Moderate deviations for systems of slow-fast stochastic reaction-diffusion equations

Here, $\epsilon$ is considered a small parameter, $\delta=\delta(\epsilon)\rightarrow 0$ as $\epsilon\to 0$ and $L>0$. The operators $\mathcal{A}_{1},\mathcal{A}_{2}$ are second-order uniformly elliptic differential operators which encode the diffusive behavior of the dynamics, while the reaction terms are given by the (nonlinear) measurable functions $f,g:[0,L]\times\mathbb{R}^{2}\rightarrow\mathbb{R}$. The operators $\mathcal{N}_{1},\mathcal{N}_{2}$ correspond to either Dirichlet or Robin boundary conditions and the initial values $x_{0},y_{0}$ are assumed to be in $L^{2}(0,L)$.

The system is driven by two independent space-time white noises $\partial_{t}w_{1},\partial_{t}w_{2},$ defined on a complete filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq 0},\mathbb{P})$. These are interpreted as the distributional time-derivatives of two independent cylindrical Wiener processes $w_{1},w_{2}$. The coefficient $\sigma:[0,L]\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a measurable function multiplied by the noise $\partial_{t}w_{1}$.

Since $1/\delta$ is large as $\epsilon\to 0$, we see that the first equation is perturbed by a small multiplicative noise of intensity $\sqrt{\epsilon}$ while the second contains large parameters and, at least formally, runs on a time-scale of order $1/\delta$. Thus, one can think of the solution $X^{\epsilon}$ of the former as the "slow" process (or slow motion) and the solution $Y^{\epsilon}$ of the latter as the "fast" process (or fast motion). Note that, since $\delta$ has a functional dependence on $\epsilon$, the $\delta$-dependence is suppressed from the notation.

As $\epsilon$ (and hence $\delta$) are taken to $0$ one expects, on the one hand, that the small noise will vanish. On the other hand, assuming that the fast dynamics exhibit ergodic behavior, $Y^{\epsilon}$ will converge in distribution to an equilibrium and its contribution to the limiting dynamics of $X^{\epsilon}$ will be averaged out with respect to the invariant measure. In [10], Cerrai demonstrated the validity of such an averaging principle for a system of reaction-diffusion equations in spatial dimension $d\geq 1$, perturbed by multiplicative (colored) noise in both components. The setting of the present paper is closer to that of [12], where Cerrai and Freidlin proved an averaging principle in spatial dimension $d=1$ and with (additive) noise only in the fast equation. In particular, letting $x\in L^{2}(0,L)$ and assuming that the coefficients are sufficiently regular, the fast process $Y^{\epsilon,x}$ with "frozen" slow component $x$ admits a unique strongly mixing invariant measure $\mu^{x}$ and the slow process $\{X^{\epsilon}\}_{\epsilon}$ converges in probability, as $\epsilon\to 0$, to the unique (deterministic) solution $\bar{X}$ of the averaged PDE

The nonlinearity $\bar{F}$ is given by the averaged reaction term

The averaging principle describes the typical dynamics of the slow process and thus can be viewed as a "Law of Large Numbers" for $X^{\epsilon}$. One may then study the problem of characterizing large deviations from the averaging limit. In the Large Deviation theory of multiscale stochastic dynamics, the relative rate at which the intensity of the small noise and the scale separation parameter vanish plays a significant role. In particular, we distinguish the following asymptotic regimes:

The problem of Large Deviations for slow-fast systems of stochastic reaction-diffusion equations has been considered in [36] in dimension one, with additive noise in the fast motion and no noise component in the slow motion. In [25], the authors proved a Large Deviation Principle (LDP) in Regime 1, for a system with spatial dimension $d\geq 1$ and multiplicative noise, using the weak convergence approach developed in [7].

Moderate deviations characterize the decay rates of rare event probabilities that lie on an asymptotic regime between the Central Limit Theorem (CLT) and the corresponding LDP. The goal of the present paper is to prove a Moderate Deviation Principle (MDP) for system (1) in Regimes 1 and 2. The latter is equivalent to deriving an LDP for the process

with speed $h^{2}(\epsilon)$. The scaling factor $h(\epsilon)$ is such that

Note that if we set $h\equiv 1$ and let $\epsilon\to 0$ we would observe the behavior of normal deviations (CLT) around $\bar{X}$ while if we naively set $h(\epsilon)=1/\sqrt{\epsilon}$ we would observe the Large Deviations behavior. Hence, the MDP fills an asymptotic gap between the CLT and the LDP and, as such, it inherits characteristics of both.

One of the most effective methods in proving statements about the behavior of rare events (such as LDPs and MDPs) is the weak convergence method (see [4], [7], as well as the books [6] and [17]) which is the method we are using in this paper. The core of this approach lies in the use of a variational representation of exponential functionals of Wiener processes (see [4] for SDEs and [7] for SPDEs). Roughly speaking, one can represent the exponential functional of the moderate deviation process $\eta^{\epsilon}$ that appears in the Laplace Principle (LP) (which is equivalent to an MDP) as a variational infimum of a family of controlled moderate deviation processes $\eta^{\epsilon,u}$, plus a quadratic cost, over a suitable family of stochastic controls $u$. In particular, for any bounded continuous function $\Lambda:C([0,T];L^{2}(0,L))\rightarrow\mathbb{R}$:

where $u=(u_{1},u_{2})$ and $\mathcal{P}^{T}(L^{2}(0,L)^{2})$ is the family of $L^{2}(0,L)^{2}$-valued progressively-measurable control processes, where $u_{i}$ is measurable with respect to the filtration $\mathcal{F}^{w}_{T}$ generated by
$\{(w_{1}(t),w_{2}(t))\;,t\in[0,T]\}$ ($i=1,2$) and has finite $L^{2}([0,T];L^{2}(0,L))$-norm.

The process $\eta^{\epsilon,u}$ that appears on the right hand side of (6) is defined by

Here, $X^{\epsilon,u}$ corresponds to a controlled slow-fast system $(X^{\epsilon,u},Y^{\epsilon,u})$ (see (25) below) which results from (1) by perturbing the paths of the noise by an appropriately re-scaled control. It is due to the latter that this representation is called variational.

In light of (6), we see that in order to obtain a limit as $\epsilon\to 0$ of the Laplace functional (i.e. to prove an MDP), one needs to analyze the limiting behavior of $\eta^{\epsilon,u}$ and, before doing so, obtain a priori estimates for the underlying controlled slow-fast system given in $\eqref{controlledsystem}$. The latter is the first technical part of the current work (Section 4). As in the LDP case, the difficulty in these estimates is in that the stochastic controls are only known to be square integrable.

Compared to the corresponding LDP, the essential source of additional complexity in Moderate Deviations lies in the proof of tightness of the family $\{\eta^{\epsilon,u};\epsilon,u\}$. What complicates the analysis is the singular moderate deviation scaling $1/\sqrt{\epsilon}h(\epsilon)$. We overcome this difficulty by following, in spirit, the general method developed by Papanicolaou, Stroock and Varadhan in [29]. This involves the study of fluctuations with the aid of an elliptic Kolmogorov equation, associated to the fast dynamics and posed on the infinite-dimensional space $L^{2}(0,L)$. After projecting the controlled fast process $Y^{\epsilon,u}$ to an $n$-dimensional eigenspace of the elliptic operator $\mathcal{A}_{2}$, we are able to apply Itô’s formula to the solution $\Phi^{\epsilon}$ of the Kolmogorov equation and derive an expression for $\eta^{\epsilon,u}$ that is free from asymptotically singular coefficients. Using the a priori estimates from Section 4 along with regularity results for $\Phi^{\epsilon}$ from [12] and [8] we are then able to show tightness (Section 6).

Regarding the characterization of the limit in distribution of the process $\eta^{\epsilon,u}$, note that the presence of stochastic controls $u$ leads to a limiting invariant measure of the controlled fast process $Y^{\epsilon,u}$ which a priori depends on $u$. In order to deal with this in a unified manner across regimes we use the so-called “viable pair” construction (see [25] and [19], [32] for the finite and infinite-dimensional settings respectively) to characterize the limit. The latter is a pair of a trajectory and measure $(\psi,P)$ that captures both the limit averaging dynamics of $\eta^{\epsilon,u}$ and the invariant measure of the controlled fast process $Y^{\epsilon,u}$. In particular, the function $\psi$ is the solution of the limiting averaged equation for $\eta^{\epsilon,u}$ and the probability measure $P$ characterizes both the structure of the invariant measure of $Y^{\epsilon,u}$ and the control $u$. Although, in general, these two objects are intertwined and coupled together into the measure $P$, Regimes 1 and 2 lead to a decoupling of the form $P(dudydt)=\nu_{t}(du|y)\mu^{\bar{X}(t)}(dy)dt$, where $\nu_{t}(du|y)$ is a stochastic kernel characterizing the control and $\mu^{\bar{X}(t)}$ is the local invariant measure.

The measure P is obtained as the limit of a family of occupation measures $P^{\epsilon,\Delta}$, that live on the product space of fast motion and control, with $\Delta=\Delta(\epsilon)\rightarrow 0$ to be specified later on. The result on the weak convergence of the pair $(\eta^{\epsilon,u},P^{\epsilon,\Delta})$ in Regimes 1 and 2 is the content of Theorem 3.2.

With the analysis of the limit and the construction of a viable pair, we then prove the Laplace Principle (equivalently LDP) for the moderate deviation process $\eta^{\epsilon}$ in Regimes 1 and 2 (Section 7). The main result of the paper is stated in Theorem 3.3. Proving the Laplace principle amounts to finding an appropriate functional $S$ such that for any bounded and continuous function $\Lambda:C([0,T];L^{2}(0,L))\rightarrow\mathbb{R}$

As is common in the relevant literature, the Laplace principle upper bound can be proven using the weak convergence of the pair $(\eta^{\epsilon,u},P^{\epsilon,\Delta})$ per Theorem 3.2. The situation is more complicated for the Laplace principle lower bound for which we need to construct nearly optimal controls in feedback form (i.e. they are functions of both time and the fast motion) that achieve the bound.

In finite dimensions, the Large Deviation theory for multiscale diffusions with periodic coefficients has been established in all three interaction Regimes and with the use of the weak convergence approach (see [19], [32] and the references therein). The problem of Moderate Deviations in finite dimensions has been treated in [13, 18, 23, 24, 28] under different settings and assumptions. Specifically, the finite-dimensional work of [28] makes use of solutions to associated elliptic equations to treat Regimes 1 and 2. While the well-posedness and regularity theory of such equations are well-studied in finite dimensions (see e.g. [30]), their analysis on infinite-dimensional spaces becomes quite more involved and the relevant literature is more limited. The absence of available regularity results for a general class of such equations is the main reason why we only consider the fast equation with additive noise.

To the best of our knowledge, the problem of moderate deviations for systems of slow-fast stochastic reaction-diffusion is being considered for the first time in this paper. Its contribution is twofold:

On a theoretical level, it provides a way to study rare events for the infinite-dimensional dynamics in both Regimes 1 and 2. In the LDP setting, Regime 2 remains open as it does not lead to a decoupling of the limiting invariant measure of $Y^{\epsilon,u}$ and the control $u$. The regularity of the optimal controls has been studied in finite dimensions using their characterization through solutions to Hamilton-Jacobi-Bellman equations (see [32]). Such techniques have not been established on an infinite-dimensional setting. However, as shown in this paper, Regime 2 can be studied in the context of Moderate Deviations. In this regime the control of the fast equation survives in the limit. This reflects the fact that we are studying fluctuations very close to the CLT and a certain derivative of the Kolmogorov equation (see the term $\Psi_{2}^{0}u_{2}$ in Theorem 3.2) captures the contribution of these fluctuations. It is worth noting that normal deviations from the averaging limit for slow-fast stochastic reaction-diffusion equations have been studied in [11]. This was done with different techniques and no explicit connection was drawn between the covariance of the limiting Gaussian process and the solution of the Kolmogorov equation. More recently, the authors of [31] generalized the results of [11] and studied normal deviations from the averaging limit using the Kolmogorov equation approach.

On a computational level, the solution to the stochastic control problem gives vital information for the design of efficient Monte Carlo methods for the approximation of rare event probabilities on the moderate deviation range. In particular, the fact that the limiting equation is affine in $\eta^{\epsilon,u}$ is expected to make moderate deviation-based importance sampling for stochastic PDE easier to implement than its large deviation-based counterpart, see [33] for the related situation in finite dimensions. We plan to explore this in a future work.

The outline of this paper is as follows: in Section 2 we give background definitions, set-up as well as our assumptions. In Section 3 we review basic facts about the weak convergence method in infinite dimensions and we define viable pairs and occupation measures as well as state our main results on averaging for the controlled moderate deviation process $\eta^{\epsilon,u}$ and the MDP. In Section 4 we prove a priori bounds for the solution of the controlled system $(X^{\epsilon,u},Y^{\epsilon,u})$. In Section 5 we prove a priori bounds for the process $\eta^{\epsilon,u}$ with the aid of the elliptic Kolmogorov equation while Section 6 is devoted to the analysis of the limit of the pairs $(\eta^{\epsilon,u},P^{\epsilon,\Delta})$. In Section 7 we prove the MDP. Finally, Appendix A contains some classical regularity results for stochastic convolutions adapted to our multiscale setting while Appendix B contains the proof of Lemma 5.4.

We denote by $\mathcal{H}$ the Hilbert space $L^{2}(0,L)$ endowed with the usual inner product $\langle\cdot,\cdot\rangle_{\mathcal{H}}$. The norm induced by the inner product is denoted by $\|\cdot\|_{\mathcal{H}}$. Throughout this paper, $\oplus$ denotes the Hilbert space direct sum. The closed unit ball of any Banach space $\mathcal{X}$, i.e. the set $\{x\in\mathcal{X}:\|x\|_{\mathcal{X}}\leq 1\}$, will be denoted by $B_{\mathcal{X}}$. The lattice notation $\wedge,\vee$ is used to indicate minimum and maximum respectively.

For $\theta>0$, we denote by $H^{\theta}(0,L)$ the fractional Sobolev space of $x\in\mathcal{H}$ such that

where $\uplambda_{2}$ denotes Lebesgue measure on $[0,L]^{2}$. $H^{\theta}(0,L)$ is a Banach space when endowed with the norm $\|\cdot\|_{H^{\theta}}:=\|\cdot\|_{\mathcal{H}}+[\cdot]_{H^{\theta}}$.

Moreover, for $T>0$ and $\beta\in[0,1)$, we denote by $C^{\beta}([0,T];\mathcal{H})$ the space of $\beta$-Hölder continuous $\mathcal{H}$-valued paths defined on the interval $[0,T]$. $C^{\beta}([0,T];\mathcal{H})$ is a Banach space when endowed with the norm

For any two Banach spaces $\mathcal{X},\mathcal{Y}$ and $k\in\mathbb{N}$ we denote the space of $k$-linear bounded operators $Q:\mathcal{X}^{k}\rightarrow\mathcal{Y}$ by $\mathscr{L}^{k}(\mathcal{X};\mathcal{Y})$. The latter is a Banach space when endowed with the norm

When the domain coincides with the co-domain, we use the simpler notation $\mathscr{L}^{k}(\mathcal{X})$ while for $k=1$ we often omit the superscript and write $\mathscr{L}(\mathcal{X};\mathcal{Y})\equiv\mathscr{L}^{1}(\mathcal{X};\mathcal{Y})$.

The spaces of trace-class and Hilbert-Schmidt linear operators $B:\mathcal{H}\rightarrow\mathcal{H}$ are denoted by $\mathscr{L}_{1}(\mathcal{H})$ and $\mathscr{L}_{2}(\mathcal{H})$ respectively. The former is a Banach space when endowed with the norm

while the latter is a Hilbert space when endowed with the inner product

The class of (globally) Lipschitz real-valued functions on $\mathcal{H}$ is denoted by $Lip(\mathcal{H})$ and the space of $k$-times Fréchet differentiable real-valued functions on $\mathcal{H}$ with bounded and uniformly continuous derivatives up to the $k$-th order ($k\in\mathbb{N}$) is denoted by $C_{b}^{k}(\mathcal{H})$. The latter is a Banach space when endowed with the norm

For $k=0$ we often omit the superscript and write $C_{b}(\mathcal{H})\equiv C_{b}^{0}(\mathcal{H})$ for the space of bounded uniformly continuous functions on $\mathcal{H}$.

The operators $\mathcal{A}_{1},\mathcal{A}_{2},$ appearing in (1), are uniformly elliptic second-order differential operators with continuous coefficients on $[0,L]$. The operators $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ act on the boundary $\{0,L\}$ and can be either the identity operator (corresponding to Dirichlet boundary conditions) or first-order differential operators of the type

for some $b,c\in C^{1}[0,L]$ such that $b\neq 0$ on $\{0,L\}$ (corresponding to Neumann or Robin boundary conditions).

For $i=1,2,$ $A_{i}$ denotes the realization of the differential operator $\mathcal{A}_{i}$ in $\mathcal{H}$, endowed with the boundary condition $\mathcal{N}_{i}$. It is defined on the dense subspace

and generates a $C_{0}$, analytic semigroup of operators $S_{i}=\{S_{i}(t)\}_{t\geq 0}\subset\mathscr{L}(\mathcal{H})$.

Let $i=1,2$ and $\theta\geq 0$. In view of Hypotheses 1(a) and 1(c), along with the previous remark, it follows that $0$ is in the resolvent set of $A_{i}$. Hence the operator $-A_{i}$, restricted to its image, has a densely defined bounded inverse $(-A_{i})^{-1}$ which can then be uniquely extended to all of $\mathcal{H}$. One can then define $(-A_{i})^{-\theta}$ via interpolation and show that it is also injective.

Letting $(-A_{i})^{\frac{\theta}{2}}:=((-A_{i})^{-\frac{\theta}{2}})^{-1}$ we define $\mathcal{H}_{i}^{\theta}:=Dom(-A_{i})^{\frac{\theta}{2}}=Range(-A_{i})^{-\frac{\theta}{2}}\subset\mathcal{H}$. The latter is a Banach space when endowed with the norm

This norm is equivalent, due to injectivity, to the graph norm (see [27], Chapter 2.2).

The analytic semigroups $S_{i}$ possess the following regularizing properties (see e.g. section 4.1.1 in [8]) :

(i) For $0\leq s\leq r\leq\frac{1}{2}$ and $t>0$, $S_{i}$ maps $H^{s}(0,L)$ to $H^{r}(0,L)$ and

for some positive constants $c_{r,s},C_{r,s}$.

(ii) $S_{i}$ is ultracontractive, i.e. for $t>0,$ $S_{i}(t)$ maps $\mathcal{H}$ to $L^{\infty}(0,L)$ and furthermore, for any $1\leq p\leq r\leq\infty$,

The next set of assumptions concerns the regularity of the nonlinear reaction terms in (1). In particular, we assume that $f,g:[0,L]\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ are measurable functions and:

or:

The reaction terms $f,g$ induce nonlinear superposition (or Nemytskii) operators denoted, respectively, by $F,G:\mathcal{H}\times\mathcal{H}\rightarrow\mathcal{H}$ and defined by

In view of Hypotheses 2(a) and 2(b), $F$ and $G$ are (globally) Lipschitz continuous. Moreover, $F$ and $G$ are Gâteaux differentiable with respect to both variables and along the direction of any $\chi\in\mathcal{H}$. Their Gâteaux derivatives are given by

and

for $\xi\in[0,L]$. Furthermore, for each fixed $y\in\mathcal{H}$ and $\chi_{1}\in\mathcal{H}$, the map

is Gâteaux differentiable along the direction of any $\chi_{2}\in\mathcal{H}$. Equivalently, the nonlinear operator $F$, when considered as a map from $\mathcal{H}$ to $L^{1}(0,L)$, is twice Gâteaux differentiable with respect to $x$, along any direction in $\mathcal{H}\times\mathcal{H}$. Its second partial Gâteaux derivative is given by

The diffusion coefficient $\sigma$ is considered as a function multiplied by the noise and hence induces, for each $x,y\in\mathcal{H}$, a multiplication operator

In view of Hypothesis 3(a) it follows that $\Sigma(x,y)\in\mathscr{L}(L^{\infty}(0,L);\mathcal{H})\cap\mathscr{L}(\mathcal{H};L^{1}(0,L))$. Moreover, under Hypothesis 3(a’), we have $\Sigma(x,y)\in\mathscr{L}(\mathcal{H})$.

For the purposes of this paper we consider a Polish space to be a completely metrizable, separable topological space. For a given topological space $\mathcal{E}$ we denote the Borel $\sigma$-algebra by $\mathscr{B}(\mathcal{E})$ and the space of Borel probability measures on $\mathcal{E}$ by $\mathscr{P}(\mathcal{E})$. If $\mathcal{E}$ is Polish then $\mathscr{P}(\mathcal{E})$, endowed with the topology of weak convergence of measures, is also a Polish space.

In this section we review the weak convergence approach to large and moderate deviations (see [17] as well as the more recent [6]) and then we state our main results of the paper on the averaging principle for the controlled process $\eta^{\epsilon,u}$ (see (7)) and on the moderate deviations for $\{X^{\epsilon}\}$.

Let $j=1,2$ and consider the cylindrical Wiener process $w_{j}:[0,\infty)\times\mathcal{H}\rightarrow L^{2}(\Omega)$ appearing in (1). For each fixed $t$, $\{w_{j}(t,\chi)\}_{\chi\in\mathcal{H}}$ is a Gaussian family of random variables and for each $t_{1},t_{2}\geq 0$, $\chi_{1},\chi_{2}\in\mathcal{H}$

The first step of the weak convergence method relies on a variational representation for functionals of the driving noise. For the infinite-dimensional setting of this paper, we will use the variational representation for $Q$-Wiener processes that was proved in [7], Theorem 3. In order to apply this result in the context of space-time white noise, we introduce a separable Hilbert space $(\mathcal{H}_{1},\langle.\;,.\rangle_{\mathcal{H}_{1}})$ such that $\mathcal{H}$ is a linear subspace of $\mathcal{H}_{1}$ and the inclusion map $\mathcal{H}\overset{i}{\rightarrow}\mathcal{H}_{1}$ is Hilbert-Schmidt (for more details on this construction we refer the reader to [21], [22]). Given a complete orthonormal basis $\{e_{j,n}\}_{n\in\mathbb{N}}\subset\mathcal{H}$, the process

is an $\mathcal{H}_{1}$-valued $Q$-Wiener process with trace-class covariance operator $Q=ii^{*}\in\mathscr{L}_{1}(\mathcal{H}_{1})$. In particular, for each $t,s\geq 0$ and $\chi_{1},\chi_{2}\in\mathcal{H}_{1}$ we have

This construction allows us to use the following representation.