LARGE DEVIATIONS AND TRANSITION BETWEEN EQUILIBRIA FOR STOCHASTIC LANDAU-LIFSHITZ EQUATION ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T.

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1 LARGE DEVIATIONS AND TRANSITION BETWEEN EQUILIBRIA FOR STOCHASTIC LANDAU-LIFSHITZ EQUATION ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ Abstract. We study a stochastic Landau-Lifshitz equation on a bounded interval and with finite dimensional noise; this could be a simple model of magnetisation in a needle-shaped domain in magnetic media. We obtain a large deviation principle for small noise asymptotic of solutions using the weak convergence method. We then apply this large deviation principle to show that small noise in the field can cause magnetisation reversal and also to show the importance of the shape anisotropy parameter for reducing the disturbance of the magnetisation caused by small noise in the field. Contents 1. Introduction Notations 4. Preliminaries 4 3. Existence of solutions 6 4. Uniqueness and the existence of a strong solution 1 5. Further regularity Small noise asymptotics Application to a model of a ferromagnetic needle Stable stationary states of the deterministic equation Noise induced instability and magnetization reversal 38 Appendix A. Uniqueness of strong solutions 43 References Introduction The aim of this paper is to study the stochastic Landau-Lifshitz model of magnetization for needle-shaped ferromagnetic domains. It is natural to describe such a domain as a bounded interval Λ R filled in by a ferromagnetic material. Let mx) R 3 denote the magnetisation vector at the point x Λ. For temperatures below the Curie point the length of the Date: March 11, Mathematics Subject Classification. 35K59, 35R6, 6H15, 8D4. Key words and phrases. stochastic Landau-Lifschitz equation, strong solutions, maximal regularity, large deviations, Freidlin-Ventzell estimates. The work of Zdzis lw Brzeźniak and of Ben Goldys was partially supported by the ARC Discovery Grant DP The research on which we report in this paper was begun at the Newton Institute for Mathematical Sciences in Cambridge UK) during the program Stochastic Partial Differential Equations. The INI support and excellent working conditions are gratefully acknowledged by all three authors. The first named author wishes to thank Clare Hall Cambridge) and the School of Mathematics, UNSW, Sydney for hospitality. 1

2 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ magnetisation vector is constant over the domain, hence it can be assumed that mx) = 1, x Λ. In this paper we will always assume that m satisfies the Neumann boundary condition m t, x) =, x x Λ. This assumption is standard in physical models of ferromagnetism. According to the Landau theory of ferromagnetism any stable magnetisation vector m should minimise the Landau energy functional E m). In this paper we will consider the energy functional of the form E m) = 1 mx) dx + β m x) + m 3x) ) dx F mx) dx, 1.1) Λ } {{ } exchange energy Λ } {{ } anistropy energy Λ }{{} energy of applied field where F R 3 is a fixed vector and F m is an inner product in R 3. H 1 Λ, R 3) of the energy E satisfy the conditions m L E m) =, mx) = 1, x =, Λ Stable points m where the gradient L E of E is understood with respect to the Hilbert space L Λ, R 3 ). If m is not a stable configuration then according to the theory of Landau and Lifschitz modified later by Gilbert, the magnetisation will evolve in time subject to the equation LLG in what follows): m t t, x) = mt, x) L E m)t, x) + αmt, x) mt, x) LE m)t, x)), where α >. One may expect that the solution to this equation tends to a stationary point of the energy for t. Let us recall that some magnetic memories for more information see for example [1]) are made of single-domain, needle-shaped particles of iron or chromium oxide about 3 to 7 nm long and about 5 nm in diameter. Each magnetic domain has two stable states of opposite magnetisation and these states are used to represent the bits and 1. The anisotropy energy is minimised when the magnetisation points along the needle axis. It is observed that under the influence of thermal noise the magnetisation can spontaneously reverse, causing loss of data. To investigate thermally induced magnetisation reversal, it is natural to use a stochastic version of the LLG equation. Informally speaking, we want to consider the LLG equation which corresponds to the energy functional with the applied field K = F + noise. More precisely, our aim will be to analyse transitions between equilibria under the influence of a small noise ɛdξt, x), where dξt, x) = 3 h i x)dw i t), i=1 ε represents dimensionless temperature, h i : Λ R 3 and W = W i ) is a standard Wiener process in R 3. Then it is natural to model the dynamics of the magnetisation M using the Stratonovitch stochastic differential equation dm = M L E M)dt ɛdw ) + αm M L E M)dt ɛdw, ) ) 1.)

3 where denotes the Stratonovitch integral. Let us note that the Stratonovitch integral allows us to preserve the constraint Mt, x) = 1 for all times. In order to formulate the final version of the equation we are going to study, we need some additional notations. Let f i ) be an orthonormal basis in R 3, and fy) = y f ) f + y f 3 ) f 3 y R 3, GM)h = M h αm M h), h : Λ R 3, see Section for more details. Then, using 1.1) we obtain the stochastic LLG to be studied in this paper: dm = GM) [ M βfm) + F ] dt + ɛgm) dξ, t, T ], M x t) Λ =, t, T ], M) = M. Note that since Gy)h is orthogonal to y R 3 for every h R 3, the formal application of the Ito formula yields Mt, x) = 1. To the best of our knowledge the stochastic LLG in the form 1.3) has not been studied before. The existence of weak martingale solutions is proved for a similar equation in a three-dimensional domain in our earlier work [3]. Kohn, Reznikoff and vanden-eijnden [14] modelled the magnetization M in a thin film, assuming that M is constant across the domain for all times and the energy functional contains the applied field only. In this case equation 1.) reduces to an ordinary stochastic differential equation dm = M F + ε dw ) αm M F + ε dw )) ) where W is now a standard Wiener process in R 3. Kohn, Reznikoff and Vanden-Eijnden used the large deviations theory to make a detailed computational and theoretical study of the behaviour of the solution. At the end of their paper, they remark that little is known about the behaviour of the solutions of stochastic Landau-Lifshitz equations which do not assume that the magnetization is uniform on the space domain. In this work we study a stochastic LLG equation 1.3) on a bounded interval of the real line. We start with the existence of weak martingale solution stated in Theorem 3.1. The main ideas in the proof of Theorem 3.1 are essentially the same as those in the proof of the existence theorem in [3]. Since the full proof of Theorem 3.1 is rather long and adds little to [3], in the proof of Theorem 3.1 we only outline the main ideas and points of difference; a detailed proof of Theorem 3.1 is in [4]. Since the domain Λ is one-dimensional, we are able to prove that weak solutions are in fact strong and unique. In Theorem 4. in Section 4, we state a pathwise uniqueness result for solutions of equation 1.3); the proof is straightforward and details are omitted. In Theorem 4.4, we assert the uniqueness in law of weak martingale solutions of equation 1.3) with paths in S := C[, T ]; H) L 4, T ; H 1, Λ, R 3)) ; we also assert the existence of a measurable mapping J : C [, T ]; R 3) S which maps the Wiener process W to a solution, y := JW ), of 1.3). Theorem 4.4 is a consequence of a very general version of the Yamada and Watanabe theorem proved in [16].

4 4 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ Next, we prove strong regularity properties of solutions to 1.3). Namely, we show that Λ Mt) 4 dx dt + see Theorem 5. for a precise formulation. Λ Mt) dx dt <, In Section 6 we prove the Large Deviations Principle for equation 1.3) with F =. We first identify the rate function and prove in Lemma 6.1 that it has compact level sets in the space X = C [, T ]; H 1, Λ; R 3)) L, T ; H, Λ; R 3)). The proof of this result follows from the maximal regularity and ultracontractivity properties of the heat semigroup generated by the Neumann Laplacian and the estimates for weak solutions of equation 1.3) obtained in Theorem 3.1 below. The Large Deviations Principle is proved in Theorem 6.5. To prove this theorem, we use the weak convergence method of Budhiraja and Dupuis [5, Theorem 4.4]. Following their work we show that the two conditions of Budhiraja and Dupuis, see Statements 1 and, Section 6, are satisfied and then Theorem 6.5 easily follows. A long proof that these two conditions are satisfied is split into a number of lemmas. We note that our proof is simpler than the corresponding proofs in [7] and [1]. In particular, we do not need to partition the time interval [, T ] into small subintervals. In Section 7, we apply the Large Deviations pprinciple to a simple stochastic model of magnetization in a needle-shaped domain. We show that small noise in the applied field causes magnetization reversal with positive probability. We also obtain an estimate which shows the importance of the shape anisotropy parameter β, for reducing the disturbance of the magnetization caused by small noise in the field. The results we obtain partially answer a question posed in [14] and provide a foundation for the computational study of stochastic Landau-Lifshitz models with one dimensional domain Notations. The inner product of vectors x, y R 3 will be denoted by x y and x will denote the Euclidean norm of x. We will use the standard notation x y for the vector product in R 3. For a domain Λ we will use the notation L p for the space L p Λ; R 3), W 1,p for the Sobolev space W 1,p Λ; R 3) and so on. For p = we will often write H instead of L, H 1 instead of W 1, and H instead of W,. We will always emphasize the norm of the corresponding space writing f H, f H 1 and so on. We will also need the spaces L p, T ; E) and C[, T ]; E) of p-integrable, respectively continuous, functions f : [, T ] E with values in a Banach space E. If E = R then we write simply L p, T ) and C[, T ]). Throughout the paper C stands for a positive real constant whose actual value may vary from line to line. We include an argument list, Ca 1,..., a m ), if we wish to emphasize that the constant depends only on the values of the arguments a 1 to a m.. Preliminaries We start with some basic concepts and notation that will be in constant use throughout the paper.

5 Let us recall that Λ R is a bounded interval. We define a linear operator A : DA) H H by { DA) := {u H : ux) = for x Λ},.1) Au := u for u DA). Let us recall that the operator A is selfadjoint and nonnegative and D A 1/) when endowed with the graph norm coincides with H 1. Moreover, the operator A + I) 1 is compact. In what follows we will need the following, well known, interpolation inequality: where the optimal value of the constant k is u L k u H u H 1 u H 1,.) k = max 1, ) 1. Λ For v, w, z H 1 by the expressions w v and z w v) we understand the unique elements of the dual space H 1 ) of H 1 such that for any φ H 1 and H 1 ) w v, φ H 1 = φ w), v H.3) H 1 ) z w v), φ H 1 = φ z) w), v H,.4) respectively. Note that the space H 1 Λ) is an algebra, hence for v, w, z H 1, linear functionals H 1 φ RHS of.3) or.4)) are continuous. In particular, since a b, a = for a, b R 3, we obtain H 1 ) w v, v H 1 = v w, v H.5) H 1 ) z v v), φ H 1 = φ z) v, v H,.6) and since a a = for a R 3, equation.3) yields H 1 ) φ v, φ H 1 = φ φ), v H =..7) The maps H 1 y y y H 1 ) and H 1 y y y y) H 1 ) are continuous homogenous polynomials of degree, resp. 3 hence they are locally Lipschitz continuous. For any real number β >, we write X β for the domain of the fractional power operator D A β) endowed with the norm x X β = I + A) β x and X β denotes the dual space of X β so that X β H = H X β is a Gelfand triple. Let C T = C [, T ]; R 3 ), F W T, F T W, ) µw denote the classical Wiener space, where µ W stands for the Wiener measure on C [, T ]; R 3 ) and F W T = Ft W ) is a µ W -completion of the natural filtration F T = Ft ) t [,T ] of the Wiener process. We will say that an F T predictable function F : [, T ] C [, T ]; R 3 ) R 3 belongs to P T if F T = sup ω C [,T ];R 3 ) F t, ω) dt 1/ <. 5 Let us recall that if a given process Z is F W T version that is F T -predictable. -predictable then it possesses an indistinguishable

6 6 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ We will denote by g the mapping g : R 3 L R 3) defined as gy)h = y h αy y h)..8) The function g is of class C. In particular, we have [ g y)h ] z = [gy)h] z = z h α[z y h) + y z h)], h, y, z R 3,.9) and for every r > sup gy) + g y) ) <..1) y r Clearly, we can define a mapping f, h) gf)h for f, h from functions spaces of R 3 -valued functions. If f L and h L then gf)h is a well defined element of L. We will use the notation G for a mapping Gf) : L L given by For fixed h i L, i = 1,, 3, let B : R 3 L be defined as 3 By = y i h i. Gf)h = f h αf f h)..11) i=1 In the next lemma we use the notation h = h i ) and h L = max i 3 h i L. Lemma.1. Assume that f i L, i = 1, and f i L r. Then there exists C r > such that G f 1 ) h G f ) h L C r h L f 1 f L, h L. 3. Existence of solutions We will be concerned with the following stochastic integral equation Mt) = M + + ε β Note, that the expresssion t ε [M M) αm M M)] ds GM)B dw s) + ε GM)fM) ds + GMs))B dw s) + ε 3 i=1 can be identified with the Stratonovich integral t ε 3 i=1 GMs))B dw s) [ G M)h i ] GM)hi ) ds GM)BF s, W ) ds, t [, T ]. [ G Ms))h i ] GMs))hi ) ds 3.1)

7 7 but we will not use this concept in the paper. We will now formulate the main result of this Section. Theorem 3.1 Existence of a weak martingale solution in H 1 ). Assume that h = h i ) 3 i=1 H 1 ) 3. Let T > be fixed and assume that F PT. Then for every M H 1 there exists a filtered probability space Ω, F, F t ), P), an F t )-adapted Wiener process W in R 3 and an F t )-adapted process M such that 1) for each β < 1 the paths of M are continuous Xβ -valued functions P-a.s.; ) For every p 1 E [ ] sup Mt) p C T, p, α, F H 1 T, M, h ) ; t [,T ] 3) For almost every t [, T ] we have Mt) Mt) L and E Ms) Ms) H ds p C T, α, F T, M, h ) 4) Mt)x) R 3 = 1 for all x Λ and for all t [, T ], P-a.s.; 5) For every t [, T ] equation 3.1) holds P-a.s. Note that in Theorem 3.1, M is an H 1 -valued process, hence the expressions Ms) Ms) and Ms) Ms) Ms)) are interpreted in the sense of.3) and.4) respectively. Proof. The proof of Theorem 3.1 is very similar to the proof of Theorem.7 in [3] and here we only sketch the main arguments putting for simplicity h 1 = h and h = h 3 =. We start with some auxiliary definitions. Let e 1, e, e 3,... be an orthonormal basis of H composed of eigenvectors of A. For each n 1, let H n be the linear span of {e 1,..., e n } and let π n u = n u, e i H e i, u H. i=1 Let G n f) = π n G π n f) π n, f L and G nf) = π n G π n f) π n, f L. Formally, G n is a derivative of G n. For each n 1, we define a process M n : [, T ] Ω H n to be a solution of the following

8 8 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ equation in H n : M n t) = π n M + α + ε β π n M n M n ) ds 3.) π n M n M n M n )) ds G n M n ) BdW s) + ε G n M n ) f M n ) ds + 3 i=1 [ G n M n ) h i ] Gn M n ) h i ) ds G n M n ) BF s, W ) ds. The proof of Theorem 1.6 in [8] can be used to show that, for each n 1, equation 3.) has a unique strong in the probabilistic sense) solution. Applying the Itô formula and the Gronwall lemma to the processes M n ) H and M n ) H, one can obtain the following, 1 uniform in n 1, estimates. Lemma 3.. Let the assumptions of Theorem 3.1 be satisfyied. Then for each n 1 M n t) H = π n u H, for all t [, T ] P a.s. Moreover, for each p [1, ) there exists a constant C α, T, R, p, M, h) such that for every n 1 E sup M n t) p C α, T, R, p, M H 1, h), 3.3) t [,T ] and E E M n s) M n s) H ds M n s) M n s) M n s)) H ds p C α, T, R, p, M, h) p C α, T, R, p, M, h). Proposition 3.3. There exists a probability space Ω, F, P ) and there exists a sequence W n, M n) of C[, T ]) C[, T ]; X 1 )-valued random variables defined on Ω, F, P ) such that the laws of W, M n ) and W n, M n) are equal for each n 1 and W n, M n) converges pointwise in C[, T ]) C[, T ]; X 1 ), P -a.s., to a limit W, M ) with distribution P W,M. Proof. Lemma 3. and the two key results of Flandoli and G atarek [11, Lemma.1 and Theorem.] can be used to obtain a subsequence of W, M n )) n 1 which converges in law on C[, T ]) C[, T ]; X 1/ ) to a limiting distribution P W,M. Then the proposition follows from the Skorohod theorem see [1, Theorem 4.3]). It remains to show that the pointwise limit W, M ) defined on the probability space Ω, F, P ) satisfies all the claims of Theorem 3.1. For each n 1, W n, M n) satisfies an equation obtained from 3.) by replacing W and M n by W n and M n, respectively. Then the

9 processes M n satisfy the estimates of Lemma 3.. These estimates together with the pointwise convergence of the sequence W n, M n)) n 1 imply that the W, M ) satisfies equation 3.1). The proof of part 5 of Theorem 3.1 is analogous to the proofs of [3, Lemma 5.1 and Lemma 5.] but an additional care must be taken to prove that for every t T G n M n ) BF s, W n ) ds To this end we note first that the processes and t [, T ] Ω s, ω ) F s, W ω )) R 3 [, T ] Ω s, ω ) F s, W n ω )) R 3 G n M ) BF s, W ) ds P a.s. 3.4) are well defined and predictable on the space Ω, F, F, P ). For any t, T ] we have G n M n ) BF s, W n ) ds = + t Gn M n ) G M n )) BF s, W n ) ds ) G M n G M )) BF t s, W n ) ds + 9 G M ) BF s, W n) ds. Using the estimate 3.3) and the definition of G it is easy to check that in the above identity the first two terms on the right-hand side converge to zero P -a.s. To complete the proof of 3.4) we use the fact that, given η >, there exists a bounded function F η : [, T ] C [, T ]; R 3 ) R 3 measurable with respect to the completed Borel σ-algebra B [, T ] C [, T ], R 3 )) such that F η s, ) is continuous on C [, T ]; R 3 ) for each fixed s [, T ] and C [,T ];R 3 ) F s, f) F η s, f) ds µ W df) < η. This fact follows form the assumption that the associated mapping F : C [, T ]; R 3 ) L [, T ]; R 3) is Borel-measurable and bounded and µ W is a Radon measure. Finally, G M ) BF t s, W n ) ds = + G M ) BF s, W n) BFη s, W n )) ds G M ) BF η s, W n ) ds. Since each process W n has distribution µ W on the space C [, T ]; R 3 ) convergence 3.4) follows by taking first n and then η.

10 1 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ 4. Uniqueness and the existence of a strong solution The main results in this section are Theorem 4., on pathwise uniqueness of solutions of equation 3.1) and Theorem 4.4 which is a version of the well known Yamada and Watanabe theorem, on uniqueness in law and the existence of a strong solution to equation 3.1). We start with a simple Lemma 4.1. Let u be an element of H 1 such that Then where the equality holds in H 1 ). ux) = 1 for all x Λ. u u u) = u u u, 4.1) Proof. Using definition.4) and invoking a well known identity we obtain for any φ H 1 : a b c) = a c)b a b)c, a, b, c R 3, H 1 ) u u u), φ H 1 = [φ u) u], u H = u φ, u H [u φ)u], u H. Since by assumption u u = 1 u ) = for a.e. x Λ, we obtain H 1 ) u u u), φ H 1 = = φ, u H u u, φ H H 1 ) u u u, φ H 1 Theorem 4. Pathwise uniqueness). Assume that Ω, F, F T, P) is a filtered probability space with the filtration F T = F t ) t [,T ] and an R 3 -valued F-Wiener process W t)) t [,T ] defined on this space. Assume also that F P T. Let M 1, M : [, T ] Ω H be F-progressively measurable continuous processes such that, for i = 1,, the paths of M i lie in L 4, T ; H 1), satisfy property 4) from Theorem 3.1 and each M i satisfies the equation M i t) = M + + ε β M i M i ds α G M i ) BdW s) + ε G M i ) f M i ) ds + for all t [, T ], P-almost everywhere. Then M i M i M i ) ds 3 j=1 M 1, ω) = M, ω), for P a.e. ω Ω. [ G M i ) h j ] G Mi ) h j ds G M i ) BF s, W ) ds 4.)

11 11 Proof. Let us fix F P T and let R > be such that Note that the above implies that F t) dt R, P a.s. F t) dt R T, P a.s. 4.3) First, we note that by Lemma 4.1 the following equality holds in X 1/. M i s) M i s) M i s)) = M i s) M i s) M i s). Let us assume that M 1 and M are two solutions. Because M i are uniformly bounded by assumption, by the local Lipschitz property of G, G and f, as well by the assumptions that each h i L, there exists a constant C 1 >, such that for all t [, T ], 3 GM t))h i GM 1 t))h i L C 1 h L M t) M 1 t) L, 4.4) i=1 3 i=1 G M t))h i GM t))h i G M 1 t))h i GM 1 t))h i L C 1 h L M t) M 1 t) 4.5) L. [ G M t)) G M 1 t)) )] BF s, W ), M t) M 1 t) L 4.6) C F t) h L M t) M 1 t) L G M t)) f M t)) G M 1 t)) f M 1 t)), M t) M 1 t) L 4.7) C 1 M t) M 1 t) L Let Z = M M 1. Then by Lemma 4.1 the process Z belongs to M, T ; V ) L Ω, C[, T ]; H) and Z is a weak solution of the problem dzt) = αaz dt + α M M M 1 M 1 ) dt 4.8) + M M M 1 M 1 ) dt + ε G M ) G M 1 ) ) BdW s) + ε 3 ] [G M ) h j G M ) h j G M 1 ) h j G M 1 ) h j dt j=1 [ ] β G M ) f M ) G M 1 ) f M 1 ) dt + [ G M t)) G M 1 t)) )] BF s, W ) dt

12 1 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ We can check that all assumptions of the Itô Lemma from [17] are satisfied to be done!) and therefore we have 1 d Zt) H = AZ, Z dt + α M t) M t) dt M 1 t) M 1 t), Z dt + α M 1 t) + M t))m 1 t) Z, Z dt [ ] + M t) Z, Z Z M 1 t), Z dt + ε 3 G M t)) h j G M t)) h j G M 1 t)) h j G M 1 t)) h j, Z dt j=1 β G M t)) f M t)) G M 1 t)) f M 1 t)), Z dt + [ G M t)) G M 1 t)) )] BF s, W ), Z dt ε G M t)) G M 1 t)) ) h j Hdt j=1 + ε 3 G M t)) G M 1 t)) ) h j, Z dw j s) j=1 = 8 3 I i t) dt + I 9,j t) dw j t) 4.9) i=1 j=1 We will estimate all the terms in 4.9). In what follows we will often use inequality.) and k is the constant from that inequality. Let us start with the 1 st term: As for the nd term we have Next, M M M 1 M 1, Z I 1 t) = AZt), Zt) = Zt). = M Z, Z + M 1 + M )M 1 Z, Z =: I + II i. i=1 I M L Z L k M L Z L Z H 1 k M ) L Z L Z L + Z L k M L Z L + k M L Z L Z L k M L Z L + k4 η M 4 L Z L + η Z L,

13 13 and II i M i L M 1 L Z L Z L M i L Z L Z L k M i L Z L Z 1 1 L Z + Z 1 ) L L k M i L Z L Z L + k M i L Z 1 L Z 3 L Hence, k η M i L Z L + η Z L + k4 4η 6 M i 4 L Z L η Z L. I t) = M M M 1 M 1, Z k [ M L + k η M 4 L 1 + η M i L + k 4η 6 M i 4 L ] Z L + 5 η Z L i=1 i=1 Let us note now that by.7), the nd part of the 4 th term, i.e. Z M 1, Z is equal to. Next, by definition.5), similarly as the estimate of II i above, we have the following estimates for the 1st part of the 4 th term using the bound Z L, we get M Z, Z = Z M, Z Z L M L Z L k η M i L Z L + η Z L + k4 4η 6 M i 4 L Z L η Z L Therefore, we get the following inequality for the 4 th term [ ] I 4 t) = M t) Z, Z Z M 1 t), Z k η M i L Z L + η Z L + k4 4η 6 M i 4 L Z L η Z L Next, we will deal with the 3 rd term. Since M 1 L = 1, the Hölder inequality yields M j M 1 Z, Z M j L M 1 L Z L Z L M j L Z L Z L k η M i L Z L + η Z L + k4 4η 6 M i 4 L Z L η Z L. Therefore, we get the following inequality for the 3 rd term I 3 t) = M 1 + M )M 1 Z, Z = M j M 1 Z, Z k η M i L ) Z L j=1 i=1 + η Z L + k4 4η 6 M i 4 L ) Z L + 3 η Z L By inequalities 4.4), 4.6) and 4.7) we get the following bound for the 5 th, 6 th and 8 th terms I i t) C 1 Zt) L. i=5,6,8 i=1

14 14 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ Finally, for the last term we get by 4.6) Finally, let us define an R-valued process ξ 9 t) := I 7 t) C 1 F t) Zt) L. 3 I 9,j s) dw j s) t [, T ]. j=1 Obviously, ξ 9 is an L -valued martingale. Next we add together the terms containing η Z L to obtain 19 4 η Z L 5η Z L. Choosing η in such a way that 5η = 1, we introduce a process ϕt) = C + F t) + k [ M L + k η M t) 4 L + 1 η M it) L + k ] 4η 6 M i t) 4 L i=1 i=1 + k η M it) L + k4 4η 6 M it) 4 L + k4 4η 6 M i t) 4 L, t [, T ]. Therefore, from all our inequalities we infer that for some constant C > Zt) L i=1 ϕs) Zs) L ds + ξ 9 t), t [, T ] 4.1) Then by the Itô Lemma applied to an R-valued process see [] for a similar idea) we infer that Y t) e = ε 3 j=1 Y t) := Zt) L e ϕs) ds dξ9 s) e ϕs) ds, t [, T ], ϕs) ds G M t)) G M 1 t)) ) h j, Z dw j s), t [, T ]. Since M 1, M and Z are uniformly bounded and G is locally Lipschitz the process defined by the RHS of the last inequality is an F-martingale. Thus, we infer that EY t), t [, T ], and since Y is nonnegative, we deduce that Y t) =, P-a.s., for every t [, T ]. Finally, the definition of Y yields This completes the proof. Zt) = P a.s., for every t [, T ].

15 Remark 4.3. Our uniqueness result should also hold in a stronger sense as follows. Suppose that M 1 is a solution in the sense of the last theorem and M a solution in the sense of Theorem 3.1, defined on the same filtered probability space. Then M 1 = M as in Theorem 4.. By an infinite-dimensional version of the Yamada and Watanabe theorem [16] pathwise uniqueness and the existence of weak solutions implies uniqueness in law and the existence of a strong solution. In Theorem 4.4 below, we state such a result for equation 3.1). In what follows we say weak solution instead of weak martingale solution and, to simplify notation, we set S T := C[, T ]; H) L 4, T ; H 1 ). 4.11) Theorem 4.4. Let assumptions of Theorem 4. be satisfied. Then uniqueness in law and the existence of a strong solution holds for equation 3.1) in the following sense: 1) if W, M) and W, M ) are two weak solutions of equation 3.1) with W and W being two Wiener processes, defined on possibly different probability spaces and M and M are S T -valued random variables, then M and M have the same law on S T ; ) there exists a measurable function J : C [, T ]; R 3), B C[,T ];R )) 3 ST, B ST ) with the following property: for any filtered probability space Ω, F, Ft ) t [,T ], P), where the filtration F t ) is such that F contains all sets in F of P-measure zero, and for any F t )-Wiener process W t)) t [,T ] defined on this space, the process J W ) is F t )-adapted and the pair W, J W )) is a weak solution of equation 3.1). By Theorem 4. it is enough to apply in the space S T the Yamada-Wantanbe theorem in the version proved in [16]. 5. Further regularity In this section, we assume that W, M) is a given weak solution of equation 3.1) such that M has paths in the space S T defined in 4.11). Recall that, by Theorem 4.4, the law of M is unique. Some regularity properties of M are listed in Theorem 3.1. The main result of this section is Theorem 5., where we prove stronger regularity of the solution. In Proposition 5.4, we use this estimate to show that paths of M lie in C[, T ]; H 1 ), P-almost everywhere; this improves upon the continuity property in Theorem 3.1. We start with a lemma that expresses M in a form which allows us to exploit the regularizing properties of the semigroup e ta ). The proof of this well known fact is omitted. Lemma 5.1. For each t [, T ] P-a.s. Mt) = e αta M + + β e αt s)a M M)ds + α e αt s)a GM)BF s, W ) ds + ε 1 e αt s)a fm) ds + ε 3 i=1 e αt s)a M M ) ds e αt s)a GM)B dw s) e αt s)a G M)h i GM)h i ds )

16 16 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ Theorem 5.. Assume that M H 1, F P T and h = ) 3 h i i=1 H1 ) 3 and let M be the corresponding weak solution. Then there exists a constant C = C α, T, F, M, h ) such that T E Mt) H dt + Mt) 4 L dt C. 4 Proof. We will use repeatedly the following well known properties of the semigroup e ta). The semigroup e ta), where A is defined in.1, is ultracontractive see, for example, []), that is, there exists C > such that for 1 p q e ta f L q t 1 C ) 1 f L p, f L p, t >. 5.) p 1 q It is also well known that A has maximal regularity property, that is, there exists C > such that for any f L, T ; H) and we have ut) = e t s)a fs)ds, t [, T ], T Aut) H dt C ft) H dt. 5.3) Without loss of generality we may assume that h = h 3 = and put h 1 = h. Then by Lemma 5.1 M can be written as a sum of seven terms: 6 Mt) = m i t), i= and we will consider each term separately. Let us fix for the rest of the proof T > and δ 5 8, 3 4). In what follows, C stands for a generic constant that depends on T only. In order to simplify notation, we put without loss of generality) ɛ = α = 1. We will show first that E To this end we will prove a stronger estimate: Mt) 4 W 1,4 dt C α, T, R, M, h). 5.4) E A δ Mt) 4 dt C α, T, R, M, h). 5.5) H Then 5.4) will follow from the Sobolev imbedding X δ W 1,4. We start with m. For each t, T ], we have A δ e ta M 4 H C t 4δ M 4 H 1,

17 and therefore, We will consider m 1. Putting f = M M we have A δ m t) 4 dt C M 4 H H ) A δ e t s)a fs) H Ct s) δ fs) H, < s < t < T, hence applying the Young inequality we obtain T A δ m 1 t) 4 H dt C C s 4δ/3 ds Thereby, since 4 3δ < 1, part 3) of Theorem 3.1 yields E t s) δ fs) H ds 3 4 fs) H ds A δ m 1 t) 4 H dt CR, T, M, h). 5.7) Since for every t [, T ] we have Mt, x) = 1- x-a.e. and h H 1, the estimate??) implies that there exists deterministic c > such that GM) + G M)hGM)h c. Therefore, the same arguments as for m 1 yield and E E A δ m 3 t) 4 H dt CR, T, M, h). 5.8) A δ m 5 t) 4 H dt CR, T, M, h). 5.9) We will consider the next term m using the fact that f = M M L 1, T ; H). Invoking the semigroup property of e ta and the ultracontractive estimate 5.) with p = 1 and q = we find that there exists C > such that obtain for s, t such that P-a.s. Therefore, A δ e t s)a C fs) H t s) δ+ 1 4 A δ e t s)a fs) ds 4 H dt sup M r) H, < s < t [, T ]. 1 r T dt C sup M r) 8 H 1 r T 4 ds t s) δ+ 1 4 ds dt. 17

18 18 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ hence since δ < 1) Theorem 3.1 yields In order to estimate m 4 we recall that E A δ m t) 4 dt C R, T, M, h). 5.1) H GMs))h a + b Ms), 5.11) where a, b > are constants depending on h only. Invoking Lemma 7. in [9] we find that for any t [, T ] 4 E A δ e t s)a GMs))h dŵ s) H CT )E = CT )E CT )E Theorem 3.1 now yields A δ e αt s)a GMs))h H ds A δ 1 e t s)a A 1 GMs))h H ds GMs))h H 1 t s) δ 1 ds CT )E sup M r) 8 H. 1 r T E A δ m 4 t) 4 H dt CT, R, M, h). 5.1) Finally, combining estimates 5.6) to 5.1) we obtain 5.5) and 5.4) follows. It remains to prove that E AMt) H dt CT, R, M, h). 5.13) To this end we note first that using the maximal inequality 5.13) and the first part of the proof it is easy to see that The estimate E Am i t) H dt CT, R, M, h), i = 1,, 3, ) Am t) H dt C M ), 5.15)

19 is an immediate consequence of the fact that M H 1 = D A 1/). We will consider now the stochastic term m 4. Using 5.11), a result of Pardoux in [17] and part 1 of Theorem 3.1 we find that E T Am 4 t) H dt E a Mt)) H + b ) dt C α, T, R, M, h). Combining 5.14), 5.15) and 5.16) we obtain 5.13) and the proof is complete ) Remark 5.3. By estimate 5.4) the vector AMt, x) R 3 is well defined t, x-a.e. and Mt, x) = 1 t, x-a.e., hence Therefore, an elementary identity yields Mt, x) Mt, x) = Mt, x), t, x a.e. a b + a b = a b, a, b R 3, Mt, x) Mt, x) + Mt, x) 4 = Mt, x), Proposition 5.4. Paths of M lie in C[, T ]; H 1 ), P-a.s. Proof. The proposition follows easily from the results in [17]. t, x a.e. Corollary 5.5. Let h i W 1,3, i = 1,, 3, and let F P T. Let W be an F T -adapted Wiener process probability space Ω, F, F T, P) Then, for every M H 1 and ɛ >, there exists a unique pathwise solution M ɛ C [, T ]; H 1) L, T ; DA)) of the equation Mt) = M + α ε 3 i=1 β M ds + α G M)h i GM)h i ds + ε GM)fM) ds + M M ds + GM) dŵ s) GM) F s, W ) ds, M M ds where all the integrals,except the Itô integral, are the Bochner integrals in H. In what follows we will denote by X T the Banach space X T = C [, T ]; H 1) L, T ; DA)). 6. Small noise asymptotics In this section we will prove the large deviation principle for the family of laws of the solutions of equation 3.1) with F identically zero and the parameter ε, 1] approaching zero.

20 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ The setting in this section is given by a probability space Ω, F, P) carrying an R 3 -valued Wiener process W t)) t [,T ]. We will denote by Ft ) the natural filtration of this Wiener process. For each ε [, 1] and F P T, let J ε F : C [, T ]; R 3) S denote the Borel mapping defined in Theorem 4.4 that acts on a Wiener process to give a weak solution of equation 3.1). By Corollary 5.5 the image of the Wiener process under the map JF ε has paths in X T almost surely. In what follows, we consider JF ε as a Borel map J ε F : C [, T ]; R 3) X T. If F t, W ) = vt) with v L, T ; R 3) is deterministic, then by Corollary 5.5 the function y v = J v is the unique solution of the equation y v t) = M + α β y v ds + α G y v ) f y v ) ds + y v y v ds + G y v ) Bvs) ds, y v y v ds 6.1) where the mapping G has been defined in.11) and h i W 1,3 for i = 1,, 3. Hence the map L, T ; R 3) v J v X T is well defined. We will define now the rate function I : X T [, ] by the formula 1 T Iu) := inf φs) ds : φ L, T ; R 3) and u = Jφ where the infimum of the empty set is taken to be infinity. For R >, define B R := φ L, T ; R 3) : φs) ds R with the weak topology of L, T ; R 3). In order to show that the family of laws {L J ε W )) : ε, 1]} satisfies the large deviation principle with the rate function I we will follow the weak convergence method of Budhiraja and Dupuis [5], see also Duan and Millet [1] and Chueshov and Millet [7]. To this end we need to show that the following two statements are true. Statement 1. For each R >, the set { J v : v B R } is a compact subset of XT. Statement. Let ε n ) be any sequence from, 1] convergent to and let v n ) be any sequence of R-valued Ft ) -predictable processes defined on [, T ] such that v T R for a certain R >. Then if v n ) converges in law on B R to v, then Jv εn n W ) converges in law on X T to Jv. The remaining part of this section is devoted to the proof of thsese two statements.

21 Lemma 6.1. Suppose that w n ) L, T ; R 3) is a sequence converging weakly to w. Then the sequence y wn converges strongly to y w in X T. In particular, the mapping is Borel. B R w J w X T Proof of Lemma 6.1. To simplify notation, we write y n for y wn, y for y w and set u n = y n y. Let R = sup w n s) ds. n N Let us recall the notation h = h i ). By Theorem 3.1, Theorem 5. and uniqueness of solutions, there exists a finite constant C T, α, R, M, h) such that for all n N: and clearly The same estimates hold for y. sup y n t) H 1 CT, α, R, M, h), 6.) t [,T ] ) y n s) H + y n 4 L 4 ds CT, α, R, M, h) 6.3) y n t)x) = 1, x Λ, t [, T ]. 6.4) Step 1. We note first that each subsequence of y n ) has a further subsequence that converges in C[, T ]; H) strongly. Indeed, since H 1 is compactly embedded in H, the estimate 6.) implies that for each fixed t [, T ], {y n t) : n 1} is relatively compact in H. Hence, it is enough to show that {y n : n 1} is a uniformly equicontinuous subset of C[, T ]; H). To this end we write for t < t T : y n t ) y n t) = t β y n y n ds + α t G y n ) f y n ) ds + t y n ds + α t t y n y n ds G y n ) Bw n ds. Then 6.), 6.3) and the Cauchy-Schwarz inequality yield immediately y n t ) y n t) H 1 + α) CT, α, R, M, h) + R M H h L ) t t. This completes the proof of Step 1. Step. Let q L, T ; H). Recall that w n w weakly in L, T ; R 3). We will show that lim sup n qs), u n s) H w n s) ws)) ds =. 6.5) t [,T ] 1

22 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ By Step 1 we can assume that u n u in C[, T ]; H). For n N { } we define operators K n : L, T ; R 3) C [, T ]; R 3) by the formula K n vt) = qs), u n s) H vs) ds, t [, T ], v L, T ; R 3). The operators K n, are compact because the functions q ), u n ) H belong to L, T ; R). Moreover, since the sequence q ), u n ) H converges strongly in L, T ; R) to a function q ), u ) H we infer that lim K n K =. n Since K n w n w) C[,T ];R 3 )) K n K w n w L [,T ];R 3 ) + K w n w) C[,T ];R 3 ), the claim follows immediately by the compactness of K because w n w weakly in L [, T ]; R 3). Step 3. We will establish uniform estimates on u n. More precisely, we will show that [ ] sup sup u n t) H + α u 1 n H ds n N t [,T ] <. 6.6) Without loss of generality we may assume that h = h 3 = and put h 1 = h and w 1 n = w n R. For each n 1, we have u n t) = + α + α β + u n y n ds + α y u n ds y n y ) y n + y )y n ds y u n ds + α G y n ) f u n ) ds β u n h) w n ds + u n ds u n y n h)w n ds + G y n ) Gy)) fy) ds Gy)hw n w) ds y u n h)w n ds. Note that each integrand on the right hand side of 6.7) is square integrable in H, because of 6.), 6.3) and 6.4). Therefore, d d dt u nt) H = 1 dt u nt), I + A)u n t), t [, T ]. 6.8) H 6.7)

23 Let N be an arbitrary positive integer. The functions s ys), s ys) π N h and s ys) ys) π N h) all belong to L, T ; DA)). Therefore, from 6.8) we have: 1 d dt u nt) H 1 = u n y n, u n H + ys) u n, u n H + α y n y ) y n + y )y n, u n u n H + α y u n, u n u n H ds α u n H α u n H β G y n ) f u n ), u n u n H β G y n ) Gy)) fy), u n u n H 6.9) u n h, u n H w n + Gy)h, u n H w n w) y π N h), u n H w n w) y h π N h), u n H w n w) + α u n y n h), u n H w n α y u n h), u n u n H w n ds + α y y π N h)), u n H w n w) + α y y h π N h)), u n H w n w) 6.1) for all t [, T ]. In the rest of this proof, C and C 1 denote positive real constants whose value may depend only on α, T, R, M and h; the actual value of the constant may be different in different instances. For each η > we estimate the terms on the right hand side of 6.1) using the Cauchy-Schwarz inequality and.) and the Young inequality to obtain: u n t) H + α 1 u n H ds C 1η + C h π N h H where, for s [, T ], we put 1 u n H ds + b n,n + C u n s) H ds 1 u n H 1 ψ n s) ds w n s) ws)) ds ) ψ n s) = η + 1 η y ns) H η ) [ y n s) H 1 y n s) H 1 + y n s) H ) + ys) H 1 ys) H 1 + ys) H ) + ys) H ys) 1 H 1 + ys) )] 1 H + η w n + w n s) 1 Note that the functions yn satisfy the uniform estimates 6.3) and so. sup ψ ns) ds <. n

24 4 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ and b n,n = sup r T + sup r T r r + C sup r T + C Gys))hs), u n H w n s) ws)) ds ys) π N hs)), u n s) H w n s) ws)) ds r ys) ys) π N hs))), u n s) H w n s) ws)) ds u n s) H ds. We choose η > such that C 1 η = α in 6.11) and use the boundedness of the sequence w n in L, T ; R 3) together with the Hölder inequality to estimate the product of square roots in the second line of 6.11) to obtain u n t) H + α 1 t u n s) H ds C ψ n s) u n s) H 1 ds + b n,n. 6.1) By estimates 6.) and 6.3) γ := sup ψ n s) ds < n 1 and γ depends on α, T, R, M and h only. Applying the Gronwall lemma we obtain sup u n t) H + α u 1 n H ds b n,ne γt 6.13) t [,T ] what completes the proof of estimates 6.13). Therefore, since by Claim6.5) b n,n as N we infer that lim sup n sup u n t) H + α 1 t [,T ] u n H ds C h π N h H. 6.14) We complete the proof of Lemma 6.1 by taking the limit as N. Note, that Statement 1 follows Lemma 6.1. Now we will occupy ourselves with the proof of that Statement. For this purpose let us chhose and fix the following processes: Let N > M H 1 Y n = J ɛn v n and y n = J v n. be fixed. For each n 1 we define an F t )-stopping time τ n = inf {t > : Y n t) H 1 N} T. 6.15)

25 5 Lemma 6.. For τ n as defined in 6.15) we have lim E n sup t [,T ] τn Y n t τ n ) y n t τ n ) H + Y n y n H 1 ds =. Proof. Let X n = Y n y n. We assume without loss of generality that β =, h = h 3 = and h 1 = h. Then for any n 1 we have dx n = α X n dt + α X n ) Y n + y n )) Y n dt + α y n X n dt + X n Y n dt + y n X n dt + G Y n ) G y n )) hv n dt + ɛ n G Y n ) hdw + ɛ n G Y n ) G Y n ) hdt Using a version of the Itô formula given in [17] and integration by parts we obtain where z n is a process defined by Therefore t X n t) H + α 1 d X n H = α X n H 1 dt + α y n X n H dt + α X n, X n ) Y n + y n )) Y n H dt X n, X n y n H dt + G Y n ) G y n )) h, X n H v n dt + ɛ n z ndt + ɛ n G Y n ) h, X n H dw z n = G Y n ) G Y n ) h, X n H + G Y n) h H. t X n H 1 ds C + C + C + C X n H X n H 1 y n H 1 ds X n 3/ X H 1 n H y n H 1 + y n H 1) ds X n 3/ X H 1 n H y n H 1 ds X n H v n ds + Cɛ n + ɛ n G Y n ) h, X n H dw. 6.16)

26 6 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ By 6.) we have sup n y n H 1 < and therefore, using repeatedly the Young inequality we find that there exists C > such that for all t [, T ] t X n t) H + α t X n H 1 ds C X n H + Cɛ n + ɛ n 1 + v n + β y n 4 H 1 ) ds G Y n ) h, X n H dw. Denoting the left hand side of the above inequality by L t and using the definition of τ n we have t τ n ) L t τn C X n H 1 + v n + β y n 4 H 1 ds where Since sup n + Cɛ n + ɛ n t τ n t τ n X n H ψ n,nds + Cɛ n + ɛ n t τ n G Y n ) h, X n H dw G Y n ) h, X n H dw, ψ n,n s) = 1 + v n s) + βn 4, s T. sup t [,T ] the Burkoholder-Davis-Gundy inequality yields G Y n t)), X n t) C, P a.s., and therefore Clearly, E sup L s τn C ɛ n + s t E sup X n r τ n ) H C ɛ n + r t E sup X n r τ n ) H ψ n,nds, 6.17) r s sup ψ n,n ds <, n 1 E sup X n r τ n ) H ψ n,nds. r s

27 7 hence the Gronwall Lemma implies E sup r T X n r τ n ) H C ɛ n e Returning now to 6.17), we also have E τ n X n s) H 1 ds C ε n e T T ψ n,n ds as n. ψ n,n ds as n. This completes the proof of Lemma 6.. Lemma 6.3. For the stopping time τ n defined in 6.15) we have lim E n sup t [,T ] τn Y n t τ n ) y n t τ n )) H + Proof. By a version of the Itô formula, see [17], Y n y n ) H ds =. 1 d Y nt) y n t)) H = Y n y n ), d Y n y n ) H + ɛ n G Y n ) h H dt. Therefore, putting X n = Y n y n and invoking equality 6.16) we obtain for any η > 1 d X nt) H = α X n H α X n, X n Y n + y n ) Y n H dt α X n, y n X n dt H X n y n, X n H dt G Y n ) G y n )) h, X n H F n dt ɛ n G Y n ) h, X n H dw ɛ n G Y n ) G Y n ) h, X n H dt + ɛ n G Y n ) h H dt. 6.18) We will estimate the terms in 6.18). First, noting that we find that X n Y n, X n H = X n u n, X n H X n Y n, X n H Cη X n H + C η X n H X n H )

28 8 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ Next, by the Young inequality and the interpolation inequality.) X n, X n Y n + y n ) Y n H Cη X n H + C η X n Y n + y n ) dx Λ Cη X n H + C η X n Λ Y n + y n ) dx, Cη X n H + C ) η X n H 1 X n H 1 + X n H ) Y n H 1 + y n H 1 and thereby X n, X n Y n + y n ) Y n H Cη X n H + C η X n H 1 Y n H 1 + y n H 1 ) + C η 6 X n H 1 Y n 4 H 1 + y n 4 H 1 ) Cη X n H + C η X n H Y n 4 H 1 ). 6.) Finally, using.) we obtain X n, y n X n X n H y n L y n H X n L H Cη X n H 6.1) + C η y n H 1 y n H 1 + y n H ) y n H 1 X n H X n H 1.

29 9 Taking into account 6.19), 6.) and 6.1) we obtain from 6.18) X n t) H + α t X n H ds Cη X n H ds ) + C η sup 1 + Y n 4 H 1 X n H 1 ds r t + Cη + Cη + Cη + ɛ n + Cɛ n X n H ds + C η X n H ds + C η sup r t sup r t X n H ds + C η sup X n r) H r t G Y n ) h, X n H dw ) 1 + X n H ds. Choosing η in such a way that 4Cη = α we obtain t τn E sup X n t τ n ) H + α X n H ds C η 1 + N 4 ) E t [,T ] ) ) X n r) H sup X n r) H 1 r t ) ) X n r) H sup X n r) H 1 r t τ n X n H 1 ds 6.) + C η 1 + N) E sup X n t τ n ) H t [,T ] + t τ n ɛ n E sup G Y n ) h, X n H dw t [,T ] + Cɛ n E ) 1 + X n H ds. By Theorem 5. there exists a finite constant C, depending on T, α, R, M and h only, such that for each n 1 E Y n s) H ds CT, α, M, u, h),

30 3 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ hence invoking the Burkholder-Davis-Gundy inequality we find that t τn E sup X n t τ n ) H + α X n H ds C η 1 + N 4 ) E t [,T ] τ n X n H 1 ds + C η 1 + N) E sup X n t τ n ) H t [,T ] + C1 + N) ɛ n. Finally, Lemma 6.3 follows from Lemma 6.. Lemma 6.4. Let ɛ n ), 1] with ɛ n and let F n ) P T be such a sequence that sup F n T R, for every ω Ω, n 1 and F n converges to F in law on B R. Then the sequence of X T -valued random elements J εn F n W ) JF n ) converges in probability to. Proof. We will use the same notation as in the proof of Lemma 6.3). Let δ > and ν >. Invoking part ) of Theorem 3.1) we can find N > M H 1 such that 1 N sup E sup Y n t) H 1 < ν n 1 t [,T ]. Then invoking Lemma 6.3 we find that for all n sufficiently large P sup Y n t) u n t) H + 1 t [,T ] P Y n u n DA) ds δ sup Y n t τ n ) u n t τ n ) H + 1 t [,T ] ) + P 1 δ E sup Y n t) H 1 N t [,T ] sup Y n t τ n ) u n t τ n ) H + 1 t [,T ] + 1 N E sup Y n t) H 1 t [,T ] < ν. τ n Y n u n DA) ds δ, τ n = T τ n Y n u n DA) ds Theorem 6.5. The family of laws {L J ε W )) : ε, 1]} on X T satisfies the large deviation principle with rate function I. Proof. We interpret each process J εn F n W ) as the solution of equation 3.1) with ε n and F n in place of ε and F. In order to use Theorem 4.4 we note that any Ft ) -predictable process

31 31 u : [, T ] Ω R such that can be written as u t, ω) dt R, for all ω Ω ut, ω) = F t, W ω)) t, ω) [, T ] Ω, 6.3) for some H t )-predictable function F : [, T ] C [, T ]) R satisfying T F s, z) ds R z C [, T ]). Statement 1 follows from Lemma 6.1. Next, we will show that Statement holds true. Let ε n ) be a sequence from, 1] that converges to and let F n : [, T ] Ω R) be a sequence of Ft ) -predictable processes that converges in law on B R, for some R >, to F. By Lemma 6.4 J εn F n W ) JF n converges in probability as a sequence of random variables in X T ) to. Lemma 6.1 implies that JF n converges in distribution on X T to JF. Indeed, since B R is a separable metric space, the Skorohod Theorem see, for example, [1, Theorem 4.3]) allows us to work with a sequence of processes that converges in B R almost surely. These two convergence results imply that J ε n F n W ) ) converges in law on X T to JF. Thus, for any uniformly continuous and bounded function f : X T R we have f J εn F n W ) ) dp fx) dl J F )x) Ω X T fj εn F n W )) fjf n ) dp Ω + X T as n. fx) dl JF n )x) X T fx) dl JF )x) Therefore, Statement is true as well and thus we conclude the proof of Theorem Application to a model of a ferromagnetic needle In this section we will use the large deviation principle established in the previous section to investigate the dynamics of a stochastic Landau-Lifshitz model of magnetization in a needleshaped particle. Here the shape anisotropy energy is crucial. When there is no applied field and no noise in the field, the shape anisotropy energy gives rise to two locally stable stationary states of opposite magnetization. We add a small noise term to the field and use the large deviation principle to show that noise induced magnetization reversal occurs and to quantify the effect of material parameters on sensitivity to noise. The axis of the needle is represented by the interval Λ and at each x Λ the magnetization ux) S is assumed to be constant over the cross-section of the needle. We define the total

32 3 ZDZIS LAW BRZEŹNIAK, BEN GOLDYS, AND T. JEGARAJ magnetic energy of magnetization u H 1 of the needle by E t u) = 1 ux) dx + β Φux)) dx Λ Λ Λ K t, x) ux) dx, 7.1) where Φu) = Φ u 1, u, u 3 ) = 1 u + u ) 3, β is the positive real shape anisotropy parameter and K is the externally applied magnetic field, such that K t) H for each t. With this magnetic energy, the deterministic Landau-Lifshitz equation becomes: y t) = y y αy y y) + Gy) βfy) + K t)) 7.) t where fy) = Φy), y R 3. We assume, as before, that the initial state u H 1 and u x) R 3 = 1 for all x Λ. We also assume that the applied field K t) : Λ R 3 is constant on Λ at each time t. Equation 7.) has nice features: the dynamics of the solution can be studied using elementary techniques and, when the externally applied field K is zero, the equation has two stable stationary states, e 1 = 1,, ) and e 1. We now outline the structure of this example. In Proposition 7., we show that if the applied field K is zero and the initial state u satisfies 1 α u ± e 1 H 1 < k Λ 1 + α, then the solution yt) of 7.) converges to ±e 1 in H 1 as t goes to. In Lemma 7.3, we show that if H = H 1, H, H 3 ) T R 3 has norm exceeding a certain value depending on α and β) and the applied field is K = H + β H fh ) and u H H H 1 < 1 k, then yt) converges in H 1 to H H as t goes to. Lemma 7.3 is used to show that, given δ, ) and T, ), there is a piecewise constant in time) externally applied field, K, which drives the magnetization from the initial state e 1 to the H 1 -ball centred at e 1 and of radius δ by time T ; in short, in the deterministic system, this applied field causes magnetization reversal by time T see Definition 7.4). What we are really interested in is the effect of adding a small noise term to the field. We will show that if K is zero but a noise term multiplied by ε is added to the field, then the solution of the resulting stochastic equation exhibits magnetization reversal by time T with positive probability for all sufficiently small positive ε. This result, in Proposition 7.5, is obtained using the lower bound of the large deviation principle. Finally, in Proposition 7.7, the upper bound of the large deviation principle is used: we obtain an exponential estimate of the probability that, in time interval [, T ], the stochastic magnetization leaves a given H 1 -ball centred at the initial state e 1 and of radius 1 α less than or equal to. This estimate emphasizes the importance of a large value k Λ 1+α of β for reducing the disturbance in the magnetization caused by noise in the field Stable stationary states of the deterministic equation. In this subsection, we identify stable stationary states of the deterministic equation 7.) when the applied field K does not vary with time.

33 Let ζ S. Since the time derivative dy dt of the solution y to 7.), belongs to L, T ; H) and y belongs to L, T ; DA)), we have for all t : yt) ζ H = u ζ H 7.3) + y ζ, y y αy y y) t = u ζ H + t yt) H = u H Lemma 7.1. Let u H 1 be such that ux) S and 1 α u ± e 1 H 1 k Λ 1 + α. Then for all x Λ 1) 1 u 1 x) u 1 x) + α 1 u 1 x)) u 1 x) αu 1 x) for all x Λ, ) ux), ±e and 3) 7 8 ux) ± e 1 ux) e 1. Proof. By.) Invoking 7.5), we find that +Gy) βfy) + K H ds ζ, y y αy y y) +Gy) βfy) + K H ds y, Gy) βfy) + K ) 7.4) αy y y) H ds sup ux) ± e 1 k u ± e 1 H u ζ H 1, x Λ k 1 α Λ k Λ 1 + α = α 1 + α. 7.5) u 1x) = 1 u x) + u 3x)) 1 ux) ζ 1 + α, x Λ. 7.6) 1 + α Hence one can use 7.6) and straightforward algebraic manipulations to verify that 1 u 1 x) u 1 x) + α 1 u 1 x)) u 1 x) αu 1x). Statements and 3 of Lemma 7.1 follow easily from 7.5). Proposition 7.. Let the applied field K be zero and let u H 1 satisfy 1 α u ± e 1 H 1 < k Λ 1 + α. 7.7) Let the process y be the solution to 7.). Then yt) converges to ±e 1 in H 1 as t. 33