Sharp interface limit for the stochastic Allen-Cahn equations (Probability Symposium)

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(1)194. 数理解析研究所講究録第2030巻 2017年 194-201. Sharp. interface limit for the stochastic Allen‐Cahn. equations. Kai Lee Graduate School of Mathematical Sciences. University of Tokyo [email protected]‐tokyo.ac.jp. E‐mail:. Abstract In this paper, we treat our recent results about Allen‐Cahn equations in several settings. Especially, of interface.. Finally,. we. sharp interface limit for the stochastic we focus on the generation and motion. show the simulation concerned with these models.. Introduction. 1. Allen‐Cahn equation is. a. reaction‐diffusion. equation which has. a. bistable reaction term;. \left\{ begin{ar y}{l u^{\dot{$\epsilon$}(t,x)=\triangleu^{$\epsilon$}(t,x)+\frac{1} $\epsilon$}f(u^{$\epsilon$}(t,x) ,t>0,\primex\in\mathb {R},\ u^{$\epsilon$}(0,x)=u_{0}^{$\epsilon$}(x) \end{ar y}\right. where \triangle. has the. :=\displaystyle\frac{\partial}{\partialx}. .. uniqueness for. take. parametrized by. a. small parameter $\epsilon$>0. .. The reaction term. conditions;. The conditions. (iii). This equation is. (1.1). (i). and. \left{bginary} (mh)f\tr{}amhs\tr{o}manhl\trm{y}a h\mtr{}aehm\tr{z}aemh\tr{o}amsp1\thr{}amn d0,\ (athrm{i} )f'\p1<0,(> mathr{i}\ )fmathr{i}\smoathr{d}\ (Af):=int_-1^{}ud0,\ (mathri {v})fu\leqC(1+|^mathr{w}\imtahr{}\smtoahr{}\meC,q>0 (athr{v})f'u\leqcmwathr{i}\m athr{s}\mo athrm{e}c>\O. ndary}ight (ii). that the reaction term is bistable.. The existence and the. impose the condition by (iv) (v). however, the condition A(f)=0 is rather important. We can typical example.. of the solution. some. mean. (1.2). are. the conditions. assured. and. We. technical reasons,. f(u)=u-u^{3}\cdot \mathrm{a}s. a. (1.1). dynamics by ignoring the diffusion term, Hence, the solution tends to \displaystyle \frac{1}{$\epsilon$}f(u) \pm 1 in an early time, and interfaces appear between the two phases \pm 1 We call this process After the generation, the generation of interface which occurs in the time of order O( $\epsilon$|\log $\epsilon$ interfaces move slowly. The constant A(f) corresponds to a speed of traveling waves. However, the waves become standing waves because A(f)=0 from the condition (iii) of (1.2). Thus, the interface motion becomes extremely slow. Indeed, Carr‐Pego [2] proved that the proper time We. can. regard. the PDE. because the reaction term. as a. is. one‐dimensional. larger. than the other term.. .. . scale for the interface motion is of order . super slow motion. .. O(\displaystyle \exp(-\frac{c}{ $\epsilon$}). .. In the article of Chen. We note that the width of interface is of order. o($\epsilon$^{\frac{1}{2} ). .. [1],. it is called.

(2) 195. The annihilation of interface is also studied once. reaches smaller than. a new. phase.. o($\epsilon$^{\frac{1}{2} ). ,. interfaces. are. by. Chen. [1].. When the width of two interfaces. annihilated in the. speed. of order. o(1). ,. and form. Reminding that the width of the interfaces is of order o($\epsilon$^{\frac{1}{2} ) the shape of interface becomes sharp as $\epsilon$\rightarrow 0 Our goal is to specify the dynamics of the interface and its proper time scale when we take the hmit $\epsilon$\rightarrow 0 in the stochastic case. We call this limit sharp interface limit. ,. .. 2. Generation of interface in one‐dimensional stochastic. Now. we. consider. a. stochastic Allen‐Cahn. equation;. \left\{ begin{ar y}{l u^{$\epsilon$}(t,x)=\partial_{x }u^{$\epsilon$}(t,x)+\frac{1} $\epsilon$}f(u^{$\epsilon$}(t,x)+$\epsilon$^{$\gam a$} (x)\dot{W}(t,x),t>0,x\in\mathb {R},\ u^{$\epsilon$}(0,x)=u_{0}^{$\epsilon$}(x),x\in\mathb {R},u^{$\epsilon$}(t,\pm\infty)=\pm1,t>0, \end{ar y}\right. where. a\in C_{0}^{\infty}(\mathbb{R}). and. \dot{W}(t, x). is. a. case. space‐time white. noise which. formally. has. (2.1) a. covariance. structure;. E[\dot{W}(t, x)\dot{W}(s, y)]= $\delta$(t-s) $\delta$(x-y). .. by a mild solution or a solution in the sense of generalized function. by Funaki [3]. He considered the case that u_{0}^{ $\Xi$}\rightar ow$\chi$_{$\xi$_{0} in L^{2}(\mathbb{R}) where the function $\chi$_{ $\xi$} is a step function defined by $\chi$_{ $\xi$}(x)=1 if x\geq $\xi$ and $\chi$_{ $\xi$}(x)=-1 if x< $\xi$ In other word, an interface is generated at the initial time. He proved that \overline{u}^{ $\epsilon$}\Rightar ow$\chi$_{$\xi$_{t} as $\epsilon$\rightarrow 0 Thus, This. the solution is defined. case was. well studied. .. where. \overline{u}^{ $\epsilon$}(t, x):=u^{ $\epsilon$}($\epsilon$^{-2 $\gamma$-\frac{1}{2} t, x). ,. and the process. $\xi$_{t} obeys. an. SDE;. d$\xi$_{t}=$\alpha$_{1}a($\xi$_{t})dB_{t}+$\alpha$_{2}a($\xi$_{t})a'($\xi$_{t})dt and start at. O($\epsilon$^{-2 $\gamma$-\frac{1}{2} ). $\xi$_{0}. where $\alpha$_{1} and. $\alpha$_{2}\in \mathbb{R} depend. on. and the interface motion is described. ,. f Namely, the proper by the SDE (2.2). .. (2.2) time scale is of order.

(3) 196. We consider. value. for. and main result. Settings. 2.1. a more. u_{0}^{ $\epsilon$}\in C^{2}(\mathbb{R}). general. initial. and prove the. value,. generation of interface. in. The initial. \left{bginary} (mh{)\Vertu_0}^$psilon {\fty}+Veru_0^$psilon;}\Vert{fy+u_0'1\int}leqc{^$pson,\ (mathr{i} )\mTathr{} e\mrath{}me\rxath{i}ms\rtahm{} \trumah{n} i\trmqah{u} e$\xi_0n[-1,]mathr{}\ n mdathr{e}\ pm athr{n}\md eathr{n}\m oathrm{f}$\epsiln>0athrm{} u\ cmathr{} \mhatr{} u_0^$\epsilon}(x{)=0, \mathri} { m)|u_0}^$\epsilon(xgqC_{1} $^\frac2(|x-i_{0}geqC$\pslon^{frac1}2),\ (mth{iarv})|u_0^$\epsilon(x-1+|u_{0J}^$\epsilon(x)+|u_{0 $\prime}(x)|lqson$^{\kap}C_3ex(-frc{\sqt$mu}2)(xge1,\ mathr{v})|u_0^$\epsilon(x+1|u_{0}^$\epsilonrm(x)|+u_{0}^$\epsilonrm }(x)|\eq$psilon^{ka}C_3\exp(frc{sqt$mu}2)(x\le-1, nd{ary}ight. some. $\kappa$,. C_{0}, C_{1}, C_{2}, C_{3}>0 and. If u_{0}^{\mathcal{E} satisfies (2.3). Theorem 2.1.. C( $\omega$). random variable. $\mu$. :=f'(0). and. \overline{u}^{ $\epsilon$}(t, x). for all. :=u^{ $\epsilon$}($\epsilon$^{-2 $\gamma$-\frac{1}{2} t, x). $\xi$_{t}^{$\epsilon$}. ,. then there exist. This result. means. positive. t\in[C( $\omega$)$\epsilon$^{2 $\gamma$+\frac{3}{2} |\log $\epsilon$|, T])\rightarrow 1. ( $\epsilon$\rightar ow 0) on. that the generation of interface occurs until the time of order by the SDE (2.2).. .. C([0, T],\mathbb{R}) O( $\epsilon$|\log $\epsilon$. of interface is described. dynamics. Outline of the. 2.2. a.s.. such that. Moreover, if$\xi$_{0} is a unique zero ofu_{0}^{ $\epsilon$} as in (2.3), the distribution of the process $\xi$_{t}^{$\epsilon$} weakly converges to that of $\xi$_{t} and $\xi$_{t} obeys the SDE (2.2) starting at $\xi$_{0}. and the. (2.3). .. and stochastic processes. P(\Vert\overline{u}^{ $\epsilon$}(t, \cdot)-$\chi$_{$\xi$_{\mathrm{t} ^{ $\epsilon$} (\cdot)\Vert_{L^{2}(\mathb {R}) \leq $\delta$. We. [6].. satisfies. proof. an outline of the proof. The Allen‐Cahn equation is also described by an L^{2} ‐gradient Ginzburg‐Landau free energy \mathcal{H}^{ $\epsilon$}(u) :=\displaystyle \int_{\mathb {R} \displaystyle \{\frac{1}{2}|\nabla u|^{2}+\frac{1}{ $\epsilon$}F(u)\}dx where F'=-f The. explain. flow of. .. minimizers of \mathcal{H}^{ $\Xi$} with the. where. m. satisfies. an. boundáry. condition. u(\pm\infty)=\pm 1. is M^{ $\epsilon$}. ODE;. :=\{m($\epsilon$^{-\frac{1}{2} (x- $\xi$) | $\xi$\in \mathbb{R}\}. \left{\begin{ar y}{l \trianglem+f( )=0,m()=0,m(\p infty)=\pm1,\ m\ athrm{i}\ athrm{s}\mathrm{ }\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{}\ athrm{o}\mathrm{n}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{i}\ athrm{n}\mathrm{c}\mathrm{}\ athrm{e}\mathrm{a}\mthrm{s}\mathrm{i}\ athrm{n}\mathrm{g}, \end{ar y}\right. We consider. a. scale. v\in L^{2}(\mathbb{R})+m. change. of the solution. v(t, x)=u^{ $\epsilon$}(t, $\epsilon$^{\frac{\mathrm{i} {2} x) and M :=M^{1}. .. We. can. decompose. s(v)+m(\cdot- $\eta$(v)) such that \Vert s(v)\Vert_{L^{2}}=dist_{L^{2}}(v, M) where $\eta$\in \mathbb{R} and this is a unique decomposition if dist_{L^{2}}(v, M)\leq\exists $\beta$ We call the coordinate (s(v), $\eta$(v)) Fermi coordinate. Then, we can consider that the interface is generated when v(t, x) goes into a tubular neighborhood of M Thus, we prove the decay of \Vert s(v)\Vert_{L^{2}} by using an energy estimate. After into. ,. .. .. entering the neighborhood of M the scaled solution never goes out of this neighborhood with high probability, and we can connect it to the result of Funaki [3]. ,. 3. Generation of interface in multi‐dimensional. In this. section,. we. consider. a. case. multi‐dimensional stochastic Allen‐Cahn equation with Neumann. boundary condition;. \left{\begin{ar y}{l u^{$\epsilon$}(t,x)=\triangleu^{$\epsilon$}(t,x)+\frac{1}$\epsilon$}f(u^{$\epsilon$}(t,x)+\dot{W}_t^{$\epsilon$}(x),t>0x\inD,\ u^{$\epsilon$}(0,x)=u_{0}(x), \inD,\ frac{\partilu}{\partil$\nu}(t,x)=0t>,x\inpartilD, \end{ar y}\right.. (3.1).

(4) 197. where. D\subset \mathbb{R}^{d}. is. a. domain with. external noise is defined. a. smooth. boundary. \dot{W}_{t^{ $\zeta$}}(x) :=$\epsilon$^{ $\gamma$}\dot{W}_{t}^{Q_{d} (x). and. $\nu$. is. unit normal vector. a. W_{t}^{Q_{d}}(x) E[W_{\mathrm{t}}^{Q_{d}}(x)W_{s}^{Q_{d}}(y)]=(t\wedge s)Q_{d}(x, y). by. ,. where. a. is. positive, symmetric and compactly supported smooth function.. covariance structure. .. \partial D. .. The. a. has a. on. Q‐Brownian motion which The function Q_{d}:D\times D\rightarrow \mathbb{R}. is. Main result. 3.1 Now. we. state the. of interface for multi‐dimensional Allen‐Cahn. generation. Theorem 3.1. Assume that u_{0} C_{1}>0, $\kappa$ and $\alpha$ satisfying constants. equation.. satisfies \Vert u_{0}\Vert_{\infty}+\Vert u_{0}'\Vert_{\infty}+\Vert u_{0}' \Vert_{\infty}\leq C_{0} If there .. $\kappa$> $\alpha$>\displaystyle \frac{1}{2},. \tilde{ $\gamma$}_{d}>0 and, for all $\gamma$\geq\overline{ $\gamma$}_{d}. ,. we. $\kappa$>1 and. have that. \displaystyle \frac{ $\alpha$}{ $\mu$}+\frac{ $\kap a$}{p}<C_{1}<\frac{1}{ $\mu$}. ,. exist constants. then there exist. positive. (\displaystyle \mathrm{i})\lim_{\in\rightar ow 0}P(-1-$\epsilon$^{ $\kappa$}\leq u^{ $\epsilon$}(x, C_{1} $\epsilon$|\log $\epsilon$|)\leq 1+$\epsilon$^{ $\kappa$} for all x\in D)=1, (\displaystyle \mathrm{i}\mathrm{i})\lim_{\in\rightar ow 0}P(u^{ $\epsilon$}(x, C_{1} $\epsilon$|\log $\epsilon$|)\geq 1-e^{ $\kap a$} for x\in D s.t. u_{0}(x)\geq$\epsilon$^{ $\beta$})=1, (iii) \displaystyle \lim_{ $\epsilon$ i\rightar ow 0}P(u^{ $\epsilon$}(x, C_{1} $\epsilon$|\log $\epsilon$|)\leq-1+$\epsilon$^{ $\kap a$} for x\in D s.t. u_{0}(x)\leq-$\epsilon$^{ $\beta$})=1, where. $\beta$ :=1-C_{1} $\mu$.. Also in the multi‐dimensional case, the interface is. Outline of the. 3.2 The. generated until the time of order O( $\epsilon$|\log $\epsilon$. proof. comparison argument. Before term. We ignore the dimensional dynamics which is represented by an SDE; proof is based. term is. on. the. greater than the other. the. generation. of. interface,. the reaction. diffusion term, and consider the. one‐. \left\{ begin{ar y}{l \dot{Y}^{$\epsilon$}( \tau$, \xi$,x)=f(Y^{$\epsilon$}( \tau$, \xi$,x)+$\epsilon$^{$\gam a$+\frac{1}2 \dot{W}_{$\tau$}^{Q_d}(x), $\tau$>0,\ Y^{\in}(0, $\xi$,x)=$\xi$\n[-2C_{\mathrm{Q},2C_{0}]. \end{ar y}\right. where. $\tau$. := $\epsilon$ t. .. We set. w_{ $\epsilon$}^{\pm}(t, x)=Y^{ $\epsilon$}(\displaystyle \frac{t}{ $\epsilon$}, u_{0}^{\pm}(x)\pm $\epsilon$ C_{2}(e^{\mathrm{A}^{\underline{t} } $\epsilon$-1), x) and prove that w_{ $\epsilon$}^{\pm}(t, x). super and sub solutions of the SPDE principle of PDEs to that of SPDEs.. Lemma 3.2. For every P as. 0<C_{1}<\displaystyle \frac{1}{ $\mu$}. ,. (3.1).. we. In this process of. proof,. we. are. the. extend the comparison. have that. ( w_{ $\xi$ j}(t, x)\leq u^{ $\epsilon$}(t, x)\leq w_{ $\epsilon$}^{+}(t, x). for. every. t\in[0, C_{1} $\epsilon$|\log $\epsilon$ x\in D ). \rightarrow 1,. e\rightarrow 0.. The main result is. and sub solutions. implied by. w_{$\epsilon$}^{\pm}. a. behavior of Y^{ $\epsilon$} , because the solution u^{ $\epsilon$} exists between the super by using Y^{ $\epsilon$}.. which is constructed.

(5) 198. Stochastic Allen‐Cahn. 4. with Dirichlet. equation. boundary. con‐. ditions Next,. we. consider the stochastic Allen‐Cahn. equation. with Dirichlet. boundary conditions;. \left\{ begin{ar y}{l \partil_{\mathrm{t}u^{$\epsilon$}(t,x)=\partil_{x}u^{$\epsilon$}+\frac{1} $\epsilon$}f(u^{$\epsilon$})+\sqrt{2}$\epsilon$^{ \gam a$}\dot{W}_{t(x),t>0,x\in[-1, ]\ u^{\mathcal{E}(0,x)=u_{0}^ $\epsilon$}(x), \in[-1, ]u^{$\epsilon$}(t,\pm1)=\pm1,t>0, \end{ar y}\right. \dot{W}_{t}(x). where. fixed. is. a. $\xi$ 0\in[-1, 1]. space‐time white noise. on. focus. on. In. .. boundary conditions,. particular,. the solution is. (2.2), $\xi$_{t}. that a\equiv 1 in the SDE constant.. In. our. boundary boundary, 4.1. we can. case,. Brownian motion. on. we. [−1, 1]. conditions.. pinned. moves as a. [−1, 1]. and. u_{0}^{ $\Xi$}\rightar ow$\chi$_{$\xi$_{0}. \mathrm{e}\rightarrow 0 in. as. L^{2}[-1, 1]. at the. boundary. x=\pm 1. If. .. we. formally consider multiplied by a. one‐dimensional Brownian motion. expect that the dynamics of interface should be described. which has. However, it is singularity.. an. for. the motion of interface in this section. From the. reflected wall. x=\pm 1 because. on. the behavior of interface. analyze. not easy to. as a. impose Dirichlet. we. near. the. because of its. Main result. Let P^{ $\xi$ j} be. a. probability. C([0, T], L^{2}[-1,1 B(t) is a reflected. Theorem 4.1. O($\epsilon$^{-2 $\gamma$-\frac{1}{2} ). As. of the solution scaled in time. Brownian motion. If $\gamma$>\displaystyle \frac{19}{4}. Theorem 4.1.. .. measure. and let P be that of Markov process. implies we. ,. on. [−1, 1] starting. that. discussed. \overline{u}^{ $\epsilon$}\Rightar ow$\chi$_{\sqrt{2}B($\alpha$_{1}^{2}t)} above,. as. the. on. $\chi$_{\sqrt{2}B($\alpha$_{1}^{2}t)} $\xi$_{0}\in[-1, 1]. ffom. weakly. then P^{ $\Xi$} converges to P. \overline{u}^{ $\epsilon$}(t, x)=u^{ $\epsilon$}($\epsilon$^{-2 $\gamma$-\frac{1}{2} t, x). on. same. and $\alpha$ 1. on. space, where. :=1\nabla m. C([0, T], L^{2}[-1,1]). as. $\epsilon$\rightarrow 0.. $\epsilon$\rightarrow 0 and the proper time scale is of order. the interface motion at the limit is. a. reflected Brownian. motion.. 4.2. Outline of the. proof. By considering. the solution. Dirichlet form. (\mathcal{E}^{ $\epsilon$}, \mathcal{D}^{ $\epsilon$}). ,. u^{ $\epsilon$}(t). as a. L^{2}[-1, 1] ‐valued Marko\acute{\mathrm{v} process,. and it is defined. there is. a. corresponding. by. \mathcal{E}^{ $\epsilon$}( $\varphi$, $\psi$):=E^{$\mu$^{ $\epsilon$}}[\{D $\varphi$, D $\psi$\rangle]=E^{$\mu$^{ $\epsilon$}}[ $\varphi$(-\mathcal{L}^{ $\epsilon$}) $\psi$|, where $\varphi$, $\psi$\in \mathcal{D}^{ $\epsilon$}\subset L^{2}(L^{2}\vdash 1,1], $\mu$^{ $\epsilon$}) and \mathcal{L}^{ $\epsilon$} is a generator of \prime u^{ $\epsilon$} ;. $\mu$^{$\epsilon$}. is. an. invariant. measure. of the SPDE. The operator. \displaystyle \mathcal{L}^{ $\epsilon$}F(u)=\langle DF(u) , u' +\frac{1}{ $\epsilon$}\ve f(u)\rangle+$\epsilon$^{2 $\gamma$}Tr(D^{2}F)(u) and \mathcal{L}^{ $\epsilon$} generates Markov semigroup \{T_{t}^{ $\epsilon$}\} of u^{ $\epsilon$} is associated with Dirichlet form (\mathcal{E}, D) ;. .. On the other. hand, Brownian. \displaystyle \mathcal{E}( $\varphi$, $\psi$):=\frac{1}{2}\int_{-1}^{1}$\varphi$'( $\xi$)$\psi$'( $\xi$)d $\xi$, ( $\varphi$, $\psi$\in \mathcal{D}) and the. proved. measure. \displaystyle \frac{1}{2}1_{[-1,1]}( $\xi$)d $\xi$. that the invariant. can. be. regard. measure. $\mu$^{$\epsilon$}. concentrates. as. a. ,. S. on. [−1, 1]. ,. uniform distribution on. motion. on. :=\{$\chi$_{ $\xi$}\}_{ $\xi$\in[-1,1]}. [−1, 1].. as. $\epsilon$\rightarrow 0. Weber .. [8]. Otto et..

(6) 199. al.. [7]. also. proved that $\mu$^{$\epsilon$}. this observation, if. converges to $\mu$ which is. uniform distribution. a. on. S. weakly.. From. characterize tbe convergence (\mathcal{E}^{ $\epsilon$}, \mathcal{D}^{ $\epsilon$})\rightar ow(\mathcal{E}, \mathcal{D}) as $\epsilon$\rightarrow 0 , it is natural to prove the result through this convergence, and this is our motivation. We consider Mosco convergence of the quadratic form which is determined by Dirichlet form. Indeed, Mosco convergence and the. Theorem 4.2. forms,. and let. we. can. strong convergence of Markov semigroup is equivalent.. (Kuwae, Shioya [5], Kolesnikov [4]). Let (\mathcal{E}^{ $\epsilon$}, D(\mathcal{E}^{ $\epsilon$}) T_{t}^{ $\epsilon$} and T_{t} be semigroups which is associate with these. convergence (\mathcal{E}^{ $\epsilon$}, \mathcal{D}(\mathcal{E}^{ $\xi$ j}) \rightar ow(\mathcal{E}, \mathcal{D}(\mathcal{E}) is all t>0.. equivalent. to the. strong. (\mathcal{E}, \mathcal{D}(\mathcal{E}). and. closed. be Dirichlet. forms. Then Mosco of operator T_{t}^{ $\epsilon$}\rightar ow T_{t}. convergence. for. The domain D of the limit is also important. If \mathcal{D}=H^{1}(S) , then Dirichlet form (\mathcal{E}, \mathcal{D}(\mathcal{E}) to a reflected Brownian motion. However, if \mathcal{D}=H_{0}^{1}(S) , then Dirichlet form. corresponds corresponds. \mathcal{D}=H^{1}(\mathcal{S}). to. Brownian motion absorbed in. a. Now. .. we. a. Thus,. boundary.. need to prove that corresponds to the. we. state Mosco convergence of Dirichlet form which. solution \overline{u}^{ $\epsilon$}.. Lemma 4.3.. The Dirichlet. \displaystyle\neg2\Vert\nablam\Vert1\langle\frac{d}{d$\xi$}\cdot, \displaystle\frac{d} $\xi$}:\rangle. Markov process. and. form. ($\epsilon$^{-2 $\gamma$-\frac{1}{2} \mathcal{E}^{ $\epsilon$}, \mathcal{D}(\mathcal{E}^{ $\epsilon$}). \mathcal{D}(\mathcal{E}):=H^{1}(S). \{$\chi$_{\sqrt{2}B($\alpha$_{1}^{2}t)}\}. where. in Mosco. B(t). is. a. sense.. reflected. converges to (\mathcal{E}, \mathcal{D}(\mathcal{E}) where \mathcal{E} ) := In particular, (\mathcal{E}, \mathcal{D}(\mathcal{E}) associates with. Brownian motion. on. [−1, 1].. Theorem 4.2 concludes the convergence of Markov semigroup, and this implies the weak convergence of finite dimensional distribution of u^{ $\epsilon$} on L^{2}[-1, 1] Combining with the tightness .. which follows from Funaki Dirichlet. [3],. we. forms. complete. the. Markov. proof. of the main result.. semigroups. Solutions. ($\epsilon$_{1}^{-2 $\gamma$-\frac{1}{2} \mathcal{E}^{ $\epsilon$}, D^{ $\epsilon$}) T_{\mathrm{t} ^{ $\epsilon$} \overline{u}^{ $\epsilon$}(t, x). Mosc |\underline{\mathrm{K}\mathrm{u}\mathrm{w}\mathrm{a}\mathrm{e}-\mathrm{S}\mathrm{h}\mathrm{i}\mathrm{o}\mathrm{y}\mathrm{a}\downarow-\downarowfin. .. dim. dist.. (2$\alpha$_{1}^{2}\mathcal{E}, H^{1}) T_{t} $\chi$_{\sqrt{2}B($\alpha$_{1}^{2}t)}(x). 5. Simulations. In this. section,. we. simulate the one‐dimensional stochastic Allen‐Cahn equation;. \dot{u}(t, x)=\triangle u(t, x)+af(u(t, x))+b\dot{W}_{t}(x) , t>0, x\in[-1, 1], \dot{W}_{t}(x). where a>0, b\in \mathbb{R}, f(u)=u-u^{3} and is a space‐time white noise. We impose Dirichlet conditions We use the boundary u(\pm 1)=\pm 1 discretizing method for this simulation. .. 5.1. Reflection of interface at the. boundary. Before the stochastic case, we consider the deterministic case (b=0) We change the time as \overline{u}(t):=u(ct) The initial value takes value -1 on x=-1 and takes 1 on x\neq-1 Now we .. .. simulate the. case. .. that. a=10^{3}, b=0, c=10^{4}. and N=150..

(7) 200. s. S ,. (1). We. can see. \triangleleft s. 0.. . (2). t=0. that the interface almost stops. B. .. t=1000. immediately although. we. take very. long. time scale.. this moves, however, the speed of interface is extremely. slow. This is the super slow motion. On the other hand, the motion of interface becomes totally different if we take b=2.. Actually,. S,. (1). We. s. . (2). t=0. In this case, the solution becomes singular, and the interface perturbs randomly and fast. can observe a reflected Brownian motion as an interface motion. Moreover, we can expect. that. we can. take the value $\gamma$ to be smaller than. Section 4.. 5.2. t=1000. \displaystle\frac{19}{4. which is lower bound of $\gamma$ in Theorem. 4.1,. Annihilation of interfaces. We also consider the annihilation of interface. the initial value. u_{0}(x). 1 \cdot. simulate the deterministic. we. (s. \triangleleft.. The annihilation. (2). occurs. f. .. symmetrically. Next. we. . ,. s. t=0. of the reaction term. case.. We set. \mathrm{w} $\pi$,0-. $\nu$ \mathrm{b}. B,. (1). First,. :=\displaystyle \sin\frac{21 $\pi$ x}{2}.. because of the. consider the stochastic. B. t=10. boundary conditions and the case.. We set the initial value. definition. u_{0}(x). :=.

(8) 201. \displaystyle \sin\frac{1 $\pi$ x}{2} We change u_{0}(x):=\displaystyle \sin\frac{21 $\pi$ x}{2}. ‐. .. the initial value because the annihilation. 5. occurs. too fast if. we. take. \mathrm{m}\mathrm{r} $\alpha$ 0-. \bullet S. \mathrm{t}5\mathrm{t}. (1). . 0. \mathrm{o}s. (2). t=0. The interfaces. move. like the. independent. Brownian. t=50. motions, and the annihilation randomly. occurs.. Acknowledgements The author would like to thank Professor T. Funaki for his tremendous supports and incisive was supported by the Program for Leading Graduate Schools, MEXT, Japan. advices. This work. and. Japan society. for the. promotion of science, JSPS.. References. [1]. X.. [2]. J.. [3]. T.. [4]. Chen, Generation, propagation, and Equations, 206 (2004), no. 2, 399‐437.. Carr, R. L. Pego, Metastable patterns Appl. Math. 42 (1989), no. 5, 523\mapsto 576.. Funaki, The scaling Theory Related Fields, A. V.. K. its. [6]. K.. [7]. F.. [8]. H.. limit 102. for. a. (1995),. of. in solutions. metastable patterns, J. Differential. of u_{t}=$\epsilon$^{2}u_{xx}-f(u). stochastic PDE and the separation no. 2, 221‐288.. Kolesnikov, Convergence of Dirichlet forms. (English summary). [5]. annihilation. Forum Math. 17. (2005),. no.. 2,. with. ,. Comm. Pure. of phases,. changing speed. Probab.. measures. on. \mathb {R}^{d}. 225‐259.. Kuwae, T. Shioya, Convergence of spectral structures: applications to spectral geometry, Comm. Anal. Geom.. a. functional analytic theory. 11. (2003),. no.. and. 4, 599‐673.. Lee, Generation and motion of interfaces in one‐dimensional stochastic Allen‐Cahn equation, To appear in Journal of Theoretical Probability.. Otto, H. Weber, M. G. Westdickenberg, Invariant measure of the stochastic Allen‐Cahn equation: the regime of small noise and large system size, Electron. J. Probab. 19 (2014), no. 23, 76 pp. Weber, Sharp interface Appl. Math.. Comm. Pure. limit for invariant 63. (2010),. no.. 8,. measures. 1071‐1109. of a. stochastic Alten‐Cahn equation,.

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