Topics in stochastic partial differential equations

Transcription

1 Topics in stochastic partial differential equations UK-Japan Winter School Classic and Stochastic Geometric Mechanics Imperial College London, January 4 January 7, 2016

2 Plan I. SPDEs, basic notion and facts II. Sharp interface limit for stochastic Allen-Cahn equation Random motion of wave fronts III. Kardar-Parisi-Zhang equation IV. Fluctuating hydrodynamics and stochastic Rayleigh-Plesset equation T. Funaki, Lectures on Random Interfaces, to appear in SpringerBriefs in Probability and Mathematical Statistics (SBPMS), Springer, 2016.

3 Outline of I I. SPDEs, basic notion and facts 1 Examples of stochastic PDEs TDGL equation (Dynamic P(ϕ)-model, Stochastic Allen-Cahn equation) KPZ equation 2 White noise, colored noise, stochastic integrals 3 Stochastic PDEs of parabolic type with additive noises Concepts of Solutions Regularity of Solutions Invariant measures

4 1. Examples of stochastic PDEs Time-dependent Ginzburg-Landau (TDGL) equation (cf. Hohenberg-Halperin 77, Kawasaki-Ohta 82, Langevin eq) t u = 1 δh 2 ( )α δu(x) (u) + ( )α/2 Ẇ (t, x, ω), x R d, Ẇ (t, x, ω) = space-time Gaussian white noise with covariance structure formally given by E[Ẇ (t, x)ẇ (s, y)] = δ(t s)δ(x y), (1) { } 1 H(u) = 2 u(x) 2 + V (u(x)) dx. R d α = 0: Model A (non-conservative system), α = 1: Model B (conservative system). Heuristically, Gibbs measure 1 Z e H du is invariant under these dynamics, where du = x R d du(x).

5 Since the functional derivative is given by δh δu(x) = u + V (u(x)), TDGL eq of non-conservative type has the form: t u = 1 2 u 1 2 V (u) + Ẇ (t, x), (2) while TDGL eq of conservative type has the form: t u = u {V (u)} + Ẇ (t, x). (3) Ẇ law = Ẇ (i.e., covariances are the same)

6 Stochastic PDEs used in physics are sometimes ill-posed. For example for (2), Noise is very irregular: Ẇ C d+1 2 := δ>0 C d+1 2 δ a.s. Linear case (without V (u)): u(t, x) C 2 d 2 d, 4 2 a.s. Well-posed only when d = 1. Similar for (3): Linear case: u(t, x) C 2 d 2 d, 8 2 a.s. Well-posed only when d = 1.

7 Martin Hairer: Theory of regularity structures, systematic renormalization TDGL equation with V (u) = 1 4 u4 : =Stochastic quantization (Dynamic P(ϕ) d -model): t ϕ = ϕ ϕ 3 + Ẇ (t, x), x Rd For d = 2 or 3, replace Ẇ by a smeared noise Ẇ ε and introduce a renormalization factor C ε ϕ. Then, the limit of ϕ = ϕ ε as ε 0 exists (locally in time). The solution map is continuous in Ẇ ε and their (finitely many) polynomials. Another approaches Gubinelli and others: Paracontrolled calculus (harmonic analytic method) Kupiainen

8 Non-nearest neighbor interactions (for better regularity) Replace the Hamiltonian by { } 1 H(u) = Au(x) u(x) + V (u(x)) dx, 2 R d where A is a higher order elliptic differential operator: Au(x) = ( 1) α D α {a αβ D β u}(x), α, β m and a αβ = a βα, positive definite. Originally A =, but we can take A = ( ) m for example. We have δh δu(x) = Au + V (u(x)), Corresponding TDGL equation has a solution with better regularity.

9 When Ẇ = 0 (noise is not added) and V = double-well type, non-conservative TDGL eq (2) is known as Allen-Cahn equation or reaction-diffusion equation of bistable type, whereas conservative TDGL eq (3) is known as Cahn-Hilliard equation. We will discuss sharp interface limit for Allen-Cahn equation with noise (Stochastic Allen-Cahn equation).

10 Dynamic phase transition, Sharp interface limit as ε 0 for TDGL equation (=stochastic Allen-Cahn equation): t u = u + 1 ε f (u) + Ẇ (t, x), x Rd (4) f = V, Potential V is of double-well type: e.g., f = u u 3 if V = 1 4 u4 1 2 u2 1 +1

11 The limit is expected to satisfy: u(t, x) ε 0 { Γ t A random phase separating hyperplane Γ t appears and the goal is to determine its dynamics under proper time scaling.

12 KPZ equation: growing interfaces with fluctuation, h(t, x) = height of interfaces, t h = xh + 1( 2 xh) 2 + Ẇ (t, x), x R Or renormalized KPZ eq: { ( x t h = x h h ) } 2 δx (x) + Ẇ (t, x), x R. Linear case (without 1( 2 xh) 2 ): h(t, x) C 1 4, 1 2 a.s. (by K. Takeuchi)

13 2. White noise, colored noise, stochastic integrals Construction of the space-time Gaussian white noise Ẇ (t, x): Take {ψ k } k=1 : CONS of L2 (R d, dx) and {Bt k } k=1 : independent 1D BMs, and consider a formal Fourier series: W (t, x) = Bt k ψ k (x), (5) k=1 called the white noise process. Ẇ (t, x) is its time derivative. Indeed, one would expect to have that E[W (t, x)w (s, y)] = E[Bt k Bs]ψ j k (x)ψ j (y) = k,j=1 (t s)ψ k (x)ψ k (y) = (t s)δ(x y), k=1 and thus its time derivative is expected to satisfy the relation (1).

14 In fact, t (t s) = 1 [0,s)(t) = 1 (t, ) (s), so that (t s) = δ(t s). s t We can also formally construct as Ẇ (t, x) = Ḃ(t)Ẇ 1 (x 1 ) Ẇ d (x d ) with independent (two-sided) BMs B, W 1,..., W d.

15 Stochastic integrals w.r.t. W (t, x): For f = f (t, x, ω): F t -adapted, L 2 ([0, T ] R d Ω) (for every T > 0), t t ( ) M t (f ) f (s, x)w (dsdx) := f (s, ), ψk L 2 (R d ) dbk s, 0 R d 0 where F t = σ{w (s, ); s t}. Itô isometry: i.e., k=1 M 2 T (f ) L2 (Ω) = f L2 ([0,T ] R d Ω) E [ M 2 T (f ) ] = T 0 R d E[f (t, x) 2 ]dtdx Burkholder-Davis-Gundy s inequality: C = C p > 0 s.t. [ ] [( T ) p/2 ] E sup M t (f ) p C E f (t, x) 2 dtdx, p > 0. 0 t T 0 R d

16 The series (5) does not converge in L 2 (R d, dx) (determines only so-called cylindrical BM). If we add a damping factor {λ k > 0} k=1 s.t. TrQ k=1 λ k < and consider W (t, x) = λk Bt k ψ k (x), (6) k=1 then it converges in L 2 (R d, dx) and defines Q-BM, where Q is an operator s.t. Qψ k = λ k ψ k. Its time derivative is called a colored noise. One can discuss on more general separable Hilbert spaces, see the book of DaPrato-Zabczyk 92.

17 3. SPDEs of parabolic type with additive noises Here we consider the TDGL equations (2) of non-conservative type and (3) of conservative type in general forms. Consider the SPDEs for u = u(t, x), t 0, x R d : t u = Au + B{b(x, u)} + CẆ (t, x). (7) A = α 2m a α(x)d α with a α Cb (Rd ), m N and α = (α ( 1,.. )., α d ) Z d +. The derivative is the usual α1 ( ) αd D α = x 1 x d. The coefficients satisfy the uniform ellipticity condition: ( 1) m+1 a α (x)σ α > 0, inf x,σ R d, σ =1 α =2m where σ α = σ α 1 1 σα d d for σ = (σ 1,..., σ d ).

18 b(x, u) is a nonlinear functional of u and B = α n b α(x)d α with b α C b (Rd ), n Z. C = α l c α(x)d α with c α Cb (Rd ), l Z. The integers n and l may be negative, then they are regarded as integral operators. Here we assume n, l 0. Ẇ (t, x) is the space-time Gaussian white noise. Note that m = 1, n = 0, l = 0 for the TDGL equation (2) of non-conservative type, while m = 2, n = 2, l = 1 for the TDGL equation (3) of conservative type.

19 3.1. Concepts of Solutions We take the weighted L 2 -spaces L 2 r = L 2 (R d, e 2rχ(x) dx), r > 0, as the state spaces for solutions of (7), where χ C (R d ) such that χ(x) = x for x 1. Definition 1 u(t, x) is called a solution of (7) with an initial value u 0 in the sense of generalized functions, if it satisfies u(t), φ = u 0, φ + t 0 { u(s), A φ + b(, u(s)), B φ }ds + W t (C φ), (8) for all φ C0 (R d ), where u, φ = u(x)φ(x)dx and W R d t is the white noise process on R d.

20 Another way to give a mathematical meaning to (7) is due to Duhamel s principle: Definition 2 u(t, x) is called a mild solution of (7), if it satisfies t t u(t) = T (t)u 0 + T (t s)b{b(, u(s))}ds + T (t s)cdw s, 0 0 where T (t) is the semigroup generated by operator A in L 2 r. The last term is defined as a stochastic integral w.r.t. white noise process for non-random operator as its integrand. In typical cases two notions of solutions are equivalent. If b(x, u) is Lipschitz continuous on L r, under the condition on m, n, l stated below, the (mild) solution exits uniquely. This is shown by standard successive approx.

21 3.2. Regularity of solutions Proposition 1 Assume 2m > 2l + d and in addition, for simplicity, n < 2l + d. Then, for the solution u(t, x) of (7), we have that 2 u(t, x) C α,β ((0, ) R d ), a.s., (9) with α = 2m 2l d 4m and β = 2m 2l d. 2

22 In particular, for the TDGL equation (2) of non-conservative type, u(t, x) C 2 d 2 d, 4 2 ((0, ) R d ). For the TDGL equation (3) of conservative type, u(t, x) C 2 d 2 d, 8 2 ((0, ) R d ). Therefore the solutions live in the usual function spaces, only when d = 1.

23 Proof of Proposition 1: q(t, x, y): fundamental solution of t A. Then, td j x α Dy β q(t, x, y) t α + β 2m j q(t, x, y), for t (0, T ], x, y R d, where { ( ) 1 } q(t, x, y) = K 1 t d x y 2m 2m 1 2m exp K 2. t We consider the mild solution and set u(t, x) = u 1 (t, x) + u 2 (t, x) + u 3 (t, x), (10) u 1 (t, x) = q(t, x, y)u 0 (y)dy, R d t u 2 (t, x) = Cy q(t s, x, y)dw s (y)dy, 0 R d t u 3 (t, x) = By q(t s, x, y)b(y, u(s))dsdy. R d 0

24 Then, for the term involving the stochastic integrals, Itô isometry proves E [ D α u 2 (t, x) D α u 2 (t, x ) 2] } C { t t 2m 2l d 2 α 2m + x x (2m 2l d 2 α δ) 2, as long as both exponents are positive. t, t (0, T ], x, x R d, δ > 0, Noting that u 2 (t, x) is Gaussian, applying Kolmogorov- Čentsov s theorem, we obtain (9) for u 2 (t, x). Other terms u 1 and u 3 have better regularity at least if n < 2l + d 2.

25 3.3. Invariant measures Next problem: the existence and uniqueness of invariant measures and ergodicity. The following general methods are known: (1) strong-feller property (DaPrato-Zabczyk) (2) asymptotic strong-feller property (Hairer-Mattingly) (3) e-property (Komorowski-Peszat-Szarek)

26 F 91 studied the invariant or reversible measures of the TDGL equation. It is shown that the (grandcanonical) Gibbs measure associated with the Hamiltonian H(u) is reversible under the TDGL equation (2) of non-conservative type and the uniqueness of (tempered) invariant measure is shown under the convexity of the potential V. Contrarily, the reversible measures of the TDGL equation (3) of conservative type are not unique. This equation has a family of canonical Gibbs measures (those associated with the Hamiltonian H λ( ) = H(u) λ(x)u(x)dx with external fields λ which satisfy λ = 0) as its reversible measures. Based on this fact, the hydrodynamic limit was discussed.

27 Summary of I. 1 Examples of stochastic PDEs TDGL equation (Dynamic P(ϕ)-model, Stochastic Allen-Cahn equation) KPZ equation 2 White noise, colored noise, stochastic integrals 3 Stochastic PDEs of parabolic type with additive noises Concepts of Solutions Regularity of Solutions Invariant measures

28 Outline of II II. Sharp interface limit for stochastic Allen-Cahn equation Random motion of wave fronts 1 Overview of results 2 Known results without noises 3 Results with noises comparison between random and non-random cases 3.1 Stochastic Allen-Cahn equation 3.2 Mass-conserving stochastic Allen-Cahn equation

29 1. Overview of results Stochastic Allen-Cahn equation u t = u + 1 ε f (u) + Ẇ ε (t, x), t > 0, x D, where Ẇ ε (t, x) is a space-time noise depending on a small parameter ε > 0, D is a domain in R d and f is bistable, i.e., f (= V ) C (R) and u ( 1, 1) s.t. f (±1) = f (u ) = 0, f (±1) < 0, f (u ) > 0. e.g., f (u) = u u 3

30 We expect to have lim ε 0 u ε (t, x) = +1 or 1. Our problem is to find the evolutional law of the random interface Γ t separating two phases +1, 1 under proper time scale Γ t For stochastic case, we always assume A(f ) = 0, where A(f ) := 1 1 f (u) du ( = V ( 1) V (1) ). This is satisfied if f is symmetric: f (u) = f ( u).

31 Overview d = 1 (F 95, 97) Take Ẇ ε (t, x) = small space-time white noise ε γ Ẇ (t, x) with γ > 19 (only 1D case is 4 well-posed). Then, if u ε (0, x) χ η0 (x), u ε (t, x) := u(ε 2γ 1 2 t, x) χηt (x) = 1 (,ηt ) 1 (ηt, ) where η t behaves as a Brownian motion with diffusion coefficient = inverse surface tension. Numerical simulation (by Yoshiki Otobe): (1) ε = 0.01, γ = 0.25: strong force to ±1 compared to (2), BM (2) ε = 0.1, γ = 0.25: BM (3) ε = 0.1, γ = 0.375: bit small fluctuation compared to (2), BM (4) ε = 0.1, γ = 2.0: very small but some fluctuation is observed, time is too short to observe BM

32 d 2 (F 99, Weber 10) Take Ẇ ε (t, x) = 1 ε ξ ε (t) with ξ ε (t) Ẇ t (time-dependent white noise) as ε 0. Then, the dynamics of the phase separating hyperplane Γ t appearing in the limit is given by v = κ + cẇt, where v: inward normal velocity of Γ t, κ: mean curvature of Γ t, c = V (u)du Mass conserving stochastic Allen-Cahn equation, F-Yokoyama 16 t u ε = u ε + 1 ( ) f (u ε ) f (u ε ) + αẇ ε (t). ε The limit is governed by v = κ κ + α D Γ t 2 Γ t Ẇ (t), where means the average over U. U D

33 2. Known results without noises We give a survey of results on sharp interface limit for Allen-Cahn equation without noise: u t = u + 1 f (u), t > 0, x D, (11) ε where D is a domain in R d and f is bistable.

34 A traveling wave solution m = m(y), y R with speed c = c(f ) R is determined by { m + cm + f (m) = 0, y R, m(± ) = ±1, namely, v(t, y) = m(y ct) is a solution of v t = 2 v + f (v), t > 0, y R. (12) y 2 We normalize m as m(0) = 0. A(f ) := 1 1 f (u) du ( = V ( 1) V (1) ) and c(f ) has the same signs, in particular, A(f ) = 0 c(f ) = 0.

35 The case A(f ) 0: The proper time scale is O(ε 1/2 ), i.e., for the solution u ε of (11), we have ū ε (t, x) := u ε (ε 1/2 t, x) χ Γt (x) (ε 0), where Γ t is a hyperplane in D and χ Γt (x) = 1(x outside of Γ t ), χ Γt (x) = 1(x inside of Γ t ). Γ t evolves according to the Huygens principle: waves with speed c(f ) are created from each point of Γ t to all outward directions, and Γ t is determined as the envelope of the wave fronts (Gärtner 83). The case A(f ) = 0 (standing wave), d 2: The proper time scale is O(1), i.e., u ε (t, x) χ Γt (x) (ε 0) and Γ t moves according to the motion by mean curvature. The case A(f ) = 0, d = 1: This is a plane wave so that the proper time scale is much longer than O(1). In fact, Carr-Pego 89 showed that the order is O(exp Cε 1/2 ).

36 3. Results with noises comparison between random/non-random cases 3.1. Stochastic Allen-Cahn equation u t = u + 1 ε f (u) + Ẇ ε (t, x), t > 0, x D, (13) where Ẇ ε (t, x) is a space-time noise depending on a small parameter ε > 0, f is bistable and D is a domain in R d. We only consider the cases (b) and (c) with noises, so that our assumptions are: f is bistable, A(f ) = 0 together with a technical condition: C, p > 0 s.t. f (u) C(1 + u p ), sup f (u) <. u

37 d = 1, D = R, Ẇ ε (t, x) = ε γ a(x)ẇ (t, x): γ > 0, a C0 2 (R) is an intensity of the noise (we assume it has a compact support from technical reason), and Ẇ (t, x) is a space-time Gaussian white noise. The SPDE (13) has a unique solution which is Hölder continuous: u ε (t, x) C 1 4, 1 2 ((0, ) R), a.s.

38 Theorem 2 (F, PTRF 95, Proc. Taniguchi symposium 97) If the initial value has the form u ε (0, x) = m((x ξ)/ ε) and the reaction term has the symmetry f (u) = f ( u), then for γ > 19 4, ū ε (t, x) := u ε (ε 2γ 1/2 t, x) = χ ξt (x) (ε 0), where χ ξ (x) = 1(x > ξ), χ ξ (x) = 1(x < ξ). The phase separation point ξ t moves according to the following stochastic differential equation: dξ t = α 1 a(ξ t )db t + α 2 a(ξ t )a (ξ t )dt, ξ 0 = ξ, (14) where B t is a 1D Brown motion, α 1 = m 1 α 2 = m 2 L 2 (R) 0 L 2 (R), dt xp(t, x, y) 2 f (m(y))m (y)dxdy, R 2 and p(t, x, y) is a fundamental solution of the linearized operator / t { 2 / y 2 + f (m(y))}.

39 This theorem shows the diffusion coefficient (mobility) α1 2 is given by the inverse of the surface tension m 2 L 2 (R) and this coincides with the conjecture made by Kawasaki-Ohta 82 and Spohn 93. We can also study the self-similar Gaussian space-time (colored) noise {W h, 1/2 h 1} with the covariance structure: E[Ẇh(t, x)ẇh(s, y)] = δ 0 (t s)q h (x y) where Q h is the Riesz potential kernel of (2h 1) the order: { h(2h 1) x 2h 2, 1/2 < h 1, Q h (x) = δ 0 (x), H = 1/2.

40 The time change ε 2γ 1/2 is very different from the case without noise. The intuitive reason that this is the proper time scale is stated as follows: ū = ū ε satisfies (in law): ū t = ε 2γ 1/2 { ū + 1ε } ( f (ū) + ε 2γ 1/2) 1/2 ε γ a(x)ẇ (t, x). Noise term is a(x)ε 1/4 Ẇ (t, x). The strong drift ε 2γ 1/2 pushes ū to the neighborhood of M ε := {ū; ū + 1ε } f (ū) = 0, ū(± ) = ±1 ={m ( (x ξ)/ ε ) ; ξ R} so that ū ε (t, x) m ((x ξ t )/ ε). In particular, the width of the interface is O(ε 1/2 ). The contribution of the noise Ẇ (t, x) comes only from this region, therefore its order is O({ε 1/2 } 1/2 ) = O(ε 1/4 ) by self-similarity. This balances with the factor ε 1/4 in front of the noise.

41 If the (centering) condition (f is odd) is violated (A(f ) = 0 is still assumed), we can show the LLN: u ε (ε 2γ t, x) = χ ξt (x), ξ t = α 3 a 2 (ξ t ) with the constant 1 α 3 = 2 m 2 L 2 (R) 0 dt p(t, x, y) 2 f (m(y))m (y)dxdy R 2 The centering condition implies α 3 = 0, so that we get CLT under longer time scale.

42 The proof of Theorem 2 consists of two steps. Step 1: To show that ū ε stays near M ε, we take Ginzburg-Landau-Wilson free energy { 1 H ε (u) := 2 u 2 (x) + 1 } ε V (u(x)) dx R as a Lyapunov function, where V is the integral of f (potential, i.e., f = V ). However, since u ε is not differentiable, we cannot insert u ε into H ε and require some extra trick. Step 2: We introduce a nice coordinate in the tubular neighborhood of M ε (or on M 1 under the spatial scaling x = εy). Consider the PDE (12). If its initial data v 0 is in an L 2 -tubular neighborhood of M 1, the solution v = v(t, y) converges to a certain m ζ (y) := m(y ζ) in M 1 as t. The limit ζ depends on the initial value v 0 so that we denote it by ζ = ζ(v 0 ) R. This defines a nice coordinate in an L 2 -tubular neighborhood of M 1. If we compute the time derivative of ζ(u ε (t)), the diverging factor cancels, cf. Appendix.

43 Recent results (d = 1): S. Weber ( 14): Several interfaces (multi-kinks) on [0, 1] with periodic boundary conditions. Noise is ε γ Ẇ (t, x) (space-time white noise). Annihilating BMs are obtained in the limit. Method: (1) Consider approximate slow manifold M and coordinate system around M (PDE case: Carr-Pego, Xinfu Chen). (2) Use the idea of expansion in stochastic case due to Antonopoulou-Blömker-Karali ( 12) when u is close to M. (3) Show annihilation when two interfaces touch. Antonopoulou-Blömker-Karali ( 12): Cahn-Hilliard eq with smooth noise (TDGL eq of conservative type with Q-BM in place of space-time white noise) on [0, 1] with no-flux boundary conditions of Neumann type: x u = 3 x u = 0 at x = 0, 1. SDEs are obtained in the sharp interface limit for multi-kinks before collisions (result is local in time). Bertini-Brassesco-Butta ( 14)

44 d = 2, D is bounded domain in R 2 with smooth boundary Consider (13) in higher dimensions with Neumann boundary condition: u/ n = 0 (x D). We assume A(f ) = 0, but don t require that f is odd. The noise Ẇ ε (t, x) = ξt ε / ε depends only on t, and ξt ε = ε γ ξ(ε 2γ t), 0 < γ < 1/3, where ξ(t) C 1 (R + ), a.s. is a stationary process with mean 0 and strong mixing property. We have that ξt ε αẇt (ε 0), but cannot treat the case that ξt ε = αẇt. Instead, we consider a mild noise converging to αẇt. Here, W t is 1D Brownian motion and α is a constant given by α := 2 0 E[ξ(0)ξ(t)] dt.

45 Theorem 3 (F, Acta Math Sinica 99) As long as the limit phase separation curve Γ t is strictly convex and stays inside D, we have that u ε (t, x) = χ Γt (x) (ε 0) where the curve Γ t moves according to the randomly perturbed curvature flow: V = κ + ( cα)ẇ t (15) where V denotes the inward normal velocity of Γ t, κ is the curvature of Γ t and c = 2 1 as explained above. 1 V (u)du H. Weber ( 10) extended Theorem 3 to arbitrary dimensions d 2 and established short time sharp interface limit under non-convex setting of interfaces. Convergence was shown in a.s.-sense due to the result by Dirr-Luckhaus-Novaga, who gave pathwise solution to V = κ + Ẇ (t).

46 Heuristic derivation of (15): Since ξt ε αẇ t, (13) is almost u t = u + 1 {f (u) + } εαẇ t ε In other words, the potential V is randomly perturbed to V (u) ( εαẇ t )u and this yields a small traveling wave toward the minimizer of the perturbed potential. This gives c 0 αẇ t in (15) More precisely, for a R (with small a ), define m = m(y; a), c = c(a) by { m + cm + {f (m) + a} = 0, y R, m(± ) = m ±, where m ± m ±(a) = ±1 + O(a) (a 0) are solutions of f (m ±) + a = 0. Then, since the solution of (13) behaves as (16) u ε (t, x) m(d(x, Γ t )/ ε; εαẇ t ), d(x, Γ t ) = signed distance between x and Γ t, by putting this in (13), we obtain

47 0 = uε t uε 1 ε f (uε ) α Ẇ t ε 1 ( ) { d ε d 1 m ε t ε m ( d ε 1 ε f (m) α Ẇ t ε 1 ( ) { } d ε d m ε t d c 0αẆ t. ) d + 1ε m ( d ε ) d 2 } The last line follows from (16), d = 1 near Γ t and c(a) = c(0) + c (0)a + O(a 2 ) = ca + O(a 2 ). (16) was used to cancel the terms of order O(1/ε). Thus the condition to cancel the terms of the order O(1/ ε) becomes d t = d + cαẇ t. Since d describes the curvature on Γ t, we obtain the limit equation (15).

48 Proof of Theorem 3: Since we assume the noise is mild, we can directly apply the PDE methods, in particular, we can construct super/sub solutions of (13) due to the comparison theorem. Those are given as functions close to ũ ε (t, x) := m(d(x, Γ ε t)/ ε; εξ ε t ) (assume this for t = 0), where the curve Γ ε t in D is determined by V = κ 1 ε c( εξ ε t ). (17) However, if Γ ε t is convex, in terms of the Gauss map (θ S 1 x(θ) Γ ε t), (17) can be rewritten into a PDE for the curvature function κ = κ ε (t, θ): κ t = κ2 { 2 κ θ 2 + κ 1 ε c( εξ ε t ) }.

49 Then, one can study its limit as ε 0 and obtain the following SPDE in the limit: { } κ 2 t = κ κ2 θ + κ + c 0α 2 Ẇt (18) where denotes the Stratonovich s stochastic integral. (18) gives the precise mathematical meaning to the random perturbation of the curvature flow (15).

50 Generation of interfaces: We assumed that interface is already created at initial time. One can show that, under rather general initial condition, an interface is generated in a short time, cf. Kai Lee 15, arxiv. Cahn-Hilliard case (d 2): Antonopoulou-Karali-Kossioris 11 considered Cahn-Hilliard eq with deterministic noise (under white noise scaling) and gave formal expansion of the solutions. For the deterministic Cahn-Hilliard eq, it is known that Hele-Shaw free boundary problem appears in the sharp interface limit (instead of mean curvature motion in Allen-Cahn case)

51 3.2. Mass conserving stochastic Allen-Cahn equation Equation u = u ε (t, x): solution of the SPDE (19) in a smooth bounded domain D in R n : ) t u ε = u ε + ε (f 2 (u ε ) f (u ε ) + αẇ ε (t), x D D ν u ε (19) = 0, x D u ε (, 0) = g ε ( ), where α > 0, ν is the inward normal vector on D, f (u ε ) = 1 f (u ε (t, x))dx. D D D

52 Ẇ ε (t) is a time derivative of W ε (t) W ε (t; ω) C([0, )) a.s. defined on a certain probability space (Ω, F, P) such that W ε (t) converges to 1D Brownian motion W (t) in a suitable sense. The reaction term f C (R) is bistable s.t. A(f ) = 0. Mass conservation law is destroyed by noise: u ε (t) = u ε (0) + αw ε (t) D D

53 Result (jointly with S. Yokoyama 16) Evolution of limit hypersurfaces γ t D: V = κ κ + α D Ẇ (t), t [0, σ], (20) γ t 2 γ t up to a certain stopping time σ > 0 (a.s.), where V = inward normal velocity of γ t, κ = mean curvature of γ t (multiplied by n 1), Ẇ (t) = white noise process, means Stratonovich stochastic integral. Evolution of approximating herpersurfaces γt ε D: V ε = κ γ ε t κ + α D 2 γ ε t Ẇ ε (t), t [0, σ ε ], (21) We assume γ ε t γ t in a proper sense. (This can be shown in 2D for convex curves)

54 Theorem 4 Assume that γ 0 has the form γ 0 = D 0 with some D 0 D and satisfies the same condition as in [CHL]. Suppose that a smooth local solution Γ = 0 t σ (γ t {t}) of (20) such that γ t D for all t [0, σ] uniquely exists. Then, there exist a family of continuous functions {g ε ( )} ε (0,1) satisfying { 1, x D\ lim g ε D0 (x) = ε 0 1, x D 0, (22) and stopping times σ ε such that (u ε (t σ ε, ), σ ε ) converges as ε 0 weakly to (χ γt σ ( ), σ) on C([0, T ], L 2 (D)) [0, ) and σ > 0 a.s.

55 The method is the asymptotic expansion employed by Chen-Hilhorst-Logak in the case without noise. Then, diverging terms like (Ẇ ε (t)) 2, (Ẇ ε (t)) 3 etc. appear. Usually, we cannot control such terms, but fortunately they appear only in the higher order terms in the expansion. Therefore, if the diverging speed of derivatives of W ε (t) is sufficiently slow, we can control them.

56 Appendix: Outline of the derivation of the SDE (14) in the limit. First introduce v ε (t, y) := ū ε (t, ε 1/2 y) observing under the microscopic spatial variable y. Then, from the SPDE for ū ε, v ε satisfies the SPDE in law: t v = ε 2γ 3/2 { v + f (v)} + ε 1/2 a(ε 1/2 y)ẇ (t, y). (23) The coordinate ζ(v) R defined in the L 2 -tubular neighborhood of M 1 introduced above enjoys the properties in the following Lemma 5. We denote its first and second Fréchet derivatives by Dζ(y, v) and D 2 ζ(y 1, y 2, v), respectively. The shifted standing wave m is defined by m η (y) = m(y η), y R for η R.

57 Lemma 5 (1) For every v in the neighborhood of M 1, we have that Dζ(, v), v + f (v) L 2 = 0. (2) For every η R, Dζ(y, m η ) = m η (y) m 2 L 2. (3) For every η R, D 2 ζ(y, y, m η ) = 1 m 2 L 2 0 dt p(t, y, z; m η ) 2 f (m η (z))m η(z) dz, R where p(t, y, z; m η ) is the fund. sol. of t { 2 y + f (m η (y))}. (1) follows from ζ(v(t)) = const along the sol. v(t) of the PDE (12). In fact, 0 = d ζ(v(t)) = Dζ(, v(t)), v(t) + f (v(t)), t > 0. dt

58 ξt ε := ε 1/2 ζ(v ε (t)): macroscopic phase separation point of v ε (t) Then, applying Itô s formula and from the SPDE (23), dξt ε = Dζ(y, v ε (t))a(ε 1/2 y)w (dtdy) R ε 1/2 D 2 ζ(y, y, v ε (t))a 2 (ε 1/2 y)dy dt. (24) R Diverging factor (first term in (23)) vanishes due to Lemma 5-(1). The quadratic variation of the first term in (24) is given by Dζ(y, v ε (t)) 2 a 2 (ε 1/2 y)dy dt. R We can assume that v ε (t) is close to m ε 1/2 ξ t for some ξ t, and thus, from Lemma 5-(2), this integral is close to a 2 (m ε (ξ t ) 1/2 ξ t (y)) 2 m 4 dy dt = a 2 (ξ t )α1 2 dt, L 2 R which leads to the first term in the SDE (14).

59 On the other hand, in the second term in (24), the contribution of D 2 ζ(y, y, m ε 1/2 ξ t ) comes only from the vicinity of y = ε 1/2 ξ t. Therefore, we may expand a 2 (ε 1/2 y) as a 2 (ε 1/2 y) = a 2 (ξ t ) (a2 ) (ξ t ) ε 1/2 (y ε 1/2 ξ t ) +. However, the contribution of the first a 2 (ξ t ) vanishes under the integration in y, since D 2 ζ(y, y, m η ) dy = 0. R The contribution of the second term, after cancellation of ε 1/2 and ε 1/2, gives 1 2 (a2 ) (ξ t )α 2 dt, from Lemma 5-(3), and this is just the second term in the SDE (14).

60 Summary of II and related results Allen-Cahn +noise: - Kawasaki-Ohta 82 (Physics) - F. d = 1 (space-time white noise), d = 2 (temporal noise, convex curve) - Weber: d 2, general case (additive noise) Cahn-Hilliard +noise: - d=1: Antonopoulou-Blmker-Karali 12, rather heuristic - d=1: Bertini-Brassesco-Buta 14, fractional BM Mass conserving Allen-Cahn eq: Another conservative system

61 Outline of III III. Kardar-Parisi-Zhang equation 1 KPZ equation [1] - Derivation, Ill-posedness, Renormalization, 1 3 -law - KPZ approximating equation-1: simple - Cole-Hopf solution, linear stochastic heat equation - KPZ approximating equation-2: suitable for finding invariant measures - Invariant measures of Cole-Hopf solution of KPZ eq 2 Coupled KPZ equation [2] - Coupled KPZ equation - Two approximating equations - Expansions - Convergence results due to paracontrolled calculus - Difference of two limits [1] F-Quastel, Stoch. PDE: Anal. Comp., 3 (2015) [2] F, Seminaire de Probab., LNM, 2137 (2015), special issue for M. Yor

62 1. KPZ equation The KPZ (Kardar-Parisi-Zhang, 1986) equation describes the motion of growing interface with random fluctuation. It has the form for height function h(t, x): t h = xh + 1( 2 xh) 2 + Ẇ (t, x), x T (or R). (25) where T R/Z = [0, 1). The coefficients 1 are not important, since we can change 2 them under space-time scaling. Ẇ (t, x) is a space-time Gaussian white noise with mean 0 and covariance structure: E[Ẇ (t, x)ẇ (s, y)] = δ(t s)δ(x y). (26)

63 Derivation of KPZ eq: A curve C t = {(x, y); y = h(t, x), x R} R 2 evolving upward with normal velocity V = κ + A with the (signed) curvature κ of C t and a constant A > 0 is described by the nonlinear PDE: t h = 2 x h 1 + ( x h) 2 + A(1 + ( xh) 2 ) 1/2. KPZ eq (25) is obtained by taking the leading terms in this eq (more precisely saying, in the eq for h(t, x) At rather than h(t, x) itself) under the situation that tilt x h of the interface is small and taking into account of fluctuations caused by a space-time white noise. Note that 1 is put in front of the second derivative and A is 2 chosen as A = 1. This simplification is essential in view of the scaling property or universality of the KPZ equation.

64 Ill-posedness of the KPZ eq (25) The nonlinearity and roughness of the noise do not match. The linear SPDE: t h = xh + Ẇ (t, x), obtained by dropping the nonlinear term has a solution h C 1 4, 1 2 ([0, ) R) a.s. Therefore, no way to define the nonlinear term ( x h) 2 in (25) in a usual sense. Actually, the following Renormalized KPZ eq with compensator δ x (x) (= + ) has the meaning: t h = x h {( xh) 2 δ x (x)} + Ẇ (t, x).

65 -power law (Sasamoto-Spohn 10 and others): Fluctuation of height function at a single point x = 0: 1 3 h(t, 0) c 1 t + c 2 t 1 3 ζtw, in particular, Var(h(t, 0)) = O(t 2 3 ), as t, i. e. the fluctuations of h(t, 0) are of order t 1 3. Subdiffusive behavior different from CLT (=diffusive behavior). The limit distribution of h(t, 0) under scaling is given by the so-called Tracy-Widom distribution ζ TW (different depending on initial distributions).

66 KPZ approximating equation-1: Simple Symmetric convolution kernel Let η C0 (R) s.t. η(x) 0, η(x) = η( x) and η(x)dx = 1 be given, and R set η ε (x) := 1η( x ) for ε > 0. ε ε Smeared noise Ẇ ε (t, x) = Ẇ (t) ηε (x) Approximating Eq-1: t h = xh + 2( 1 ( x h) 2 c ε) + Ẇ ε (t, x), where c ε = R η ε (y) 2 dy ( = 1 ) ε η 2 L 2 (R).

67 Cole-Hopf solution to the KPZ equation u := x h satisfies viscous stochastic Burgers equation. Motivated by this, consider the Cole-Hopf transform: Z = Z ε := e h, then Z satisfies t Z = xz + ZẆ ε (t, x). (The product ZẆ ε is defined in Itô s sense.) Indeed, apply Itô s formula for z = e h to see t Z = Z t h Z( th) 2 = 1 2 Z{ 2 xh + ( x h) 2 c ε } + ZẆ ε Zc ε = xz + ZẆ ε, since Z{ 2 xh + ( x h) 2 } = 2 xz.

68 It is not difficult to show (Bertini-Giacomin 97) that Z = Z ε converges to the sol Z of the linear stochastic heat equation (SHE) (defined in Itô s sense): t Z = xz + ZẆ (t, x), (27) with a multiplicative noise. This is a well-posed eq. This implies that the solution h = h ε of the approximating KPZ equation-1 converges to the Cole-Hopf solution of the KPZ equation defined by h CH (t, x) := log Z(t, x). (28) Comparison theorem for (27): Z(0) > 0 Z(t) > 0.

69 KPZ approximating equation-2: Suitable for studying inv meas We consider another KPZ approximating equation: t h = xh + 1 2( ( x h) 2 c ε) η ε 2 + Ẇ ε (t, x), (29) where η 2 (x) = η η(x), η ε 2(x) = η 2 (x/ε)/ε. General principle: Consider the SPDE t h = F (h) + Ẇ, and let A be a certain operator. Then, the structure of the invariant measures essentially does not change for t h = A 2 F (h) + AẆ. Recall TDGL eq of Models A and B. (F-Quastel [1]) The distribution of B η ε (x), where B is the periodic Brownian motion (in case T) or the two-sided Brownian motion (in case R), is invariant for h ε determined by (29). (cf. a similar result by DaPrato-Debussche-Tubaro 07 on T)

70 Cole-Hopf transform for SPDE (29) The goal is to pass to the limit ε 0 in the KPZ approximating equation (29): t h = x h + 1 2( ( x h) 2 c ε) η ε 2 + Ẇ ε (t, x). F-Quastel [1] considered its Cole-Hopf transform: Z ( Z ε ) := e h. Then, by Itô s formula, Z satisfies the SPDE: t Z = x Z + A ε (x, Z) + ZẆ ε (t, x), (30) where { A ε (x, Z) = 1 ( x ) 2 2 Z(x) Z η ε Z 2(x) ( ) 2 x Z (x)}. Z The complex term A ε (x, Z) looks vanishing as ε 0.

71 But this is not true. Indeed, under the average in time t, A ε (x, Z) can be replaced by a linear function 1 24 Z. The limit as ε 0 (under stationarity of tilt), t Z = xz+ 1 Z + ZẆ (t, x). 24 Or, heuristically at KPZ level, t h = xh + 1{( 2 xh) 2 δ x (x)} Ẇ (t, x). 24 Or one can say that the limit h(t, x) of the KPZ approximating eq-2 (29) is given by h(t, x) = h CH (t, x) t, where h CH (t, x) denotes the Cole-Hopf solution.

72 Taking the limit ε 0 (Similar to Boltzmann-Gibbs principle) Asymptotic replacement of A ε (x, Z ε (s)) by 1 24 Z ε (s, x). To avoid the complexity arising from the infiniteness of invariant measures, we view h ε (t, ρ) = h ε (t, x)ρ(x)dx (height averaged by ρ C0 (R), 0, ρ(x)dx = 1) in modulo 1 (called wrapped process). See Appendix below for the proof of Theorem 6. Theorem 6 For every φ C 0 (R) satisfying supp φ supp ρ =, we have that [ { t } 2 ] lim E π νε Ã ε (φ, Z ε (s))ds = 0, ε 0 where π is the uniform measure for h ε (0, ρ) [0, 1), Ã ε (φ, Z) = Ã ε (x, Z)φ(x)dx 0 R Ã ε (x, Z) = A ε (x, Z) 1 24 Z(x).

73 Invariant measures of Cole-Hopf solution As a byproduct, one can give a class of invariant measures for the Cole-Hopf solution of the KPZ equation. (We state results only on R). Let ν c be the distribution of B(x) + c x, where B(x) is the two-sided Brownian motion s.t. ν c (B(0) dx) = dx. Note that these are not probability measures. Then, {ν c } c R are invariant under the Cole-Hopf solution of the KPZ equation. c means the average tilt of the interface. We have different invariant measures for different average tilts. We use scale invariance: For every c R, h c (t, x) := h CH (t, x + ct) + cx c2 t is also a Cole-Hopf solution (with a new white noise).

74 Appendix: Proof of Theorem 6 (1) Reduction of equilibrium dynamic problem to static one: The expectation is bounded by { [ ] } 20 t sup 2E π νε Ã ε (φ, Z)Φ Φ, ( L ε 0)Φ π ν ε, Φ L 2 (π ν ε ) ( = 20t A ε (φ, Z) 2 1,ε) where L ε 0 is the symmetric part of L ε. This is a generic bound in a stationary situation. Here, [ ] [ ] 2E π νε Ã ε (φ, Z)Φ = E π Z ρ E νε [B ε (φ, Z)Φ(h(ρ), h)], where Z ρ = exp{ R log Z(x)ρ(x)dx}, Bε (x, Z) = 2Aε (x,z) Z ρ B ε (φ, Z) = R Bε (x, Z)φ(x)dx. and

75 (2) The key is the following static bound: Proposition 7 For Φ = Φ( h) L 2 ( C, ν) such that Φ 2 1,ε = Φ, ( L ε 0)Φ π ν ε <, and φ satisfying the condition of Theorem 6, we have that E νε [B ε (φ, Z)Φ] C(φ) ε Φ 1,ε, (31) with some positive constant C(φ), which depends only on φ, for all ε: 0 < ε δ Once this proposition is shown, the proof of Theorem 6 is concluded, since the sup in the last slide is bounded by 20t sup{2ec(φ) ε Φ 1,ε Φ 2 1,ε} = const( ε) 2 0.

76 Point of the proof of Proposition 7 First note that E νε [B ε (φ, Z)Φ] = E νε [ Z(x) Z ρ ( {Ψ ε η2(x) ε Ψ ε (x)} 1 ) ] Φ 12 To compute this expectation, since {Ψ ε η2(x) ε Ψ ε (x)} is 2nd order Wiener functional, we need to pick up the 2nd order and 0th order terms of the products of two Φ. We apply the diagram formula to compute the Winer chaos expansion of products of two functions. Wiener functionals Z(x) Z ρ

77 2. Coupled KPZ equation (Ferrari-Sasamoto-Spohn 13) R d -valued coupled KPZ equation for h(t, x) = (h α (t, x)) d α=1 on T (or R): t h α = xh α Γα βγ x h β x h γ + σ α β Ẇ β, x T. (32) We use Einstein s convention. Ẇ (t, x) = (Ẇ α (t, x)) d α=1 is an R d -valued space-time Gaussian white noise with the covariance structure E[Ẇ α (t, x)ẇ β (s, y)] = δ αβ δ(x y)δ(t s). (σ α β ) 1 α,β d, (Γ α βγ ) 1 α,β,γ d are given constants. From the form of the equation (32), the constants Γ α βγ ought to satisfy Γ α βγ = Γ α γβ for all α, β, γ. (33)

78 Two approximating equations Symmetric convolution kernels η C0 (R) and η ε, η2 ε are given similarly as before. Simple approximating equation with smeared noise: t h α = xh α Γα βγ( x h β x h γ c ε A βγ ) + σβ α Ẇ β η ε, (34) where A βγ = d δ=1 σβ δ σγ δ and c ε = 1 ε η 2 L 2 (R) is the same as before.

79 Approximation suitable for studying invariant measures: t h α = xh α Γα βγ( x h β x h γ c ε A βγ ) η ε 2 + σ α β Ẇ β η ε, (35) For the solution of (35), F [2] showed (on R), under the additional condition: Γ α βγ = Γ γ αβ (36) and σβ α = δα β (Kronecker s δ), the infinitesimal invariance of the distribution of B η ε (x), where B is the R d -valued two-sided Brownian motion.

80 When d = 1 and Γ α βγ = σα γ = 1 for simplicity, the approximating equations (34) and (35) have the forms: and t h = xh + 1 2( ( x h) 2 c ε) + Ẇ ηε, (37) t h = xh + 1 2( ( x h) 2 c ε) η ε 2 + Ẇ ηε, (38) respectively, for which we have already discussed.

81 Goal As we saw, the solution of (37) converges as ε 0 to the Cole-Hopf solution h CH (t, x) of the KPZ equation, while the solution of (38) converges to h CH (t, x) + 1 t under 24 the equilibrium setting (F-Quastel) and also under the non-equilibrium setting (Hoshino). The method of F-Quastel is based on the Cole-Hopf transform, which is not available for the coupled equation with multi-components in general. Our goal is to study the limits of the solutions of (34) and (35) as ε 0 based on the paracontrolled calculus (due to Gubinelli and others). In particular, we study the difference between these two limits. Jointly with Masato Hoshino (Univ Tokyo)

82 Expansion We think of the noise as the leading term and the nonlinear term as its perturbation by putting (small parameter) a > 0 in front of the nonlinear term, though we eventually take a = 1. Lh α = a 2 Γα βγ x h β x h γ + σ α β Ẇ β, where L = t x. We expand the solution h of the coupled KPZ eq (32) in a: h α = k=0 ak hk α. Then, we have a k Lhk α = σβ α Ẇ β + a 2 k=0 k 1,k 2 =0 a k 1+k 2 Γ α βγ x h β k 1 x h γ k 2.

83 Comparing the terms of order a 0, a 1, a 2, a 3,... in both sides and noting the condition (33), we obtain the followings: Lh α 0 = σ α β Ẇ β, Lh α 1 = 1 2 Γα βγ x h β 0 xh γ 0, Lh α 2 = Γ α βγ x h β 0 xh γ 1, Lh α 3 = Γ α βγ x h β 0 xh γ Γα βγ x h β 1 xh γ 1,

84 By replacing Ẇ β by Ẇ β η ε and taking care of the factor c ε A βγ, we have the expansion of the solution of the equation (34) (simple approximating eq): and the equations: Lh α 0 = σ α β Ẇ β η ε, h α,ε = k=0 a k h α,ε k Lh1 α = 1 2 Γα βγ( x h β 0 xh γ 0 cε A βγ ), Lh2 α = Γ α βγ x h β 0 xh γ 1, Lh3 α = Γ α βγ( x h β 0 xh γ 2 Bβγ,ε ) Γα βγ( x h β 1 xh γ 1 C βγ,ε ),

85 Furthermore, we have the expansion h α,ε = k=0 a k hα,ε k of the solution of the equation (35) (approximation suitable for studying invariant measures) and the equations: L h α 0 = σ α β Ẇ β η ε, L h α 1 = 1 2 Γα βγ( x h β 0 x h γ 0 cε A βγ ) η ε 2, L h α 2 = Γ α βγ x h β 0 x h γ 1 ηε 2, L h α 3 = Γ α βγ( x h β 0 x h γ 2 B βγ,ε ) η ε Γα βγ( x h β 1 x h γ 1 C βγ,ε ) η ε 2,

86 After defining h α 0,..., h α 3 (actually, +one more term h α 4 ) in the above way, we consider the equation for extra term and solve it by fixed point theorem in a suitable space (controlled by these driving terms). Similar for h. Our notation and those in [Hairer, Gubinelli] studying the case d = 1 correspond with each other as follows: h 0 =, h 1 =, h 2 =, h 3 = +, c ε A βγ =, C βγ,ε =, B βγ,ε =.

87 Convergence results due to paracontrolled calculus (rough formulation) Convergence of driving terms: hi α (i = 0, 1,..., 4; indeed, two terms in h3 α should be considered separately) s.t. h α,ε i h α α,ε i and h i hi α in C([0, T ], C κ i (T)), ε 0 ε 0 where κ 0 = µ, κ 1 = 2µ, κ 2 = µ + 1, κ 3 = 2µ + 1, κ 4 = 2µ 1(< 0) with µ ( 1, 1). 3 2 If driving terms (h α,ε i ) converges to (hi α ), then the solutions of the KPZ equations with these driving terms converge in C([0, T ], C µ (T)).

88 Example Assume the condition (36) (the condition we assumed for studying the stationarity) in addition to (33) and (σβ α ) = σi with σ R and a unit matrix I. In this case, we have h α (t, x) = h α (t, x) + c α t, 1 α d, in the limit, where c α = σ4 24 β,γ,γ 1,γ 2 Γ α βγγ β γ 1 γ 2 Γ γ γ 1 γ 2.

89 Summary of III. 1 KPZ equation: t h = x h ( xh) 2 + Ẇ (t, x), x R. 2 KPZ approximating equation with Ẇ ε (t, x) = Ẇ (t) η ε (x): t h = x h + 1 2( ( x h) 2 c ε) η ε 2 + Ẇ ε (t, x) has invariant measure ν ε (=distribution of B η ε ). 3 As ε 0, h ε converges to h CH (t, x) t. 4 To study the limits of two types of coupled KPZ approximating equations, we apply Gubinelli s paracontrolled calculus.

90 Outline of IV IV. Fluctuating hydrodynamics and stochastic Rayleigh-Plesset equation 1 Fluctuating hydrodynamics 2 Rayleigh-Plesset equation

91 Fluctuating hydrodynamics Spohn et al. JSP 14, 15 Hydrodynamic limit: From microscopic systems with random effects, passing through the so-called local equilibrium (or local average under local ergodicity), one can derive nonlinear PDEs. If the system has d (local) conserved quantities (e.g. 3D compressible Euler equation in fluid dynamics has 5 conserved quantities = mass, momenta and energy), we have a system of d coupled nonlinear PDEs. The noises (= random fluctuations) in the microscopic systems are averaged out and disappear in macroscopic limit equations.

92 However, if we consider linearization of this system around a global equilibrium, the noise terms survive under a proper scaling and we obtain linear SPDE s in the limit. At least heuristically, if we expand the equation to the second order, we can expect to obtain coupled KPZ equations in the limit. Physically interesting object is the covariance structure of the solution of the coupled KPZ equation. One can expect: h(t, x); h(0, 0) equil. (ct) κ f ( (ct) κ x ) with some function f and κ > 0.

93 Depending on κ, we call Diffusive scaling κ = 1 2, f = e cx2 KPZ scaling κ = 2 3, f = f KPZ α-lévy scaling κ = 1 α with 0 < α < 2 (anomalous behavior) (e.g. α = 3 5, κ = 5 3 ) For coupled KPZ equation (e.g. d = 2), if Γ and Γ , both h 1 and h 2 have KPZ scalings. On the other hand, if Γ γ αβ = 0 for all α, β, γ, then the system is linear, decoupled and the scaling should be diffusive. Mixed case such as Γ and Γ 2 22 = Γ 2 11 = 0 is interesting. It is expected that 2 different modes coexist (Spohn et al.).

94 Rayleigh-Plesset equation This part is related to the lectures of Professor Shibata and motivated by the works of Professor Yoshimura, one of organizers. The radial motion of spherical bubble in incompressible fluids is described by RP eq: d 2 R dt ( ) 2 dr + 4ν L dr 2R dt R 2 dt + 2σ ρ L R 2 (p V p ) p 3k G R = 0, ρ L R ρ L R3k+1 (39) where R = R(t) is radius of the bubble at time t. R: an ambient radius of the bubble, ν L : the liquid viscosity, σ W : the coefficient of the surface tension, ρ L : density of liquid, p V : the liquid pressure, p : the pressure at infinity, p G : the partial pressure of the non-condensation gas, k: the adiabatic constant (k 1).

95 Eq (39) can be derived from the equations of incompressible and spherically symmetric liquid flows surrounding an oscillating spherical gas bubble with radius R(t), cf. Shikada and Yoshimura 11. We can rewrite (39) into the first order system: Ṙ =V, ( 3V 2 V = 2R + ν V R + σ 2 R + p 2 R where b ). R 3k+1 ν = 4ν L, σ = 2σ, p = p p V, b = p G. ρ L ρ L ρ L Our goal is to formulate the stochastic Rayleigh-Plesset equation by adding natural random noises. (Micro or nano-bubbles) R 3k (40)

96 Model of stochastically perturbed flow Add noises c 1 ẇ 1 (t) and c 2 ẇ 2 (t) to the Eq (40): Ṙ = V R α c 1 ẇ 1 (t), ( 3 R V = 2VR α c 1 ẇ 1 (t) + V c 2 ẇ 2 (t) 2 V 2 V + 4ν L R + 2σ ) ρ L R A(R), where A(R) = A pv,p,ρ L,k(R) is given by A(R) = p V p ρ L + p 3k G R ρ L R 3k. The first equation means that we add a noise R α c 1 ẇ 1 at the boundary of the bubble, while the second means that the velocity field of the liquid outside of the bubble is perturbed by c 2 ẇ 2.

97 It is natural to understand these eq s in Stratonovich sense: dr = Vdt R α c 1 dw 1, R dv = 2VR α c 1 dw 1 + V c 2 dw 2 ( 3 2 V 2 V + 4ν L R + 2σ ) ρ L R A(R) dt, Since Stratonovich integral and Itô integral are related by u(t) dw(t) = u(t)dw(t) du(t)dw(t), one can rewrite as 2VR α 1 c 1 dw 1 = 2c 1 VR α 1 dw 1 + c 1 d(vr α 1 )dw 1, dw i (t)dw j (t) = δ ij dt, dw i (t)dt = 0, i, j = 1, 2.

98 Thus, we obtain stochastic Rayleigh-Plesset equation: ( dr = c 1 R α dw 1 + V + c2 1 α 2 R2α 1) dt, dv =2c 1 VR α 1 dw 1 + c ( 2V ( R dw c V 2R 2 + (3 α)c2 1 VR 2α 2) ( 3V 2 2R + ν V R 2 + σ R 2 + p R b ) ) R 3k+1 dt. where ν = 4ν L, σ = 2σ ρ L, p = p p V ρ L, b = p G 0 R3k ρ L. When c 1 = c 2 = 0, (41) is the deterministic Rayleigh-Plesset equation. (41)

99 Local Lipschitz continuity of each coefficients leads to the unique local existence of solutions. Global existence of solutions? No explosion? i.e. Our problems are: 1 R(t) > 0 for all t > 0 a.s.? 2 R(t) 2 + V (t) 2 < for all t > 0 a.s.?

100 Case 1: c 1 = c 2 = 0 (Deterministic case) Theorem 8 (F, Ohnawa, Suzuki, Yokoyama, J. Math. Anal. Appl. 15) (i) There exists a unique equilibrium point (R, V ) = (R, 0) with R > 0. (ii) For every initial datum, there exists a unique global solution. (iii) In the case ν = 0, the solution is periodic for every initial datum. (iv) In the case ν > 0, the solution converges to (R, 0) for every initial datum. In particular, if ν is greater than a certain constant determined by σ, p, b and k, V (t) changes its sign only finite times while if ν is less than this value, V (t) changes its sign infinitely many times.

101 Lyapunov argument for showing global existence in the deterministic case: Ṙ =V, V = 3V 2 Lyapunov function: 2R ν V R 2 σ R 2 p R + ϕ(r, V ) := 1 2 R3 V 2 + σ 2 R2 + p 3 R3 + b R 3k+1. b 3(k 1) R 3(k 1), k > 1. where d dt ϕ(r(t), V (t)) = L 0ϕ(R(t), V (t)) = νr(t)v (t) 2 0, L 0 ϕ := V ( R + 3V 2 2R ν V R 2 σ R 2 p R + b ) R 3k+1 V.

102 Case 2: c 1 = 0 (noise only on surface boundary) (c 2 small) Theorem 9 (FOSY 15) Let c 1 = 0 and ν > c 2 2 (c 2 0). Then, for every initial value, the SDE (41) has a unique global solution defined for all t > 0. Furthermore, Theorem 10 (FOSY 15) There exists an invariant measure of (41). Case 3: c 1, c 2 0 (c 2 small) Theorem 11 (FOSY 15) Let α = 1 and ν c 2 2. Then, for every initial value, the SDE (41) has a unique global solution defined for all t > 0.

103 Lyapunov argument for showing global existence in the stochastic case: The generator corresponding to (41) is given by L c1,c 2,α = 1 (( 4c1 2 V 2 R 2α 2 + c 2 V 2 ) R 2 V 2 4c2 1 VR 2α 1 2 ) V R + c2 1 R 2α 2 R 2 ( ( c V 3V 2R 2 + (3 α)c2 1 VR 2α 2 2 2R + ν V R 2 + σ R 2 + p R b )) R 3k+1 V ( + V + c2 1 α ) 2 R2α 1 R. If c 1 = 0 and 0 < c 2 ν, then we have If c 1 0, c 2 0 and α = 1, then L 0,c2,αϕ = (c 2 2 ν)v 2 R 0. L c1,c 2,1ϕ max ( 2 9 c2 1, c2 1 2 (3k 3)2) ϕ.

104 Summary of IV. 1 Fluctuating hydrodynamics 2 Rayleigh-Plesset equation

105 Thank you for your attention!