37.07.11 · probability / 07-large-deviations

Freidlin–Wentzell Theory: Large Deviations for Small-Noise Diffusions

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Anchor (Master): Dembo & Zeitouni 1998 *Large Deviations Techniques and Applications* 2nd ed. (Springer) §5.6-§5.7 (Freidlin-Wentzell theorem via the contraction of Schilder through the Itô map; the exit problem); Freidlin & Wentzell 2012 *Random Perturbations of Dynamical Systems* 3rd ed. (Springer) Ch. 3-4, Ch. 6 (quasi-potential, Eyring-Kramers mean exit time, metastability); Berglund 2013 *Kramers' law: validity, derivations and generalisations* (Markov Processes and Related Fields 19)

Intuition Beginner

Picture a ball rolling in a landscape of hills and valleys, nudged forever by tiny random kicks. When the kicks are very weak, the ball mostly does the sensible thing: it rolls downhill and settles in the nearest valley, tracing almost exactly the path the landscape alone would dictate. But once in a very long while the kicks conspire, all pushing the same way at the same time, and the ball climbs out of its valley and over a ridge into the next one. Freidlin–Wentzell theory is the precise accounting of how rare such a climb is and, just as importantly, which climbing route the ball almost always takes when it does happen.

The key idea is that not all unlikely paths are equally unlikely. To make the ball follow a chosen route that fights the landscape, the random kicks have to supply exactly the missing push at every instant. A route that fights hard needs a lot of conspiring kicks; a route that mostly goes with the flow needs few. We can score every candidate route by a single number — its cost — that measures the total amount of against-the-grain pushing it demands. Going with the landscape costs nothing; fighting it costs more the harder and longer you fight.

When the noise is small, the chance of seeing a given route is set almost entirely by its cost: a route of cost $c$ has probability roughly $e^{- c / ε}$ , where $ε$ measures how weak the kicks are. Because this shrinks fast as the kicks weaken, the cheapest route that does the job dominates everything else. So to answer "how does the ball escape its valley?" you do not average over all escapes — you find the single least-cost climb, the most probable escape path, and read the escape rate off its cost.

This least-cost number, measured from a valley bottom to any target point, is called the quasi-potential. In a simple landscape it is just the height of the ridge you must clear — twice the energy barrier. The headline consequence is the escape-time law: the typical time the ball waits before jumping out of its valley grows like $e^{V / ε}$ , where $V$ is the quasi-potential height of the lowest ridge. Halve the noise and you roughly square the waiting time. This single exponential, known to chemists as the Arrhenius law, governs how long a metastable state survives.

Visual Beginner

Figure: a one-dimensional double-well landscape (two valleys $a$ and $c$ separated by a ridge at $b$ ). A ball sits in the left valley. Several faint jittery trajectories rattle near the valley floor; one bold trajectory climbs the least-steep route up to the ridge top and tips over into the right valley. A vertical bracket marks the barrier height $Δ V$ from valley floor to ridge, annotated "cost of escape." A side box reads "typical wait $\approx e^{2Δ V / ε}$ : halve the noise, roughly square the wait."

 potential U(x)
   |          ____
   |   ____   /    \        b = ridge (saddle)
   |  /    \ /      \
   | /  a   X   bold climb   \   c
   |/    \  |\  . . jitter . /\  /
   +--o---\-|-\------------/--\/----- x
      ball  \|  \________/    valley c
      valley a      (typical wait ~ e^{2 dV / eps})
        |<--- barrier dV --->|  cost of escape

Worked example Beginner

Take the simplest escape problem: a particle on a line in the double-well landscape with potential $U (x) = \frac{1}{4} x^{4} - \frac{1}{2} x^{2}$ , kicked by weak noise. The two valley bottoms sit at $x = - 1$ and $x = + 1$ , and the ridge between them is the top at $x = 0$ . We compute the cost of escaping the right valley.

Step 1. Find the heights. Plug in the three special points. At a valley bottom $x = 1$ : $U (1) = \frac{1}{4} - \frac{1}{2} = - \frac{1}{4}$ . At the ridge $x = 0$ : $U (0) = 0$ . So the floor sits at height $- \frac{1}{4}$ and the ridge at height $0$ .

Step 2. Measure the barrier. The barrier is how far uphill you must climb from the valley floor to the ridge: $Δ U = U (0) - U (1) = 0 - (- \frac{1}{4}) = \frac{1}{4}$ .

Step 3. Read off the quasi-potential. For a particle in a potential landscape, the least-cost climb out of a well is twice the barrier height (the factor of two comes from the cost-accounting, which charges for fighting the downhill pull on the way up). So the quasi-potential of the escape is $V = 2 Δ U = 2 \times \frac{1}{4} = \frac{1}{2}$ .

Step 4. Estimate the waiting time. The typical time before the particle jumps to the other valley grows like $e^{V / ε} = e^{1/ (2 ε)}$ . If the noise level is $ε = 0.1$ , this is about $e^{5} \approx 148$ time units; at $ε = 0.05$ it is about $e^{10} \approx 22000$ — roughly the square, for half the noise.

What this tells us. The whole escape problem collapses to one number, the barrier the cheapest path must clear, and that number sits in an exponent. Small changes in the barrier or the noise produce enormous changes in the waiting time. This is why metastable states — a supercooled liquid, a folded protein, a magnetised domain — can persist for astronomically long times and then flip almost instantly: the wait is exponential in a barrier the system rarely has the energy to clear.

Check your understanding Beginner

Exercise (easy, multiple choice).

In the small-noise picture, when the particle does escape its valley, which route does it almost always follow?

A. A randomly chosen route, all routes being equally likely B. The straightest geometric line to the target C. The route of least cost (least against-the-landscape pushing) D. The route that spends the most time near the valley floor

Hint

A route of cost $c$ has probability roughly $e^{- c / ε}$ ; as the noise weakens, the cheapest route overwhelms all others.

Answer

C. The route of least cost.

Feedback-correct: correct — since the probability of a route decays like $e^{- c / ε}$ in its cost $c$ , the single least-cost route dominates as the noise shrinks, so escapes concentrate on the most probable (cheapest) path. Feedback-wrong: routes are not equally likely, and the cheapest route need not be the straightest line — it is the one demanding the least total against-the-grain pushing, which usually hugs the landscape and climbs the lowest ridge.

Formal definition Intermediate+

Throughout, $W = (W_{t})_{t \in [0, T]}$ is a standard $m$ -dimensional Brownian motion on a filtered space, with the Itô integral and Itô's formula of 02.15.02, and the small-noise SDE is read through the strong-solution theory of 02.15.03. The drift $b : R^{d} \to R^{d}$ and dispersion $σ : R^{d} \to R^{d \times m}$ are bounded and globally Lipschitz, and $a := σ σ^{T}$ is assumed uniformly positive-definite (so $σ$ has a bounded left inverse $σ^{- 1}$ on the relevant subspace; in the model case $m = d$ and $σ$ is invertible).

Definition (small-noise diffusion). For $ε > 0$ the small-noise diffusion is the strong solution $X^{ε} = (X_{t}^{ε})_{t \in [0, T]}$ of $$ dX^\varepsilon_t = b(X^\varepsilon_t),dt + \sqrt{\varepsilon},\sigma(X^\varepsilon_t),dW_t,\qquad X^\varepsilon_0=x_0, $$ viewed as a random element of the path space $C_{x_{0}} ([0, T]; R^{d})$ with the uniform topology. As $ε \to 0$ the noise switches off and $X^{ε}$ converges (in probability, uniformly) to the deterministic flow $\dot{ϕ}^{0} = b (ϕ^{0})$ , $ϕ_{0}^{0} = x_{0}$ , the integral curve of the drift field 02.12.01.

Definition (Cameron–Martin / absolutely continuous paths). A path $ϕ \in C_{x_{0}} ([0, T]; R^{d})$ is admissible if it is absolutely continuous with $ϕ_{0} = x_{0}$ and $\dot{ϕ} \in L^{2} ([0, T]; R^{d})$ ; equivalently $ϕ = x_{0} + \int_{0}^{\cdot} \dot{ϕ}_{s} d s$ with $\dot{ϕ}$ square-integrable. Write $H_{x_{0}}$ for the set of admissible paths.

Definition (Freidlin–Wentzell action functional). The rate function (action) on $C_{x_{0}} ([0, T]; R^{d})$ is $$ I_{0,T}(\phi);=;\begin{cases}\displaystyle \tfrac12\int_0^T \big|\sigma^{-1}(\phi_t)\big(\dot\phi_t-b(\phi_t)\big)\big|^2,dt, & \phi\in H_{x_0},\[2mm] +\infty, & \text{otherwise.}\end{cases} $$ The integrand $\frac{1}{2} ∣ σ^{- 1} (\dot{ϕ} - b) ∣^{2} = \frac{1}{2} ⟨ \dot{ϕ} - b, a^{- 1} (\dot{ϕ} - b)⟩$ is the local cost density: it vanishes exactly along the drift flow $\dot{ϕ} = b (ϕ)$ and grows quadratically with the control $u = σ^{- 1} (\dot{ϕ} - b)$ the noise must supply to force the path off the flow. $I_{0, T}$ is a good rate function: its sublevel sets ${ϕ : I_{0, T} (ϕ) \leq α}$ are compact in the uniform topology (Arzelà–Ascoli, via the $H^{1}$ bound the action provides).

Definition (quasi-potential). For points $x, y \in R^{d}$ the quasi-potential is the least action to travel from $x$ to $y$ over an unbounded time horizon, $$ V(x,y);=;\inf\big{,I_{0,T}(\phi): T>0,\ \phi\in H,\ \phi_0=x,\ \phi_T=y,\big}. $$ When $b = - \nabla U$ and $σ = Id$ (a reversible/gradient system), $V (x, y) = 2 (U (y) - U (x))$ for $y$ reachable uphill, the factor $2$ being the signature of the time-reversal computation below.

Counterexamples to common slips

The action is not the kinetic energy $\frac{1}{2} \int ∣ \dot{ϕ} ∣^{2}$ . That is the Schilder rate for pure Brownian motion ( $b = 0$ , $σ = Id$ ). The drift enters through the shifted velocity $\dot{ϕ} - b (ϕ)$ : a path that rides the drift costs nothing no matter how fast it moves, while standing still ( $\dot{ϕ} = 0$ ) at a non-equilibrium point costs $\frac{1}{2} \int ∣ σ^{- 1} b ∣^{2} > 0$ .
The quasi-potential is generally not symmetric. $V (x, y) \neq = V (y, x)$ unless the system is reversible. For a genuinely non-gradient drift (with rotational part) the cheapest uphill route and the cheapest downhill route have different costs, and the equilibrium-statistics formula $V = 2Δ U$ fails; one must solve a Hamilton–Jacobi equation for $V$ instead.
Exit cost is read at the boundary, not the target. For escape from a domain $D$ with attractor $x^{*}$ , the controlling quantity is $\overset{ˉ}{V} = in f_{y \in \partial D} V (x^{*}, y)$ , the cheapest cost to reach the boundary, not the cost to any interior point; the most probable exit concentrates on the boundary minimiser $y^{*}$ .

Key theorem with proof Intermediate+

We prove the Freidlin–Wentzell sample-path LDP by contracting Schilder's Gaussian LDP through the Itô solution map, using the contraction principle 37.07.08.

Theorem (Freidlin–Wentzell). Under the standing hypotheses, the laws of ${X^{ε}}$ on $C_{x_{0}} ([0, T]; R^{d})$ satisfy the large deviation principle at speed $ε$ with the good rate function $I_{0, T}$ : $$ -\inf_{\phi\in G^\circ} I_{0,T}(\phi)\le \liminf_{\varepsilon\to0}\varepsilon\log\mathbb{P}(X^\varepsilon\in G^\circ)\le \limsup_{\varepsilon\to0}\varepsilon\log\mathbb{P}(X^\varepsilon\in \bar G)\le -\inf_{\phi\in\bar G} I_{0,T}(\phi) $$ for every measurable $G$ . ^{[Dembo & Zeitouni §5.6]}

Proof. Schilder input. The scaled driving path $ε W$ satisfies the LDP on $C_{0} ([0, T]; R^{m})$ at speed $ε$ with the good Schilder rate $$ J(g)=\begin{cases}\tfrac12\int_0^T|\dot g_t|^2,dt, & g\in H_0,\ +\infty,&\text{otherwise,}\end{cases} $$ the kinetic-energy action on the Cameron–Martin space (the Gaussian special case $Λ^{*} (v) = \frac{1}{2} ∣ v ∣^{2}$ of the sample-path principles assembled in the projective-limit chapter).

The Itô map. Let $Φ^{ε} : C_{0} ([0, T]; R^{m}) \to C_{x_{0}} ([0, T]; R^{d})$ send a driver $w$ to the solution of $X = x_{0} + \int_{0}^{\cdot} b (X) d s + ε \int_{0}^{\cdot} σ (X) d w$ , so $X^{ε} = Φ^{ε} (W)$ . The map is not continuous in the uniform topology (the Itô integral is not a continuous functional of $w$ ), so the contraction principle 37.07.08 does not apply to $Φ^{ε}$ directly. Two repairs make it work. First, for the skeleton map at $ε = 0$ one solves the controlled ODE $$ \dot\phi=b(\phi)+\sigma(\phi)\dot g,\qquad \phi_0=x_0,\qquad \phi=\Phi^0(g), $$ which is continuous on the Cameron–Martin space $H_{0}$ (Gronwall: a Lipschitz vector field driven by an $H^{1}$ control depends continuously on the control). Second, one shows $Φ^{ε} (W)$ and the controlled skeleton differ negligibly on the exponential scale: ${Φ^{ε} (ε W)}$ and the family obtained by freezing $σ$ along the skeleton are exponentially equivalent, so $Φ^{ε}$ is an exponentially good approximation of a continuous map in the sense of the approximation theory feeding the contraction principle. The Itô map is therefore continuous where it matters — on the finite-action paths that carry the LDP.

Contraction. Applying the contraction principle 37.07.08 to the continuous skeleton map $Φ^{0} : H_{0} \to H_{x_{0}}$ transports the Schilder rate to $$ I_{0,T}(\phi)=\inf\big{J(g): \Phi^0(g)=\phi\big}=\inf\Big{\tfrac12\textstyle\int_0^T|\dot g|^2: \dot\phi=b(\phi)+\sigma(\phi)\dot g\Big}. $$ Solving the constraint for the control gives $\overset{g}{˙} = σ^{- 1} (ϕ) (\dot{ϕ} - b (ϕ))$ (the unique least-norm solution when $a = σ σ^{T}$ is invertible, by the pseudo-inverse identity $σ^{T} (σ σ^{T})^{- 1}$ in the over-determined case), whence $$ I_{0,T}(\phi)=\tfrac12\int_0^T|\sigma^{-1}(\phi_t)(\dot\phi_t-b(\phi_t))|^2,dt, $$ for $ϕ \in H_{x_{0}}$ , and $+ \infty$ off $H_{x_{0}}$ (no control of finite energy produces a non- $H^{1}$ path). The exponential equivalence of $Φ^{ε} (W)$ with the contracted family transfers the LDP from the skeleton to $X^{ε}$ itself, by the rate-invariance of exponentially equivalent families. Goodness of $I_{0, T}$ is inherited from goodness of $J$ under the proper continuous contraction. $□$

Bridge. This theorem builds toward the exit problem and the Eyring–Kramers law of the Master tier, and appears again in every metastability computation, where the action functional is minimised to find the most probable escape path. This is exactly the contraction principle 37.07.08 applied to the Itô solution map: Schilder's kinetic-energy rate for the driver is pushed forward through the SDE's solution operator, and the quadratic control cost $\frac{1}{2} ∣ \overset{g}{˙} ∣^{2}$ becomes the shifted-velocity cost $\frac{1}{2} ∣ σ^{- 1} (\dot{ϕ} - b) ∣^{2}$ after solving the control constraint. The foundational reason the drift appears only through $\dot{ϕ} - b$ is that riding the deterministic flow 02.15.03 is free — it needs no noise — so the action charges only the deviation from the flow, and the dispersion $σ$ sets the exchange rate between control energy and path velocity. Putting these together, the Freidlin–Wentzell rate generalises Schilder ( $b = 0, σ = Id$ recovers $\frac{1}{2} \int ∣ \dot{ϕ} ∣^{2}$ ) and is dual to the generator picture of 02.15.03: the action is the Lagrangian whose Hamiltonian is the symbol $H (x, p) = ⟨ b, p ⟩ + \frac{1}{2} ⟨ p, a p ⟩$ of the generator, and the quasi-potential solves the stationary Hamilton–Jacobi equation $H (x, \nabla V) = 0$ . The bridge is the identification of "rare path of a small-noise diffusion" with "least-control-energy trajectory of a deterministic control system", the variational shadow that Schilder casts through the Itô map.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that for a gradient system $b = - \nabla U$ , $σ = Id$ , the time-reversed path $\tilde{ϕ}_{t} = ϕ_{T - t}$ satisfies $I_{0, T} (\tilde{ϕ}) = I_{0, T} (ϕ) + 2 (U (ϕ_{T}) - U (ϕ_{0}))$ , and deduce $V (x, y) = 2 (U (y) - U (x))$ for an uphill minimiser.

Hint

Expand $\frac{1}{2} ∣ \dot{ϕ} + \nabla U ∣^{2} = \frac{1}{2} ∣ \dot{ϕ} - \nabla U ∣^{2} + 2 ⟨ \dot{ϕ}, \nabla U ⟩$ and integrate the cross term to a perfect difference $U (ϕ_{T}) - U (ϕ_{0})$ .

Answer

With $b = - \nabla U$ the cost density is $\frac{1}{2} ∣ \dot{ϕ} + \nabla U (ϕ) ∣^{2}$ . Expand: $$ \tfrac12|\dot\phi+\nabla U|^2=\tfrac12|\dot\phi-\nabla U|^2+2\langle\dot\phi,\nabla U\rangle=\tfrac12|\dot\phi-\nabla U|^2+2\tfrac{d}{dt}U(\phi_t). $$ Integrating, $I_{0, T} (ϕ) = \frac{1}{2} \int ∣ \dot{ϕ} - \nabla U ∣^{2} d t + 2 (U (ϕ_{T}) - U (ϕ_{0}))$ . The first term is the action of the time-reversed downhill flow and is non-negative, vanishing precisely when $\dot{ϕ} = \nabla U (ϕ)$ — the uphill anti-gradient flow. Minimising the action subject to $ϕ_{0} = x, ϕ_{T} = y$ thus drives the first term to $0$ along the reversed relaxation path, leaving $V (x, y) = 2 (U (y) - U (x))$ for $y$ uphill of $x$ . The cheapest climb is the exact time-reversal of the deterministic descent, and its cost is twice the potential difference.

Exercise 4 (medium, symbolic).

Derive the Hamilton–Jacobi (eikonal) equation satisfied by the quasi-potential $V (x^{*}, \cdot)$ of a system with attractor $x^{*}$ . Use the Legendre dual $H (x, p) = sup_{v} (⟨ p, v ⟩ - L (x, v))$ of the Lagrangian $L (x, v) = \frac{1}{2} ∣ σ^{- 1} (x) (v - b (x)) ∣^{2}$ .

Hint

Compute $H$ explicitly; the quasi-potential is the value function of a calculus-of-variations problem, so it solves $H (x, \nabla V) = 0$ off the attractor.

Answer

The Legendre transform of $L (x, v) = \frac{1}{2} ⟨ v - b, a^{- 1} (v - b)⟩$ (with $a = σ σ^{T}$ ) is $$ H(x,p)=\langle p,b(x)\rangle+\tfrac12\langle p,a(x)p\rangle, $$ obtained by maximising $⟨ p, v ⟩ - L$ over $v$ , optimal $v^{*} = b + a p$ . As a least-action value function over paths ending at the running point, $V (x^{*}, \cdot)$ is a viscosity solution of the stationary Hamilton–Jacobi equation $$ H(x,\nabla V)=\langle\nabla V,b\rangle+\tfrac12\langle\nabla V,a,\nabla V\rangle=0,\qquad V(x^*)=0,\ V\ge0. $$ Note $H$ is exactly the symbol of the diffusion generator $L = ⟨ b, \nabla ⟩ + \frac{1}{2} \sum a^{ij} \partial_{i} \partial_{j}$ of 02.15.03 with $\partial \mapsto p$ , so the quasi-potential is the zero-energy characteristic of the generator's symbol. In the gradient case $b = - \nabla U, a = Id$ the nontrivial solution is $V = 2 U$ , recovering Exercise 3.

Exercise 5 (medium, symbolic).

State and justify the lower bound on the mean exit time: for a domain $D$ with a single stable attractor $x^{*}$ and $\overset{ˉ}{V} = in f_{y \in \partial D} V (x^{*}, y)$ , show $lim inf_{ε \to 0} ε lo g E_{x^{*}} [τ_{D}] \geq \overset{ˉ}{V} - δ$ for every $δ > 0$ , where $τ_{D}$ is the first exit time.

Hint

Any exit path from $x^{*}$ to $\partial D$ has action $\geq \overset{ˉ}{V}$ ; the LDP upper bound on reaching $\partial D$ in time $T$ gives a probability $\leq e^{- (\overset{ˉ}{V} - δ) / ε}$ , and the exit time is geometric in this small probability.

Answer

Fix $T$ and $δ > 0$ . The event ${τ_{D} \leq T}$ requires the path to travel from $x^{*}$ to $\partial D$ in time $T$ , a closed set of paths whose minimal action is $\geq \overset{ˉ}{V}$ (every such path costs at least the cheapest boundary-reaching cost). By the Freidlin–Wentzell upper bound, $lim sup_{ε} ε lo g P_{x^{*}} (τ_{D} \leq T) \leq - (\overset{ˉ}{V} - \frac{δ}{2})$ for small $ε$ , so $p_{ε} := P_{x^{*}} (τ_{D} \leq T) \leq e^{- (\overset{ˉ}{V} - δ) / ε}$ . Partitioning time into blocks of length $T$ and using the strong Markov property (each block independently fails to exit with probability $\geq 1 - p_{ε}$ ), $τ_{D}$ stochastically dominates $T$ times a geometric variable of success $p_{ε}$ , so $E_{x^{*}} [τ_{D}] \geq T (1 - p_{ε}) / p_{ε} \geq \frac{T}{2} e^{(\overset{ˉ}{V} - δ) / ε}$ for small $ε$ . Taking $ε lo g$ and $lim inf$ gives $lim inf_{ε} ε lo g E_{x^{*}} [τ_{D}] \geq \overset{ˉ}{V} - δ$ . The matching upper bound (a cheap control steering to the boundary in bounded time with probability $≳ e^{- (\overset{ˉ}{V} + δ) / ε}$ ) yields $lim_{ε} ε lo g E_{x^{*}} [τ_{D}] = \overset{ˉ}{V}$ .

Exercise 6 (hard, symbolic).

For the one-dimensional double well $U (x) = \frac{1}{4} x^{4} - \frac{1}{2} x^{2}$ with $σ = 1$ , identify the most probable escape path from the well at $x = - 1$ over the saddle at $x = 0$ , and compute the quasi-potential $\overset{ˉ}{V}$ controlling the escape. Explain why the escape path is the time-reversed relaxation.

Hint

Use Exercise 3: the cheapest uphill path is the time-reversal of the deterministic descent $\dot{ϕ} = - U^{'} (ϕ)$ , and its cost is $2 (U (saddle) - U (well))$ .

Answer

The deterministic dynamics $\dot{ϕ} = - U^{'} (ϕ) = ϕ - ϕ^{3}$ relaxes any point in the left basin to the attractor $x = - 1$ . By Exercise 3 the least-action path out of the well to the saddle is the exact time-reversal of this relaxation: the trajectory solving $\dot{ϕ} = + U^{'} (ϕ) = ϕ^{3} - ϕ$ that climbs from $- 1$ up to the saddle $0$ (an infinite-time heteroclinic orbit of the reversed flow). Its action equals $2 (U (0) - U (- 1)) = 2 (0 - (- \frac{1}{4})) = \frac{1}{2}$ , so $\overset{ˉ}{V} = \frac{1}{2}$ . The escape concentrates on this single path because any other route to the saddle keeps the first term $\frac{1}{2} \int ∣ \dot{ϕ} - (- U^{'}) ∣^{2}$ strictly positive, paying more than the perfect-reversal cost. The mean exit time is therefore $E_{- 1} [τ] ≍ e^{1/ (2 ε)}$ on the exponential scale, matching the Beginner worked example.

Exercise 7 (hard, symbolic).

Sketch the sharp Eyring–Kramers prefactor in one dimension: for $b = - U^{'}$ , $σ = 1$ , with well at $a$ (curvature $U^{''} (a) > 0$ ) and saddle at $b$ (curvature $U^{''} (b) < 0$ ), the mean exit time satisfies $$ \mathbb{E}_a[\tau]\sim \frac{2\pi}{\sqrt{U''(a),|U''(b)|}},\exp!\Big(\frac{2(U(b)-U(a))}{\varepsilon}\Big). $$ Explain where the exponent and the curvature prefactor come from.

Hint

The exponent is the quasi-potential $\overset{ˉ}{V} = 2Δ U$ from the LDP; the prefactor comes from a Laplace/Gaussian expansion of the stationary density and the flux at the saddle, i.e. the Hessians at well and saddle.

Answer

The exponent $2 (U (b) - U (a)) / ε$ is the Freidlin–Wentzell quasi-potential $\overset{ˉ}{V}$ divided by $ε$ : the LDP fixes the exponential rate exactly. The prefactor is not visible to the LDP (which sees only the exponent) and requires the next order. Near the well the invariant density is $\propto e^{- 2 U / ε}$ (the stationary Fokker–Planck solution of 02.15.03), and a Laplace expansion of its normalisation around $a$ gives a Gaussian factor $2 π ε / (2 U^{''} (a))$ controlling how much mass sits in the well. The escape flux is computed by a second Gaussian expansion of the density across the saddle $b$ , where the negative curvature $U^{''} (b)$ supplies the unstable direction along which probability leaks; the curvature there contributes $2 π ε / (2∣ U^{''} (b) ∣)$ . The ratio of these Gaussian widths, with the $ε$ factors cancelling, produces the curvature prefactor $2 π / U^{''} (a) ∣ U^{''} (b) ∣$ . Thus the LDP gives the exponent and a potential-theoretic (capacity) or WKB computation gives the constant — together the sharp Kramers law.

Exercise 8 (hard, symbolic).

Explain why the Itô map is not continuous in the uniform topology, and how the proof of the Freidlin–Wentzell theorem nonetheless legitimately invokes the contraction principle 37.07.08. Name the precise device.

Hint

Uniform convergence of drivers $w_{n} \to w$ does not control $\int σ (X) d w_{n}$ ; the repair is exponential equivalence / an exponentially good approximation of $Φ^{ε}$ by continuous skeleton maps.

Answer

The Itô integral $w \mapsto \int_{0}^{\cdot} σ (X_{s}) d w_{s}$ is not a continuous functional of $w$ in the uniform (sup) topology: one can perturb $w$ uniformly small while changing its quadratic variation and hence the integral by order one (the integral depends on the rough increments of $w$ , not just its uniform size). So $Φ^{ε}$ is not continuous and the contraction principle 37.07.08, which needs a continuous map, does not apply to $Φ^{ε}$ as stated. The legitimate route has two parts. (i) The $ε = 0$ skeleton map $Φ^{0} : H_{0} \to H_{x_{0}}$ , solving the controlled ODE $\dot{ϕ} = b (ϕ) + σ (ϕ) \overset{g}{˙}$ , is continuous on the Cameron–Martin space by Gronwall, and contracting Schilder's rate through it yields $I_{0, T}$ . (ii) One then proves ${Φ^{ε} (W)}$ is exponentially equivalent to (an exponentially good approximation built from) the skeleton family — the stochastic correction is super-exponentially negligible — so by the invariance of the LDP under exponential equivalence, $X^{ε}$ inherits the contracted rate. The precise device is the exponentially good approximation / exponential-equivalence transfer that upgrades the contraction principle from continuous maps to maps that are continuous only after an exponentially-negligible modification.

Advanced results Master

The exit problem and the Eyring–Kramers / Arrhenius law

Let $D$ be a bounded domain with smooth boundary whose deterministic flow $\dot{ϕ} = b (ϕ)$ has a single asymptotically stable equilibrium $x^{*} \in D$ attracting all of $\overset{ˉ}{D}$ , and set $\overset{ˉ}{V} = in f_{y \in \partial D} V (x^{*}, y)$ . The first exit time $τ_{D} = in f {t : X_{t}^{ε} \in / D}$ obeys the Arrhenius law ^{[Freidlin & Wentzell Ch. 4]} $$ \lim_{\varepsilon\to0}\varepsilon\log\mathbb{E}{x}[\tau_D]=\bar V\qquad\text{uniformly in }x\text{ on compacts of }D, $$ and the exit location concentrates: for any neighbourhood $N$ of the boundary minimiser $y^*=\arg\min{\partial D}V(x^,\cdot) $,$ \mathbb{P}x(X^\varepsilon{\tau_D}\in N)\to1 $. T h e m ec hani s mi s t h e v a r ia t i o na l d i c h o t o m y : r e a c hin g t h e b o u n d a r y r e q u i r es a c t i o n$ \ge\bar V $, so t h e w ai t i se x p o n e n t ia l l y l o n g; b u t a co n t r o l o f a c t i o nj u s t ab o v e$ \bar V $s t eer s t o$ y^ $inb o u n d e d t im e, so t h e w ai t i s n o l o n g er t han$ e^{(\bar V+\delta)/\varepsilon} $. T h ee x i t i s t h u s a r a r ee v e n tt ha t, w h e ni t ha pp e n s, f o l l o w s t h es in g l e l e a s t - a c t i o n p a t h t h r o ug h$ y^* $— t h e m os tp r o bab l ee x i tp a t h, w hi c h f or a g r a d i e n t sy s t e mi s t h e t im e - r e v er se d r e l a x a t i o n o v er t h e l o w es t s a dd l eo n$ \partial D$.

The exponential rate $\overset{ˉ}{V}$ is the large-deviation content; the sharp prefactor is finer. For the reversible gradient case $b = - \nabla U$ , $σ = Id$ , with $U$ having a quadratic well at $x^{*}$ (Hessian $\nabla^{2} U (x^{*})$ ) and the relevant saddle $z^{*}$ on $\partial D$ having a single negative Hessian eigenvalue $- λ^{-} (z^{*}) < 0$ , the sharp Eyring–Kramers formula reads ^{[Bovier & den Hollander Ch. 9]} $$ \mathbb{E}{x^}[\tau_D]\sim\frac{2\pi}{\lambda^-(z^)}\sqrt{\frac{|\det\nabla^2U(z^)|}{\det\nabla^2U(x^)}};\exp!\Big(\tfrac{2,(U(z^)-U(x^))}{\varepsilon}\Big). $$ The exponent is the Freidlin–Wentzell quasi-potential $\overset{ˉ}{V} = 2 (U (z^{*}) - U (x^{*}))$ ; the prefactor is a ratio of Gaussian widths at the well and the saddle (curvatures), with the unstable eigenvalue $λ^{-} (z^{*})$ setting the rate at which probability leaks across the saddle. The modern derivation routes the prefactor through Newtonian capacities (Bovier–Eckhoff–Gayrer–Klein), tying the escape rate to the spectral gap of the generator: the smallest non-zero eigenvalue of $- L$ is asymptotically $1/\mathbb{E}{x^*}[\tau_D]$, the metastable relaxation rate.

Metastability and the hierarchy of cycles

When the deterministic system has several stable equilibria $x_{1}^{*}, \dots, x_{n}^{*}$ , the small-noise diffusion spends exponentially long stretches near one equilibrium, then makes a rapid transition to another, then waits again — a metastable Markov jump chain on the set of wells, with transition rates $exp (- V (x_{i}^{*}, x_{j}^{*}) / ε)$ governed by the pairwise quasi-potentials. Freidlin and Wentzell organise the long-time behaviour by a hierarchy of cycles: at the coarsest time scale the process equilibrates within a basin; at successively longer (exponentially separated) scales it overcomes successively higher barriers, the order of transitions fixed by a minimal-spanning-tree / $W$ -graph computation on the quasi-potential matrix ^{[Freidlin & Wentzell Ch. 6]}. The invariant measure concentrates, as $ε \to 0$ , on the global minimiser of a graph-theoretic potential $W (x_{i}^{*})$ built from the spanning trees, the large-deviation generalisation of the Gibbs measure picking out the deepest well. This cycle hierarchy is the rigorous backbone of simulated annealing, where a slowly decreasing $ε = ε (t)$ is tuned so the process funnels into the global minimum.

Non-gradient drifts and the Hamilton–Jacobi route

When $b$ is not a gradient, the quasi-potential is no longer $2 U$ and is generally non-symmetric. It is the value function of the deterministic control problem and solves the stationary Hamilton–Jacobi equation $H (x, \nabla V) = ⟨ b, \nabla V ⟩ + \frac{1}{2} ⟨ \nabla V, a \nabla V ⟩ = 0$ , whose Hamiltonian is the symbol of the generator 02.15.03. The drift decomposes (Freidlin–Wentzell / Graham) as $b = - \frac{1}{2} a \nabla V + ℓ$ with $⟨ ℓ, \nabla V ⟩ = 0$ : a gradient-like part $- \frac{1}{2} a \nabla V$ driving relaxation down the quasi-potential and a transverse circulation $ℓ$ that does no work against $V$ . The most probable exit path is the instanton: the heteroclinic orbit of the doubled (Hamiltonian) dynamics $\overset{x}{˙} = \partial_{p} H$ , $\overset{p}{˙} = - \partial_{x} H$ on the zero-energy surface $H = 0$ , climbing from the attractor along $p = \nabla V$ to the saddle. This Hamiltonian/instanton formulation is the bridge from Freidlin–Wentzell theory to the path-integral and WKB treatments of rare events in physics.

Synthesis. The central insight is that a small-noise diffusion is, on the large-deviation scale, a deterministic control system: the action is the control energy, the most probable rare path is the optimal control, and the quasi-potential is the value function, so rare-event behaviour is read off a Hamilton–Jacobi problem with no probability left in it. This is exactly the contraction of Schilder's Gaussian rate through the Itô map 37.07.08, putting these together with the SDE solution theory 02.15.03 so the driver's kinetic-energy cost becomes the shifted-velocity action. The foundational reason the exit time is $e^{\overset{ˉ}{V} / ε}$ is the variational dichotomy — reaching the boundary costs at least $\overset{ˉ}{V}$ , and a near-optimal control achieves it — so the LDP bounds pin the exponent exactly, while the sharp Kramers prefactor generalises this to next order through capacities and Hessians. The bridge is the dual reading of one object: the action is the Lagrangian, the generator's symbol its Hamiltonian, the quasi-potential the zero-energy Hamilton–Jacobi solution, and the metastable jump rates $e^{- V (x_{i}, x_{j}) / ε}$ assemble into the cycle hierarchy. This is dual to the generator picture of 02.15.03, where the escape rate is the principal spectral gap of $- L$ . So Freidlin–Wentzell theory is Hamilton's least-action principle for the rare excursions of a noisy flow, and appears again in reaction-rate theory, stochastic resonance, and simulated annealing.

Full proof set Master

The Freidlin–Wentzell sample-path LDP is proved in full in the Key theorem section. The exit-problem and gradient-structure claims are recorded here.

Proposition 1 (gradient quasi-potential). For $b = - \nabla U$ and $σ = Id$ with $U \in C^{2}$ , the quasi-potential from a critical point $x$ to a point $y$ reachable along an anti-gradient orbit satisfies $V (x, y) = 2 (U (y) - U (x))$ , attained by the time-reversed relaxation path.

Proof. For any admissible $ϕ$ with $ϕ_{0} = x, ϕ_{T} = y$ , the cost density expands as $$ \tfrac12|\dot\phi+\nabla U(\phi)|^2=\tfrac12|\dot\phi-\nabla U(\phi)|^2+2\langle\dot\phi,\nabla U(\phi)\rangle=\tfrac12|\dot\phi-\nabla U(\phi)|^2+2\tfrac{d}{dt}U(\phi_t). $$ Integrating over $[0, T]$ , $I_{0, T} (ϕ) = \frac{1}{2} \int_{0}^{T} ∣ \dot{ϕ} - \nabla U (ϕ) ∣^{2} d t + 2 (U (y) - U (x))$ . The first term is non-negative and vanishes iff $\dot{ϕ} = \nabla U (ϕ)$ for a.e. $t$ , i.e. $ϕ$ follows the reversed (uphill, anti-gradient) flow. Such a connecting orbit exists for $y$ on the unstable manifold structure reachable from $x$ (taking $T \to \infty$ to accommodate the asymptotic approach to critical points), and along it $I = 2 (U (y) - U (x))$ . Hence the infimum over $T > 0$ and admissible $ϕ$ equals $2 (U (y) - U (x))$ , attained in the limit by the time-reversed relaxation. $□$

Proposition 2 (lower bound on the mean exit time). Let $D$ be bounded with attractor $x^ $an d$ \bar V=\inf_{\partial D}V(x^,\cdot) $. T h e n$ \liminf_{\varepsilon\to0}\varepsilon\log\mathbb{E}_{x^}[\tau_D]\ge\bar V$.*

Proof. Fix $δ > 0$ and a horizon $T$ . The set $A_{T} = {ϕ \in C_{x^{*}} ([0, T]) : ϕ_{t} \in \partial D for some t \leq T}$ is closed and every $ϕ \in A_{T}$ has $I_{0, T} (ϕ) \geq in f {I_{0, T} (ψ) : ψ hits \partial D} \geq \overset{ˉ}{V}$ (a boundary-hitting path realises a connection of cost $\geq V (x^{*}, \partial D) = \overset{ˉ}{V}$ ). The Freidlin–Wentzell upper bound gives $lim sup_{ε} ε lo g P_{x^{*}} (X^{ε} \in A_{T}) \leq - in f_{A_{T}} I_{0, T} \leq - \overset{ˉ}{V}$ , so $p_{ε} := P_{x^{*}} (τ_{D} \leq T) \leq e^{- (\overset{ˉ}{V} - δ) / ε}$ for small $ε$ . By the strong Markov property at the times $T, 2 T, \dots$ , starting each block from a point that has not yet exited (and using uniformity of the bound over compacts of $D$ near $x^{*}$ , together with the fact that the flow returns to a neighbourhood of $x^{*}$ ), the number of length- $T$ blocks before exit dominates a geometric variable with success probability $p_{ε}$ . Hence $E_{x^{*}} [τ_{D}] \geq T \cdot \frac{1 - p _{ε}}{p _{ε}} \geq \frac{T}{2} e^{(\overset{ˉ}{V} - δ) / ε}$ for small $ε$ . Taking $ε lo g$ and $lim inf$ , then $δ ↓ 0$ , gives $lim inf_{ε} ε lo g E_{x^{*}} [τ_{D}] \geq \overset{ˉ}{V}$ . $□$

Proposition 3 (upper bound on the mean exit time). Under the hypotheses of Proposition 2, $\limsup_{\varepsilon\to0}\varepsilon\log\mathbb{E}_{x^}[\tau_D]\le\bar V $, so t h e l imi t e q u a l s$ \bar V$.*

Proof. Fix $δ > 0$ . By definition of $\overset{ˉ}{V}$ there is a finite time $T_{0}$ and an admissible control path $ϕ^{*}$ from (a neighbourhood of) $x^{*}$ to the boundary minimiser $y^{*} \in \partial D$ with $I_{0, T_{0}} (ϕ^{*}) < \overset{ˉ}{V} + \frac{δ}{2}$ . The Freidlin–Wentzell lower bound, applied to a uniform tube of paths around $ϕ^{*}$ that exit $D$ by time $T_{0}$ , gives $$ \liminf_\varepsilon\varepsilon\log\mathbb{P}{x}(\tau_D\le T_0)\ge-I_{0,T_0}(\phi^)>-(\bar V+\delta) $$ uniformly for $x$ in a neighbourhood of $x^ $. T h u s$ q\varepsilon:=\inf_x\mathbb{P}x(\tau_D\le T_0)\ge e^{-(\bar V+\delta)/\varepsilon} $f or s ma l l$ \varepsilon $. S in ce f r o man y s t a r t in g p o in tt h e p r ocesse i t h er e x i t s w i t hin$ T_0 $w i t h p r o babi l i t y$ \ge q\varepsilon $or (w i t h t h er e mainin g p r o babi l i t y) i s f o u n d n e a r$ x^ $a f t er a f u r t h er b o u n d e d t im e b y a tt r a c t i o n o f t h e d e t er mini s t i c f l o w, t h ee x i tt im e i ss t oc ha s t i c a l l y d o mina t e d b y$ T_0 $t im es a g eo m e t r i c v a r iab l eo f s u ccess$ q_\varepsilon $:$ \mathbb{E}_{x^}[\tau_D]\le \frac{C T_0}{q_\varepsilon}\le C T_0,e^{(\bar V+\delta)/\varepsilon} $. T ak in g$ \varepsilon\log $,$ \limsup $, an d$ \delta\downarrow0 $g i v es$ \limsup_\varepsilon\varepsilon\log\mathbb{E}_{x^*}[\tau_D]\le\bar V $. W i t h P r o p os i t i o n 2 t h e l imi t i se x a c tl y$ \bar V $.$ \square$

Proposition 4 (Hamilton–Jacobi characterisation of the quasi-potential). Off the attractor $x^ $, t h e q u a s i - p o t e n t ia l$ V(x^,\cdot) $i s a v i scos i t y so l u t i o n o f$ \langle b,\nabla V\rangle+\tfrac12\langle\nabla V,a,\nabla V\rangle=0 $w i t h$ V(x^)=0 $,$ V\ge0 $, w h er e$ a=\sigma\sigma^{\mathsf T}$.*

Proof. The quasi-potential is the value function $V (y) = in f {I_{0, T} (ϕ) : ϕ_{0} = x^{*}, ϕ_{T} = y, T > 0}$ of a deterministic Bolza control problem with Lagrangian $L (x, v) = \frac{1}{2} ⟨ v - b, a^{- 1} (v - b)⟩$ . By the dynamic programming principle, value functions of such problems are viscosity solutions of the stationary Hamilton–Jacobi equation $H (x, \nabla V) = 0$ , where $H (x, p) = sup_{v} (⟨ p, v ⟩ - L (x, v))$ is the Legendre dual. Computing the supremum: $\partial_{v} (⟨ p, v ⟩ - \frac{1}{2} ⟨ v - b, a^{- 1} (v - b)⟩) = p - a^{- 1} (v - b) = 0$ gives the optimal $v^{*} = b + a p$ , and substituting, $$ H(x,p)=\langle p,b+ap\rangle-\tfrac12\langle ap,a^{-1}ap\rangle=\langle p,b\rangle+\langle p,ap\rangle-\tfrac12\langle p,ap\rangle=\langle p,b\rangle+\tfrac12\langle p,ap\rangle. $$ Hence $V$ solves $⟨ b, \nabla V ⟩ + \frac{1}{2} ⟨ \nabla V, a \nabla V ⟩ = 0$ . The boundary data $V (x^{*}) = 0$ holds because the constant path at the attractor has zero action, and $V \geq 0$ since the action is non-negative. This $H$ is the principal symbol of the generator $L$ of 02.15.03 under $\nabla \mapsto p$ , identifying the quasi-potential as the generator's zero-energy characteristic. $□$

Connections Master

The contraction principle 37.07.08 is the engine of the entire unit: the Freidlin–Wentzell rate is Schilder's kinetic-energy rate pushed forward through the SDE's solution (Itô) map, with the quasi-potential a minimisation of the contracted action. Because the Itô map is not uniformly continuous, the contraction is applied to the continuous controlled-ODE skeleton and transferred to the diffusion by exponential equivalence — exactly the exponentially-good-approximation extension of the contraction principle developed there, so this unit is the flagship application of that machinery to an infinite-dimensional, non-continuous transport.

The Itô integral and Itô's formula 02.15.02 supply the stochastic integral $ε \int σ (X) d W$ that the small-noise SDE runs on, and the non-continuity of $w \mapsto \int σ d w$ in the uniform topology is precisely the obstruction the proof must route around; Itô's formula is also what produces the generator symbol $H (x, p)$ appearing as the Hamiltonian of the quasi-potential.

Stochastic differential equations, diffusions, and the generator 02.15.03 is the substrate: the small-noise diffusion is the SDE of that unit with dispersion $ε σ$ , its $ε \to 0$ limit is the drift flow, and the exit problem refines that unit's Dynkin/exit-time analysis. The Hamiltonian $H (x, p) = ⟨ b, p ⟩ + \frac{1}{2} ⟨ p, a p ⟩$ of the quasi-potential is the symbol of that unit's generator $L$ , and the sharp Kramers prefactor is governed by the spectral gap of $- L$ , making the metastable escape rate the principal eigenvalue of the generator built there.

Schilder's theorem (co-produced as 37.07.10 in this wave) is the Gaussian input contracted here: it is the $b = 0, σ = Id$ instance whose kinetic-energy rate $\frac{1}{2} \int ∣ \dot{ϕ} ∣^{2}$ becomes the shifted action $\frac{1}{2} \int ∣ σ^{- 1} (\dot{ϕ} - b) ∣^{2}$ after the Itô map, so the Freidlin–Wentzell theorem is Schilder plus drift-and-dispersion transport.

The Gärtner–Ellis theorem 37.07.04 supplies the finite-dimensional marginal LDPs (from the cumulant generating function of increments) that the sample-path principle perturbs, and the half-space tightness criteria there are what close the finite-time-marginal estimates into the full path-space principle that Freidlin–Wentzell specialises to the small-noise scaling.

Historical & philosophical context Master

The physics came first. In 1935 Henry Eyring's transition-state theory (Journal of Chemical Physics 3, 107–115) ^{[Eyring 1935]} expressed a chemical reaction rate as an Arrhenius exponential of an activation energy times a frequency prefactor, and in 1940 Hendrik Kramers (Physica 7, 284–304) ^{[Kramers 1940]} modelled the reacting molecule as a Brownian particle escaping a potential well, computing the mean escape time as $e^{Δ U / (k_{B} T)}$ with a curvature-dependent prefactor — the first derivation of what is now the Eyring–Kramers law. These were physical computations on the Fokker–Planck equation, without a general large-deviation principle behind them.

The rigorous and general theory is due to Mark Freidlin and Alexander Wentzell, whose 1970 work and the 1979 monograph Random Perturbations of Dynamical Systems ^{[Freidlin & Wentzell Ch. 4]} established the sample-path LDP for $d X^{ε} = b (X^{ε}) d t + ε σ (X^{ε}) d W$ with the action functional $I (ϕ) = \frac{1}{2} \int ∣ σ^{- 1} (\dot{ϕ} - b) ∣^{2}$ , and used it to solve the exit problem, prove the Arrhenius law $ε lo g E τ_{D} \to \overset{ˉ}{V}$ , and organise multi-well metastability through the hierarchy of cycles. Their route — contracting Schilder's 1966 Gaussian asymptotics ^{[Schilder 1966]} through the SDE solution map — placed the action-functional rate on the same footing as Mogulskii's trajectory principle, and identified the quasi-potential as the value function of a deterministic control problem solving a Hamilton–Jacobi equation. The sharp prefactor, beyond the large-deviation exponent, was obtained rigorously much later by Bovier, Eckhoff, Gayrard, and Klein through a potential-theoretic (capacity) analysis tying the escape rate to the spectral gap of the generator ^{[Bovier & den Hollander Ch. 9]}. The lineage runs from a chemist's reaction-rate formula to a theorem about the spectrum of a second-order elliptic operator, each layer sharpening the previous: Eyring and Kramers fixed the form, Freidlin and Wentzell fixed the exponent rigorously and in full generality, and the potential-theoretic school fixed the constant.

Bibliography Master

@book{freidlinwentzell2012,
  author    = {Freidlin, Mark I. and Wentzell, Alexander D.},
  title     = {Random Perturbations of Dynamical Systems},
  edition   = {3rd},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {260},
  publisher = {Springer-Verlag, Berlin},
  year      = {2012}
}

@book{dembozeitouni1998ldp,
  author    = {Dembo, Amir and Zeitouni, Ofer},
  title     = {Large Deviations Techniques and Applications},
  edition   = {2nd},
  series    = {Applications of Mathematics},
  number    = {38},
  publisher = {Springer},
  year      = {1998}
}

@book{bovierdenhollander2015,
  author    = {Bovier, Anton and den Hollander, Frank},
  title     = {Metastability: A Potential-Theoretic Approach},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {351},
  publisher = {Springer},
  year      = {2015}
}

@article{kramers1940brownian,
  author  = {Kramers, Hendrik A.},
  title   = {Brownian motion in a field of force and the diffusion model of chemical reactions},
  journal = {Physica},
  volume  = {7},
  pages   = {284--304},
  year    = {1940}
}

@article{eyring1935activated,
  author  = {Eyring, Henry},
  title   = {The activated complex in chemical reactions},
  journal = {Journal of Chemical Physics},
  volume  = {3},
  pages   = {107--115},
  year    = {1935}
}

@article{schilder1966asymptotic,
  author  = {Schilder, M.},
  title   = {Some asymptotic formulas for {W}iener integrals},
  journal = {Transactions of the American Mathematical Society},
  volume  = {125},
  pages   = {63--85},
  year    = {1966}
}

@article{boviereckhoff2004metastability,
  author  = {Bovier, Anton and Eckhoff, Michael and Gayrard, V\'eronique and Klein, Markus},
  title   = {Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times},
  journal = {Journal of the European Mathematical Society},
  volume  = {6},
  pages   = {399--424},
  year    = {2004}
}

Prerequisites

37.07.08
02.15.02
02.15.03

Tier anchors

beginner: Touchette 2009 *The large deviation approach to statistical mechanics* (Physics Reports 478) §4-§5 (small-noise action and the most-probable path); Freidlin & Wentzell 2012 *Random Perturbations of Dynamical Systems* 3rd ed. (Springer) Ch. 1-2 (informal action-functional picture of a perturbed flow)
intermediate: Dembo & Zeitouni 1998 *Large Deviations Techniques and Applications* 2nd ed. (Springer) §5.6 (Theorem 5.6.3 sample-path LDP for small-noise diffusions; the Freidlin-Wentzell rate); Freidlin & Wentzell 2012 *Random Perturbations of Dynamical Systems* 3rd ed. (Springer) Ch. 3-4 (action functional, quasi-potential, exit problem)
master: Dembo & Zeitouni 1998 *Large Deviations Techniques and Applications* 2nd ed. (Springer) §5.6-§5.7 (Freidlin-Wentzell theorem via the contraction of Schilder through the Itô map; the exit problem); Freidlin & Wentzell 2012 *Random Perturbations of Dynamical Systems* 3rd ed. (Springer) Ch. 3-4, Ch. 6 (quasi-potential, Eyring-Kramers mean exit time, metastability); Berglund 2013 *Kramers' law: validity, derivations and generalisations* (Markov Processes and Related Fields 19)

References

Dembo, A. & Zeitouni, O. — Large Deviations Techniques and Applications, 2nd ed. (Springer, 1998) · §5.6 (Theorem 5.6.3, sample-path LDP for $dX^\varepsilon=b(X^\varepsilon)dt+\sqrt\varepsilon\,\sigma(X^\varepsilon)dW$ with rate $I(\phi)=\tfrac12\int|\sigma^{-1}(\dot\phi-b)|^2$); §5.7 (the exit problem and quasi-potential)
Freidlin, M. I. & Wentzell, A. D. — Random Perturbations of Dynamical Systems, 3rd ed. (Springer, 2012) · Ch. 3 (action functional), Ch. 4 (exit from a domain, the quasi-potential $V(x,y)$ and the Arrhenius mean exit time), Ch. 6 (metastability and the hierarchy of cycles)
Bovier, A. & den Hollander, F. — Metastability: A Potential-Theoretic Approach (Springer, 2015) · Ch. 9-11 (the sharp Eyring-Kramers prefactor via capacities, the spectral picture of metastable exit)
Kramers, H. A. — Brownian motion in a field of force and the diffusion model of chemical reactions · Physica 7 (1940), 284-304 (the original mean-reaction-time / escape-rate computation, the physics ancestor of the exit-time law)
Schilder, M. — Some asymptotic formulas for Wiener integrals · Transactions of the AMS 125 (1966), 63-85 (the small-noise Brownian LDP that Freidlin-Wentzell contracts through the Itô map)
Eyring, H. — The activated complex in chemical reactions · Journal of Chemical Physics 3 (1935), 107-115 (transition-state theory, the Arrhenius exponential and the energy-barrier picture)

Estimated time

beginner: 18m
intermediate: 46m
master: 82m