te Itô formula to te process Zt u and invoking 3.1) we obtain d Zt u = Zt u, dzt u + G i Zt u dt or equivalently = Z u t, G i Zt u dt + Zt u, G i Zt u dw i t) + =, Z u t = u, for all t, P a.s.. G i Zt u dt To prove c), we begin by letting s < t T and x, y D. For any p 1 we ave 3.7) E Z t y) Z s x) p p 1 E Z t y) Z s y) p + p 1 E Z s y) Z s x) p. It is well known tat tere exists C 1 > suc tat 3.8) E Z t y) Z s y) p C 1 t s p. If tere exists α, 1] suc tat g i x) g i y) c i x y α, x, y D, i = 1,..., q

6 6 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE ten for a certain C >, for any R 3 tere olds 3.9) Ten Z s y) Z s x) = 1 = Using 3.9) we obtain G i x) C, G i x) G i y)) C x y α, G i x) G i y) ) C x y α. G i y)z r y) dr + G i x)z r x) dr G i y) G i x) ) Z r y) dr + 1 G i y) G i x)) Z r y)dw i r) + E Z s y) Z s x) p C x y αp + C Terefore, invoking te Gronwall Lemma we obtain G i y)z r y)dw i r) G i x)z r x)dw i r) 3.1) E Z s y) Z s x) p Ce CT x y αp. Combining 3.7), 3.8) and 3.1) we obtain G i x) Z r y) Z r x)) dr E Z r y) Z r x) p dr. 3.11) E Z t y) Z s x) p c 1 t s p + c x y αp. G i x) Z r y) Z r x)) dw i r). Let β > and r = d β. Let p be cosen in suc a way tat p r and pα r. Te set [, T ] D can be covered by a finite number of open sets B k wit te property t s r + x y r < 1 on every set B k. In eac B k, 3.11) ten yields E Z t y) Z s x) p c t s r + x y r ), and te result ten follows by te Kolmogorv-Centsov teorem, see p. 57 of [16]. Lemma 3.. Assume tat g i C 1+α b D, R 3 ). Ten te following olds. a) For every t we ave Z t Cb 1D, LR3 )) P-a.s. b) For every x D te process ξ t x) = Z t x) is te unique solution of te linear Itô equation dξ t x) = 1 G i ξ t x) + H i Z t x) ) dt + Gi ξ t x) + I i Z t x) ) dw i t), wit ξ x) = and te operators H i, I i L R 3) defined as I i u = u g i and H i u = G i u g i + G i u g i ).

7 FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 7 c) For every γ < min α, 1 ) te mapping t, x) Zt x) is γ-hölder continuous. d) We ave E sup t T sup Z t <. x D Proof. a) Let E denote te Banac space of continuous and adapted processes Z taking values in te space of linear operators L R 3) and endowed wit te norm Z E = For every x D we define a mapping ) 1/ E sup Z t. t T K : D E D E, Kx, Z)t) = I + G i x)z s dw i s). It is easy to ceck tat te assumptions of Lemma 9., p. 38 in [1] are satisfied and terefore a) olds. b) Te proof is completely analogous to te proof of Teorem 9.8 in [1], and is ence omitted. c) Te proof is analogous to te proof of part c) of Lemma 3.1. d) Te estimate follows easily from c). For every u L D) we will consider te L D)-valued process Z t u) defined by Clearly, 3.1) Z t u) = u + [Z t u)]x) = Z t x)ux) x a.e. Z s u) g i dw i s), t, were te equality olds in L D). Te process Z t is now an operator-valued process taking values L L D) ) and it will still be denoted by Z t. Te next lemmas follow immediately from te properties of te matrix-valued process considered above. Lemma 3.3. Assume tat {g i : i = 1,..., q} C 1+α b D). Ten for every u L D) te stocastic differential equation 3.1) as a unique strong continuous solution in L D). Moreover, tere exists Ω Ω suc tat P Ω ) = 1 and for every ω Ω te following olds. a) For all t and every u L D), Z t ω, u) = u. b) For every t te mapping u Z t ω, u) defines a linear bounded operator Z t ω) on L D). In particular, 3.13) Z t ω, u + v) = Z t ω, u) + Z t ω, v). Moreover, for every T > tere exists a constant C T > suc tat 3.14) E sup Z t u) L D) C T u L D). t T

8 8 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE c) For every t te operator Z t ω) is invertible and te inverse operator is te unique solution of te stocastic differential equation on L D): 3.15) Zt 1 u) = u Zs 1 G i u) dw i s), u L D). Finally, 3.16) Zt 1 ω) = Zt ω). Lemma 3.4. Assume tat g i C 1+α b D, R 3 ) for i = 1,, q. Ten, for every u H 1 D) Z t u) H 1 P-a.s. Furtermore, te process ξ t u) := Z t u), is te unique solution of te linear equation dξ t u) = 1 G i ξ t u) + H i Z t u) ) dt + Gi ξ t u) + I i Z t u) ) dw i t), wit ξ u) = u. Lemma 3.5. For any u, v L D), tere olds for all t and P-a.s.: 3.17) Z t u v) = Z t u) Z t v), Proof. Let Zt u := Z t u) and Zt v := Z t v) for all t. We now prove 3.17); te property 3.13) can be obtained in te same manner. Using te Itô formula for Zt u Zt v and 3.6), we obtain dzt u Zt v ) = dzt u Zt v + Zt u dzt v + Zt u g i ) Zt v g i ) dt 3.18) = Z u t Zt v g i ) Zt v Zt u g i ) ) dw i t) + 1 Z u t Z v ) t g i ) g i Z v t Z u )) t g i ) g i dt + Zt u g i ) Zt v g i ) dt. Using an elementary identity 3.19) a b c) = a, c b a, b c, a, b, c R 3, we find tat 3.) Z u t Z v t g i ) Z v t Z u t g i ) = Z u t Z v t ) g i and 3.1) Z u t Z v t g i ) g i ) Z v t Z u t g i ) g i ) = Z u t, g i Z v t g i ) Z v t, g i Z u t g i ) Z u t g i ) Z v t g i ) = Z u t Z v t ) g i ) gi Z u t g i ) Z v t g i ).

9 FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 9 Invoking 3.) and 3.1), equation 3.18) we obtain dz u t Z v t ) = 1 Z u t Zt v ) ) g i gi dt + Z u t Zt v ) ) g i dwi t). Terefore, te process V t := Zt u Zt v is a solution of te following stocastic differential equation: { dv t = 1 q ) Vt g i gi dt + q ) Vt g i dwi t) V = u v. On te oter and, it follows from 3.6) tat te process Z t u v) satisfies te same equation. Hence, 3.17) follows from te uniqueness of solutions to 3.18). Lemma 3.6. For any u, v H 1 D), tere olds for all t and P-a.s.: 3.) Z t u), Z t v) L D) = u, v L D) + F t, u, v), wit were F t, u, v) := F 1,i s, u, v) ds + F,i s, u, v) dw i s) and F 1,i t, u, v) := Z t u), 1 H i G i I i )Z t v) L D) 1 H i G i I i )Z t u), Z t v) L D) + I i Z t u), I i Z t v) L D) ; F,i t, u, v) := Z t u), I i Z t v) L D) I iz t u)), Z t v) L D) Proof. Let ξt u := ξ t u) and ξt v := ξ t v) for all t. In addition, we consider a C function φ : L D) ) R defined by φx, y) = x, y L D). By using te Itô Lemma we obtain d ξt u, ξt v L D) = dξu t, ξt v L D) + ξu t, dξt v L D) + dξu t, dξt v L D) 1 = G i ξt u + H i Zt u, ξ v t L D) + 1 ξ u t, G i ξt v + H i Z v t L D) ) + G i ξt u + I i Zt u, G i ξt v + I i Zt v L D) dt ) 3.3) + G i ξt u + I i Zt u, ξt v L D) + ξu t, G i ξt v + I i Zt v L D) dw i t).

10 1 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE Using 3.1) and 3.), we deduce from 3.3) tat d ξt u, ξt v L D) = 1 H izt u, ξt v L D) + 1 ξu t, H i Zt v L D) + G iξt u, I i Zt v L D) = I i Z u t, G i ξ v t L D) + I iz u t, I i Z v t L D) ) dt ) I i Zt u, ξt v L D) + ξu t, I i Zt v L D) dw i t) 1 H i G i I i )Zt u, ξt v L D) + ξt u, 1 H i G i I i )Zt v L D) ) + I i Zt u, I i Zt v L D) dt ) ξt u, I i Zt v L D) + I izt u, ξt v L D) dw i t). Integrating by parts for te first and te last term in te rigt and side of te above equation and noting te omogeneous Neumann boundary condition of g i, we obtain d ξt u, ξt v L D) = 1 H i G i I i )Zt u, Z v t L D) + ξt u, 1 H i G i I i )Z v t L D) ) + I i Zt u, I i Zt v L D) dt ) 3.4) + ξt u, I i Zt v L D) I izt u ), Zt v L D) dw i t). Hence, te resutl follows from replacing t by s and intergrating 3.4) over [, t]. Remark 3.7. By using integration by parts and te omogeneous Neumann boundary conditions of g i for i = 1,, q we obtain some symmetry properties of functions F 1,i, F,i and F : for any u, v, v 1, v H 1 D), F 1,i t, u, v) = F 1,i t, v, u); F,i t, u, v) = F,i t, v, u); and ence, F t, u, v) = F t, v, u). Furtermore, it follows from 3.13) tat F t, u, v 1 + v ) = F t, u, v 1 ) + F t, u, v ). Te following lemmas state some important properties of F used trougout tis paper. Lemma 3.8. Assume tat g i W, D) for i = 1,, q. Ten for any u, v L Ω; H 1 D)) tere exists a constant c depending on T and {g i },,q suc tat 3.5) and for any ɛ >, 3.6) E sup Z t u) L D) ce u H 1 D), t [,T ] E sup F s, u, v) cɛe u H 1 D) + cɛ 1 E v L D). s [,t]

11 FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 11 Proof. It follows from 3.) tat 3.7) 3.8) E Z t u) L D) = E u L D) + E[F t, u, u)] E u L D) + E F1,i τ, u, u) dτ + E F,i τ, u, u) dw i τ). For convenience, we next estimate F τ, u, v), wic is a sligtly more general version of F τ, u, u). By using te elementary inequality 3.9) ab 1 ɛa + 1 ɛ 1 b, te assumption g i W, D) and 3.3), tere olds F 1,i τ, u, v) Z τ u), 1 H i G i I i )Z τ v) L D) + 1 H i G i I i )Z τ u), Z τ v) + I L D) i Z τ u), I i Z τ v) L D) c ɛ Z τ u) L D) + ɛ Z τ u) L D) + ɛ 1 Z τ v) ) L D) Tis implies tat 3.3) E c ɛ Z τ u) L D) + ɛ u L D) + ɛ 1 v L D)). F 1,i τ, u, v) dτ cɛte u L D) + cɛ 1 te v L D) + cɛe Z τ u) L D) dτ. Ten, by using te Burkolder-Davis-Gundy inequality, Hölder inequality, 3.3) and 3.9), we estimate E sup F,i τ, u, v) dw i τ) ce F,i τ, u, v) ) dτ 1/ s [,t] 3.31) 3.3) ce ce ce Zτ u) L D) Z τ v) L D) + Z τ u) L D) Z τ v) L D)) dτ 1/ Zτ u) L D) v L D) + u L D) v L D)) dτ 1/ [ v L D) ] Z τ u) L D) dτ) 1/ + ct 1/ E [ ] u L D) v L D) cɛt 1/ E u L D) + cɛ 1 t 1/ + 1)E v L D) + cɛe Z τ u) L D) dτ. We use 3.3) and 3.3) wit v = u and ɛ = 1 togeter wit 3.7) to deduce E Z t u) L D) ce u H 1 D) + ce Z τ u) L D) dτ. Hence, te result 3.5) follows immediately by using Gronwall s inequality.

12 1 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE To prove 3.6) we note tat 3.33) E sup F s, u, v) s [,t] E + E sup s [,t] F1,i τ, u, v) dτ F,i τ, u, v) dw i τ). Hence, it follows from 3.3), 3.3) and 3.5) tat E sup F,i τ, u, v) dw i τ) cɛe u L D) + cɛ 1 E v L D) s [,t] wic completed te proof of te lemma. + cɛe Z τ u) L D) dτ c ɛe u H 1 D) + ɛ 1 E v L D)), Lemma 3.9. For any u L Ω; L D) ), v L Ω; H 1 D) ) and s T, tere exists a constant c depending on {g i },,q suc tat E F s, u, v) cs E[ u L D) ]) 1/ E[ v L D) ]) 1/ + cs 1/ + s) E[ u L D) ]) 1/ E[ v L D) ]) 1/. Proof. From te definition of function F in Lemma 3.6 and te triangle inequality, tere olds E F s, u, v) E F 1,i τ, u, v) dτ + E F,i τ, u, v) dw i τ) 3.34). From Remark 3.7, we note tat F,i τ, u, v) = F,i τ, v, u), and terefore, by using 3.31), te last term of 3.34) can be estimated as follows: E F,i τ, u, v) dw i τ) = E F,i τ, v, u) dw i τ) 3.35) ce [ u L D) + cs 1/ E [ u L D) v L D)]. ] Z τ v) L D) dτ) 1/ We now estimate F1,i τ, u, v) by integrating by parts and ten using Hölder s inequality, te assumption g i W, D) and 3.3) as follows: F 1,i τ, u, v) = Z τ u), 1 H i G i I i )Z τ v) ) L D) + 1 H i G i I i )Z τ u), Z τ v) L D) + I i Z τ u), I i Z τ v) L D) c u L D) Zτ v L D) + v L D)),

13 and terefore, 3.36) FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 13 E F 1,i τ, u, v) [ dτ ce u L D) + cse [ u L D) v L D)]. Z τ v L D) dτ )] Hence, by using Hölder inequality we obtain from 3.34) 3.36) tat tere olds: [ E F s, u, v) ce u L D) Z τ v L D) dτ )] + cs 1/ + s)e [ ] u L D) v L D) c E[ u L D) ]) 1/ [ s E Z τ v L D) dτ) ]) 1/ 3.37) + cs 1/ + s) E[ u L D) ]) 1/ E[ v L D) ]) 1/. Via te Minkowski inequality and 3.5), we observe tat 3.38) [ s E Z τ v L D) dτ) ]) 1/ E[ Zτ v L D) ]) 1/ dτ cs E[ v L D) ]) 1/. Te required result follows from 3.37) and 3.38), wic completes te proof of tis lemma. 4. Equivalence of weak solutions In tis section we use te process Z t ) t defined in te preceding section to define a new process m from M. Let 4.1) mt, x) = Z 1 t Mt, x) t, a.e.x D. We will sow tat tis new variable m is differentiable wit respect to t. In te next lemma, we introduce te equation satisfied by m so tat M is a solution to 1.3) in te sense of.1). Lemma 4.1. If m, ω) H 1, T ; L D)) L, T ; H 1 D)), for P-a.s. ω Ω, satisfies mt, x) = 1 and for any ψ L, T ; W 1, D)) 4.) t m, ψ L D T ) + λ 1 Ten M = Z t m satisfies.1) P-a.s.. t, a.e. x D, P a.s., Z s m Z s m, Z s ψ L D) ds + λ Z s m Z s m, Z s m ψ) L D) ds =, P-a.s.. Proof. Using Itô s formula for M = Z t m, we deduce Mt) = M) + Zm g i dw i s) + = M) + M g i dw i s) + Z t m) ds Z s t m) ds.

14 14 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE Multiplying bot sides by a test function ψ C D) and integrating over D we obtain Mt), ψ L D) = M), ψ L D) + M g i, ψ L D) dw is) 4.3) + Z s t m), ψ L D) ds = M), ψ L D) + + t m, Zs 1 ψ L D) ds, M g i, ψ L D) dw is) were in te last step we used 3.16). On te oter and, it follows from 4.) tat, for all ξ L, t; W 1, D)), tere olds: 4.4) t m, ξ L D) ds = λ 1 Z s m Z s m, Z s ξ L D) ds λ Z s m Z s m, Z s m ξ) L D) ds. Using 4.4) wit ξ = Zs 1 ψ for te last term on te rigt and side of 4.3) we deduce Mt), ψ L D) = M), ψ L D) + M g i, ψ L D) dw is) It follows from 3.17) tat ) λ 1 M M, ψ λ Mt), ψ L D) = M), ψ L D) + wic complete te proof. L D) ds M M, Zs m Zs 1 ψ) L D) ds. ) λ 1 M M, ψ M g i, ψ L D) dw is) L D) ds λ M M, M ψ) L D) ds, Te following lemma sows tat te constraint on m is inerited by M. Lemma 4.. Te process M satisfies Mt, x) = 1 if and only if m defined in 4.1) satisfies mt, x) = 1 t, a.e. x D, P a.s. t, a.e.x D, P a.s..

15 FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 15 Proof. Te proof follows by using 3.16): m = m, m = Z 1 t M, Zt 1 M = M, Z t Zt 1 M = M, M = M. In te next lemma we provide a relationsip between equation 4.) and its Gilbert form. Lemma 4.3. Let m, ω) H 1, T ; L D)) L, T ; H 1 D)) for P-a.s. ω Ω satisfy 4.5) mt, x) = 1, t, T ), x D, and 4.6) λ 1 t m, ϕ L D T ) + λ m t m, ϕ L D T ) = µ Z s m, Z s m ϕ) L D) ds, were µ = λ 1 + λ. Ten m satisfies 4.). Proof. For eac ψ L, T ; W 1, D)), using Lemma 7.1 in te Appendix, tere exists ϕ L, T ; H 1 D)) suc tat 4.7) λ 1 ϕ + λ ϕ m = ψ. We can write 4.6) as 4.8) t m, λ 1 ϕ + λ ϕ m L D T ) + λ 1 From 4.5) and 3.3), we obtain tat Z s m Z s m, Z s λ 1 ϕ) L D) ds + λ Z s m, Z s λ ϕ m) L D) ds =. 4.9) Z t mt, x) = 1, t, T ) and x D. On te oter and, by using 4.9), 3.19) and a standard identity 4.1) a, b c = b, c a = c, a b, for all a, b, c R 3, we obtain λ 1 Z s m Z s m, Z s λ ϕ m) L D) ds + λ 4.11) Moreover, we ave λ Zs m Z s m, Z s λ 1 ϕ) ds =. L D) 4.1) λ Zs m Z s m, λ Z s ϕ Z s m ds =. L D) Summing 4.8), 4.11) and 4.1) gives t m, λ 1 ϕ + λ ϕ m L D T ) + λ 1 Z s m, Z s λ 1 ϕ) L D) ds Z s m Z s m, Z s λ 1 ϕ + λ ϕ m) L D) ds + λ Z s m, Z s λ 1 ϕ + λ ϕ m) L D) ds λ Zs m Z s m, Z s λ 1 ϕ + λ ϕ m) L D) ds =

16 16 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE Te desired equation 4.) follows by noting 4.7) and using 3.19), 4.1) and 4.9). Remark 4.4. By using 4.1) and 4.5) we can rewrite 4.6) as 4.13) λ 1 m t m, w L D T ) λ t m, w L D T ) = µ Z s m, Z s w L D) ds, or equivalently, tanks to Lemma 3.6, 4.14) λ 1 m t m, w L D T ) λ t m, w L D T ) =µ m, w L D T ) + µ F t, mt, ), wt, )) dt, were w = m ϕ for ϕ L, T ; H 1 D)). We note in particular tat w m =. Tis property will be exploited later in te design of te finite element sceme. We state te following lemma as a consequence of Lemmas 4.3, 4. and 4.1. Lemma 4.5. Let m, ω) H 1, T ; L D)) L, T ; H 1 D)) for P-a.s. ω Ω. If m is a solution of 4.5) 4.6), ten M = Z t m is a weak martingale solution of 1.3) in te sense of Definition.. Proof. By using Lemmas 4.1, 4. and 4.3 togeter wit te imbedding H 1, T ; L D)) C, T ; H 1 D), we deduce tat M satisfies 1), ), 3), 4) in Definition., wic completes te proof. Tanks to te above lemma, we now can now restrict our attention to solving equation 4.6) rater tan.1). 5. Te finite element sceme In tis section we design a finite element sceme to find approximate solutions to 4.6). In te next section, we prove tat te finite element solutions converge to a solution of 4.6). Ten, tanks to Lemma 4.5, we obtain a weak solution of.1). Let T be a regular tetraedrization of te domain D into tetraedra of maximal messize. We denote by N := {x 1,..., x N } te set of vertices and introduce te finite-element space V H 1 D), wic is te space of all continuous piecewise linear functions on T. A basis for V can be cosen to be {φ n ξ 1, φ n ξ, φ n ξ 3 } 1 n N, were {ξ i },,3 is te canonical basis for R 3 and φ n x m ) = δ n,m. Here δ n,m denotes te Kronecker delta symbol. Te interpolation operator from C D) onto V, denoted by I V, is defined by N I V v) = vx n )φ n x) v C D, R 3 ). n=1 Before introducing te finite element sceme, we state te following result proved by Bartels [4], wic will be used in te subsequent analysis. Lemma 5.1. Assume tat 5.1) φ i φ j dx for all i, j {1,,, J} and i j. D Ten for all u V satisfying ux l ) 1, l = 1,,, J, tere olds ) 5.) u I V dx u dx. u D D

17 FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 17 Wen d =, we note tat condition 5.1) olds for Delaunay triangulation. Rougly speaking, a Delaunay triangulation is one in wic no vertex is contained inside te perimeter of any triangle. Wen d = 3, condition 5.1) olds if all diedral angles of te tetraedra in T L D) are less tan or equal to π/; see [4]. In wat follows we assume tat 5.1) olds. To discretize te equation 4.6), we fix a positive integer J, coose te time step k to be k = T/J and define t j = jk, j =,, J. For j = 1,,..., J, te solution mt j, ) is approximated by m j) V, wic is computed as follows. Since we can define m j+1) m t t j, ) mt j+1, ) mt j, ) k from m j) by 5.3) m j+1) = m j) mj+1) + kvj), k m j) were v j) is an approximation of m t t j, ). Hence, it suffices to propose a sceme to compute v j). Motivated by te property t m m =, we will find v j) 5.4) W j) := { w V wx n ) m j) x n) =, n = 1,..., N in te space Wj) }. Given m j) V, we use 4.14) to define v j) and trial functions can be used see Remark 4.4). Hence, we define by v j) te following equation λ 1 m j) vj), wj) 5.5) L D) λ v j), wj) We summarise te algoritm as follows. Algoritm 5.1., defined by instead of using 4.6) so tat te same test LD) = µ m j) + kθvj) Wj) ), wj) + µf t j, m j), wj) ) P-a.s.. Step 1: Set j =. Coose m ) = I V m. Step : Find v j) Wj) satisfying 5.5). Step 3: Define N m j+1) m j) x) := x n) + kv j) x n) n=1 m j) x n) + kv j) x φ n x). n) satisfying L D) Step 4: Set j = j + 1, and return to Step if j < J. Stop if j = J. Since m ) x n) = 1 and v j) x n) m j) x n) = for all n = 1,..., N and j =,..., J, we obtain by induction) 5.6) m j) x n) + kv j) x n) 1 and m j) x n) = 1, j =,..., J. In particular, 5.6) sows tat te algoritm is well defined. We finis tis section by proving te following tree lemmas concerning some properties of m j) and R,k.

18 18 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE Lemma 5.. For any j =,..., J, m j) were D denotes te measure of D. L D) 1 and m j) L D) D, Proof. Te first inequality follows from 5.6) and te second can be obtained by integrating m j) x) over D. Lemma 5.3. Tere exist a deterministic constant c depending on m, {g i } q, λ 1 and λ suc tat for any θ [1/, 1] and for j = 1,..., J tere olds E m j) Proof. Taking w j) v j) λ or equivalently 5.7) µ m j) + k j 1 µ 1 λ E L D) = vj) L D) = µ, vj) v i) + j 1 L D) k θ 1) E v i) in equation 5.5) yields to te following identity: m j), vj) = λ L D) From Lemma 5.1 it follows tat m j+1) v j) c. L D) v + µkθ j) L + µf t j, m j) D) L D) v µkθ j) µf t j, m j) L D) L D) m j) L D) + kvj) ), L D) or equivalently, m j+1) m j) v + L D) L D) k j) + k L D) Terefore, togeter wit 5.7), we deduce m j+1) v + L D) kµ 1 j) λ + L D) k θ 1) v j) m j), vj), ) vj), L D). m j) L D) kf t j, m j) Tus, it follows from 3.6) tat E m j+1) + L D) kµ 1 λ E v j) + L D) k θ 1)E v j) E m j) [ + ke F t j, m j) L, )] D) vj) 1 + kcɛt )E m j) + ckɛt + T 1/ )E L D) + ckɛ 1 T + T 1/ + 1)E v j). L D), vj) L D) )., ) vj). L D) m j) L D)

19 By coosing ɛ = we deduce E FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 19 µ 1 λ ct +T 1/ +1) m j+1) in te rigt and side of tis inequality and using Lemma 5. + L D) kµ 1 λ E v j) + L D) k θ 1)E ck kc)e m j). L D) v j) L D) Replacing j by i in te above inequality and summing for i from to j 1 yeilds 5.8) E m j) + k j 1 µ 1 λ E L D) ckj + c m ) v i) + j 1 L D) k θ 1) E j 1 L D) + kc E m i) v i) L D). L D) Since m H D) it can be sown tat tere exists a deterministic constant c depending only on m suc tat 5.9) m ) L D) c. Hence, inequality 5.8) implies 5.1) E m j) + k j 1 µ 1 λ E L D) v i) By using induction and 5.9) we can sow tat + j 1 L D) k θ 1) E E m i L D) c1 + ck)i. j 1 c + kc E Summing over i from to j 1 and using 1 + x e x we obtain v i) L D) m i). L D) j 1 k E m i L D) ck 1 + ck)j 1 e ckj = c. ck i= Tis togeter wit 5.1) gives te desired result. 6. Te main result In tis section, we will use te finite element function m j) to construct a sequence of functions tat converges in an appropriate sense to a function tat is a weak martingale solution of 1.3) in te sense of Definition.. Te discrete solutions m j) and v j) constructed via Algoritm 5.1 are interpolated in time in te following definition.

20 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE Definition 6.1. For all x D, u, v V and all t [, T ], let j {,..., J} be suc tat t [t j, t j+1 ). We ten define m,k t, x) := t t j k m,k t, x) := mj) x), mj+1) v,k t, x) := v j) x), F k t, u, v) := F t j, u, v) x) + t j+1 t k P a.s.. m j) x), We note tat m,k t) is an F tj adapted process for t [t j, t j+1 ). Te above sequences ave te following obvious bounds. Lemma 6.. Tere exist a deterministic constant c depending on m, g, µ 1, µ and T suc tat for all θ [1/, 1], E m,k L D T ) + E m,k L D T ) + E v,k L D T ) + kθ 1)E v,k L D T ) c, were m,k = m,k or m,k. In particular, wen θ [, 1 ), E m,k L D T ) +E m,k L D T ) θ 1)k ) E v,k L D T ) c. Proof. It is easy to see tat J 1 m,k L D T ) = k i= J 1 L D) and v,k L D T ) = k v i) L D). m i) Bot inequalities are direct consequences of Definition 6.1, Lemmas 5., and 5.3, noting tat te second inequality requires te use of te inverse estimate see e.g. [14]) v i) L D) c v i) L D). Te next lemma provides a bound of m,k in te H 1 -norm and establises relationsips between m,k, m,k and v,k. Lemma 6.3. Assume tat and k approac, wit te furter condition k = o ) wen θ [, 1 ). Te sequences {m,k}, {m,k }, and {v,k} defined in Definition 6.1 satisfy te following properties: 6.1) E m,k H 1 D T ) c, 6.) E m,k m,k L D T ) ck, 6.3) E v,k t m,k L 1 D T ) ck, 6.4) E m,k 1 L D T ) c + k ). Proof. Te results can be obtained by using Lemma 6. and te arguments in te proof of [13, Lemma 6.3]. We now prove some properties of F and F k, wic will be used in te next two lemmas. i=

21 FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 1 Lemma 6.4. For any u, v L Ω; L, T ; H 1 D)) ), tere exists a constant c depending on T and {g i },,q suc tat 6.5) E[ F t, ut, ), vt, )) dt] c ) E[ u L D T ) ]) 1/ E[ v L D T ) ]) 1/ + E[ v L D T ) ]) 1/, ere, F = F or F k. Furtermore, 6.6) E[ F t, ut, ), vt, )) F k t, ut, ), vt, )) dt] ck 1/ ) E[ u L D T ) ]) 1/ E[ v L D T ) ]) 1/ + E[ v L D T ) ]) 1/. Proof. Proof of 6.5): Te first result of te lemma for F = F can be deduced from Lemma 3.9 by replacing s t, u ut, ), v vt, ) and using Hölder s inequality as follows: 6.7) E[ F t, ut, ), vt, )) dt] = c We first note tat E[ E[ F t, ut, ), vt, )) ] dt E[ ut, ) L D) ]) 1/ E[ vt, ) L D) ]) 1/ dt + c E[ ut, ) L D) ]) 1/ E[ vt, ) L D) ]) 1/ dt c E[ u L D T ) ]) 1/ E[ v L D T ) ]) 1/ + E[ v L D T ) ]) 1/ F k t, ut, ), vt, )) dt] = E[ F k t, ut, ), vt, )) ] dt J 1 j+1 = j= t j E[ F t j, ut, ), vt, )) ] dt, ten apply Lemma 3.9 for s t j, u ut, ) and v vt, ) to deduce E[ J 1 j+1 F k t, ut, ), vt, )) dt] c t j E[ ut, ) L D) ]) 1/ E[ vt, ) L D) ]) 1/ dt J 1 + c ct j= t 1/ + ct + T 1/ ) j + t j ) j+1 t j j= t j E[ ut, ) L D) ]) 1/ E[ vt, ) L D) ]) 1/ dt E[ ut, ) L D) ]) 1/ E[ vt, ) L D) ]) 1/ dt E[ ut, ) L D) ]) 1/ E[ vt, ) L D) ]) 1/ dt. Hence, 6.5) wit function F = F k follows by using Hölder s inequality. ).

22 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE Proof of 6.6): Noting tat E[ F t, ut, ), vt, )) F k t, ut, ), vt, )) dt ] J j+1 = E[ F t, ut, ), vt, )) F t j, ut, ), vt, )) ] dt 6.8) Here := j= J j= t j j+1 t j F j t, x, y) := E[ F j t t j, ut, ), vt, )) ] dt. F j 1,i s, x, y) ds + F j,i s, x, y) d W i s), in wic F j 1,i s, x, y) = F j 1,is + t j, x, y), F,i s, x, y) = F,is + t j, x, y) and W i s) = W i s + t j ) W i t j ). By using te same arguments as in te proof of Lemma 3.9 we obtain te same result for te upper bound of F j, namely Hence, tere olds 6.9) j+1 t j E F j s, u, v) cs E[ u L D) ]) 1/ E[ v L D) ]) 1/ E F j s, u, v) ds c + cs 1/ + s) E[ u L D) ]) 1/ E[ v L D) ]) 1/. j+1 + c ck t j j+1 t j j+1 t j + ck 1/ + k) Terefore, it follows from 6.8) and 6.9) tat E[ ck s E[ u L D) ]) 1/ E[ v L D) ]) 1/ ds s 1/ + s) E[ u L D) ]) 1/ E[ v L D) ]) 1/ ds, E[ u L D) ]) 1/ E[ v L D) ]) 1/ ds j+1 t j F t, ut, ), vt, )) F k t, ut, ), vt, )) dt ] E[ u L D) ]) 1/ E[ v L D) ]) 1/ ds + ck 1/ + k) E[ u L D) ]) 1/ E[ v L D) ]) 1/ ds. E[ u L D) ]) 1/ E[ v L D) ]) 1/ ds. Te result follows immediately by using Hölder s inequality, wic completes te proof of te lemma. Te following two Lemmas 6.5 and 6.6 sow tat m,k and m,k, respectively, satisfy a discrete form of 4.6).

23 FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 3 Lemma 6.5. Assume tat and k approac wit te following conditions k = o ) wen θ < 1/, 6.1) k = o) wen θ = 1/, no condition wen 1/ < θ 1. Ten for any ψ C, T ); C D) ), tere olds P-a.s. λ 1 m,k v,k, m,k L ψ + λ v,k, m,k ψ D T ) L D T ) 3 + µ m,k + kθv,k), m,k ψ) + µ F k t, m L,k, D T ) m,k ψ) dt = were I 1 := λ 1 m,k v,k + λ v,k, m,k ψ I V m,k ψ) I := µ m,k + kθv,k), m,k ψ I V m,k ψ)) L D T ), L D T ), I 3 := µ F k t, m,k, m,k ψ) F kt, m,k, I V m,k ψ)) dt. Furtermore, E I i = O) for i = 1,, 3. Proof. For t [t j, t j+1 ), we use equation 5.5) wit w j) j=1 I j, = I V m,k t, ) ψt, )) to see λ 1 m,k t, ) v,kt, ), I V m,k t, ) ψt, )) L D) + λ v,k t, ), I V m,k t, ) ψt, )) L D) + µ m,k t, ) + kθv,kt, )), I V m,k t, ) ψt, )) L D) + µf k t, m,k t, ), I V m,k t, ) ψt, )) ) =. Integrating bot sides of te above equation over t j, t j+1 ) and summing over j =,..., J 1 we deduce λ 1 m,k v,k, I V m,k ψ) + λ L v,k, I V m,k ψ) D T ) L D T ) + µ m,k + kθv T,k), I V m,k ψ) + µ F k t, m L,k, I V m,k ψ)) dt =. D T ) Tis implies λ 1 m,k v,k, m,k L ψ + λ v,k, m,k ψ D T ) + µ m,k + kθv,k), m,k ψ) + µ L D T ) = I 1 + I + I 3. L D T ) F k t, m,k, m,k ψ) dt

24 4 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE Hence it suffices to prove tat E I i = O) for i = 1,, 3. First, by using Lemma 5. we obtain 6.11) m,k L D T ) sup m j) L D) 1, j J and 6.1) m,k L D T ) sup m j) L D). j J Lemma 6. and 6.11) togeter wit Hölder s inequality and Lemma 7. yield [ ) ] E I 1 ce m,k L D T ) + 1 v,k L D T ) m,k ψ I V m,k ψ) L D T ) [ ] ce v,k L D T ) m,k ψ I V m,k ψ) L D T ) c E[ v,k L D T ) ]) 1/ E[ m,k ψ I V m,k ψ) L D T ) ]) 1/ c. Te bound for E I can be obtained similarly, using Lemma 6. and noting tat wen θ [, 1 ], a suitable bound on k v,k L D T ) can be deduced from te inverse estimate as follows: k v,k L D T ) ck 1 v,k L D T ) ck 1. Te bound for E I 3 can be obtained by noting te linearity of F in Remark 3.7 and using Lemmas 6.4 and 7.. Indeed, E I 3 = µe F k t, m,k, m,k ψ I V m,k ψ)) dt µe F k t, m,k, m,k ψ I V m,k ψ)) dt c E[ m,k L D T ) ]) 1/ E[ m,k ψ I V m,k ψ)) L D T ) ]) 1/ + ) E[ m,k ψ I V m,k ψ) L D T ) ]) 1/ c. Tis completes te proof of te lemma. Lemma 6.6. Assume tat and k approac satisfying 6.1)., T ); C D) ), tere olds P-a.s. C Ten for any ψ 6.13) λ 1 m,k t m,k, m,k ψ L D T ) + λ t m,k, m,k ψ L D T ) + µ m,k ), m,k ψ) L D T ) + µ F t, m,k, m,k ψ) dt = 7 I j, j=1

25 were FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 5 I 4 = λ 1 m,k v,k, m,k L ψ + λ 1 m,k t m,k, m,k ψ D T ) L D T ), I 5 = λ v,k, m,k L ψ λ t m,k, m,k ψ D T ) L D T ), I 6 = µ m,k + kθv,k), m,k ψ) µ m,k), m,k ψ) L D T ) L D T ), I 7 = µ F t, m,k, m,k ψ) F k t, m,k, m,k ψ)) dt. Furtermore, E I i = O) for i = 1,, 6 and E I 7 = O + k 1/ ). Proof. From Lemma 6.5 it follows tat λ 1 m,k t m,k, m,k ψ L D T ) + λ t m,k, m,k ψ L D T ) + µ m,k ), m,k ψ) L D T ) + µ F k t, m,k, m,k ψ) dt = I I 7. Hence it suffices to prove tat E I i = O) for i = 4,, 6. First, by using te triangle inequality and Hölder s inequality, we obtain λ 1 1 I 4 m,k m,k) v,k, m,k ψ L D T ) + m,k v,k, m,k m,k) ψ L D T ) + m,k v,k t m,k ), m,k ψ L D T ), m,k m,k L D T ) v,k L D T ) m,k L D T ) + m,k L D T )) ψ L D T ) + v,k t m,k L 1 D T ) m,k L D T ) ψ L D T ). Terefore, te required bound on E I 4 can be obtained by using 6.11), 6.1) and Lemmas 6., 6.3. Te bounds on E I 5 and E I 6 can be obtaineded similarly. In order to prove te bound for E I 7, we first use te triangle inequality ten Remark 3.7 and Lemma 6.4 to obtain E I 7 E F t, m,k, m,k ψ) F k t, m,k, m,k ψ) dt + E F k t, m,k m,k, m,k ψ) dt + E F k t, m,k, m,k m,k ) ψ) dt ck 1/ ) E[ m,k L D T ) ]) 1/ E[ m,k L D T ) ]) 1/ + E[ m,k L D T ) ]) 1/ + c ) E[ m,k m,k L D T ) ]) 1/ E[ m,k L D T ) ]) 1/ + E[ m,k L D T ) ]) 1/ c + k 1/ ), in wic 6.1) and 6.) are used to obtain te last inequality. Tis completes te proof of te lemma.

26 6 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE In order to prove te convergence of random variables m,k, we first state a result of tigtness for te family Lm,k ). We ten use te Skorood teorem to define anoter probability space and an almost surely convergent sequence defined in tis space wose limit is a weak martingale solution of equation 4.14). Te proof of te following results are omitted since tey are relatively simple modification of te proof of te corresponding results from [13]. Lemma 6.7. Assume tat and k approac, and furter tat 6.1) olds. Ten te set of laws {Lm,k )} on te Banac space C [, T ]; H 1 D) ) L, T ; L D)) is tigt. Proposition 6.8. Assume tat and k approac, and furter tat 6.1) olds. Ten tere exist: a) a probability space Ω, F, P ); b) a sequence {m,k } of random variables defined on Ω, F, P ) and taking values in C [, T ]; H 1 D) ) L, T ; L D)); and c) a random variable m defined on Ω, F, P ) and taking values in C [, T ]; H 1 D) ) L, T ; L D)), satisfying 1) Lm,k ) = Lm,k ), ) m,k m in C [, T ]; H 1 D) ) L, T ; L D)) strongly, P -a.s.. Moreover, te sequence {m,k } satisfies 6.14) E[ m,k HD T ) ] c, 6.15) E[ m,k 1 L D T ) ] c + k ), 6.16) m,k L D T ) c P -a.s., ere, c is a positive constant only depending on {g i },,q. We are now ready to state and prove our main teorem. Teorem 6.9. Assume tat T >, M H 1 D) satisfies??) and g i W, D) for i = 1,, q satisfy te omogeneous Neumann boundary condition. Ten m, te sequence {m,k } and te probability space Ω, F, P ) given by Proposition 6.8 satisfy: 1) te sequence of {m,k } converges to m weakly in L Ω ; H 1 D T )); and ) Ω, F, F t) t [,T ], P, M ) is a weak martingale solution of 1.3), were M t) := Z t m t) t [, T ], a.e. x D. Proof. From 6.16) and property ) of Proposition 6.8, tere exists a set V Ω suc tat P V ) = 1 and for all ω V tere old m,k ω ) L D T ) c and m,k ω ) m ω ) in L D T ) strongly. Hence, by using Lebesgue s dominated convergence teorem, we deduce 6.17) m,k m in L Ω ; L D T )) strongly, wic implies from 6.14) tat 6.18) m,k m in L Ω ; H 1 D T )) weakly.

27 FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 7 In order to prove Part ), by noting Lemma 4.5 and Remark 4.4 we only need to prove tat m satisfies 4.5) and 4.14), namely 6.19) m t, x) = 1, t, T ), x D, P -a.s. and 6.) Im, ϕ) = P -a.s. ϕ L, T ; H 1 D)), were Im, ϕ) :=λ 1 m t m, m ϕ L D T ) λ t m, m ϕ L D T ) µ m, m ϕ) L D T ) µ F t, m t, ), m t, ) ϕt, )) dt. By using 6.15) and 6.17), we obtain 6.19) immediately. In order to prove 6.), we first find te equation satisfied by m,k and ten pass to te limit wen and k approac. By using Lemmas 6.6 and property 1) of Proposition 6.8, it follows tat for any ψ C, T ; C D)) tat tere olds 6.1) E Im,k, ψ) = O + k1/ ). To pass to te limit in 6.1), we first using 6.17) 6.19) and te same arguments as in [13, Teorem 6.8] to obtain tat as and k tend to, 6.) m,k t m,k, m,k ϕ L Ω ;L D T )) m t m, m ϕ L Ω ;L D T )), 6.3) tm,k, m,k ϕ L Ω ;L D T )) t m, m ϕ L Ω ;L D T )), 6.4) m,k, m,k ϕ) L Ω ;L D T )) m, m ϕ) L Ω ;L D T )). Ten, by using Remark 3.7 and 6.5) wit F = F, we estimate E F t, m,k t, ), m,k t, ) ϕt, )) F t, m t, ), m t, ) ϕt, )) dt E + E F t, m,k t, ) m t, ), m,k t, ) ϕt, )) dt F t, m,k t, ) m t, ) ) ϕt, ), m t, )) dt c m,k m L Ω ;L D T )) m,k L Ω ;L D T )) + m,k L Ω ;L D T )) + m L Ω ;L D T )) + m L Ω ;L D T ))). Since m L Ω ; H 1 D T )), it follows from 6.14) and 6.17) tat 6.5) E [ F t, m,k t, ), m,k t, ) ϕt, )) dt] E [ F t, m t, ), m t, ) ϕt, )) dt ], as and k tend to. From 6.) 6.5) we deduce tat E Im,k, ψ) Im, ψ), and ence, togeter wit 6.1) E Im, ψ) =. Tis implies 6.) wic completes te proof of our main teorem.

28 8 BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE 7. Appendix For te reader s convenience we will recall te following results, wic are proved in [13]. Lemma 7.1. For any real constants λ 1 and λ wit λ 1, if ψ, ζ R 3 satisfy ζ = 1, ten tere exists ϕ R 3 satisfying 7.1) λ 1 ϕ + λ ϕ ζ = ψ. As a consequence, if ζ H 1 D T ) wit ζt, x) = 1 a.e. in D T and ψ L, T ; W 1, D)), ten ϕ L, T ; H 1 D)). Lemma 7.. For any v CD), v V and ψ C D T ), I V v L D) v L D), m,k ψ I V m,k ψ) L[,T ],H 1 D)) c m,k L[,T ],H 1 D)) ψ W, D T ), were m,k is defined in Defintion 6.1 Te next lemma defines a discrete L p -norm in V, equivalent to te usual L p -norm. Lemma 7.3. Tere exist -independent positive constants C 1 and C suc tat for all p [1, ] and u V, were Ω R d, d=1,,3. C 1 u p L p Ω) d N n=1 ux n ) p C u p L p Ω), Acknowledgements Te autors acknowledge financial support troug te ARC Discovery projects DP and DP Tey are grateful to Vivien Callis for a number of elpful conversations. References [1] F. Alouges. A new finite element sceme for Landau-Lifcitz equations. Discrete Contin. Dyn. Syst. Ser. S, 1 8), [] F. Alouges, A. de Bouard, and A. Hocquet. A semi-discrete sceme for te stocastic Landau Lifsitz equation. Stocastic Partial Differential Equations: Analysis and Computations, 14), [3] F. Alouges and P. Jaisson. Convergence of a finite element discretization for te Landau-Lifsitz equations in micromagnetism. Mat. Models Metods Appl. Sci., 16 6), [4] S. Bartels. Stability and convergence of finite-element approximation scemes for armonic maps. SIAM J. Numer. Anal., 43 5), 38 electronic). [5] L. Baňas, Z. Brzeźniak, A. Prol, and M. Neklyudov. A convergent finite-element-based discretization of te stocastic Landau Lifsitz Gilbert equation. IMA Journal of Numerical Analysis, 13). [6] W. Brown. Termal fluctuation of fine ferromagnetic particles. IEEE Transactions on Magnetics, ), [7] W. F. Brown. Termal fluctuations of a single-domain particle. Pys. Rev., ), [8] Z. Brzeźniak, B. Goldys, and T. Jegaraj. Weak solutions of a stocastic Landau Lifsitz Gilbert equation. Applied Matematics Researc express, 1), [9] I. Cimrák. A survey on te numerics and computations for te Landau-Lifsitz equation of micromagnetism. Arc. Comput. Metods Eng., 15 8), [1] G. Da Prato and J. Zabczyk. Stocastic Equations in Infinite Dimensions. Encyclopedia of Matematics and its Applications. Cambridge University Press, Cambridge, 14. [11] H. Doss. Liens entre quations diffrentielles stocastiques et ordinaires. Annales de l I.H.P. Probabilits et statistiques, ),

29 FEM FOR STOCHASTIC LANDAU LIFSHITZ GILBERT EQUATION 9 [1] T. Gilbert. A Lagrangian formulation of te gyromagnetic equation of te magnetic field. Pys Rev, ), [13] B. Goldys, K.-N. Le, and T. Tran. A finite element approximation for te stocastic Landau Lifsitz Gilbert equation. Journal of Differential Equations, 6 16), [14] C. Jonson. Numerical Solution of Partial Differential Equations by te Finite Element Metod. Cambridge University Press, Cambridge, [15] G. Ju, Y. Peng, E. K. C. Cang, Y. Ding, A. Q. Wu, X. Zu, Y. Kubota, T. J. Klemmer, H. Amini, L. Gao, Z. Fan, T. Rausc, P. Subedi, M. Ma, S. Kalarickal, C. J. Rea, D. V. Dimitrov, P. W. Huang, K. Wang, X. Cen, C. Peng, W. Cen, J. W. Dykes, M. A. Seigler, E. C. Gage, R. Cantrell, and J. U. Tiele. Hig density eat-assisted magnetic recording media and advanced caracterization progress and callenges. IEEE Transactions on Magnetics, 51 15), 1 9. [16] O. Kallenberg. Foundations of Modern Probability. Probability and Its Applications. Springer-Verlag New York, Cambridge,. [17] L. Landau and E. Lifscitz. On te teory of te dispersion of magnetic permeability in ferromagnetic bodies. Pys Z Sowjetunion, ), [18] H. J. Sussmann. An interpretation of stocastic differential equations as ordinary differential equations wic depend on te sample point. Bulletin of te American Matematical Society, ), Scool of Matematics and Statistics, Te University of Sydney, Sydney 6, Australia address: beniamin.goldys@sydney.edu.au Scool of Matematics and Pysics, Te University of Queensland, QLD 47, Australia address: j.grotowski@uq.edu.au Scool of Matematics and Statistics, Te University of New Sout Wales, Sydney 5, Australia address: n.le-kim@unsw.edu.au