Universit`
a degli Studi di Pisa
FACOLT `A DI MATEMATICA Corso di Laurea Magistrale in Matematica
Tesi di laurea magistrale
Large Deviations for SDEs
Candidato:
Umberto Pappalettera
Matricola 524622
Relatore:
Prof. Franco Flandoli
Contents
Introduction v
1 Large Deviations 1
1.1 Basic definitions . . . 1
1.2 First results . . . 4
2 Large Deviations for SDEs 9 2.1 Schilder’s Theorem for Brownian Motion . . . 9
2.2 Freidlin-Wentzell Theorem for SDEs with regular drift . . . 13
2.3 Freidlin-Wentzell Theorem for bounded continuous drift . . . 16
2.4 Freidlin-Wentzell Theorem for unbounded drift . . . 22
2.5 A counterexample for the case b ∈L∞ . . . . 24
3 Large Deviations for Peano Phenomenon 27 3.1 First order LDP . . . 28
3.2 Second order LDP . . . 30
A One-parameter Semigroups 43
Introduction
The study of asymptotical problems have always played an important role in Probability. During their study at university, a lot of students come in touch with classical theorems for a sequence of random variables of the type of Law of Large Numbers or Central Limit Theorem, which have been known (with a formulation more or less precise) for some cen-turies. Large Deviations are a bit more recent: while some basic ideas of the theory can be traced to Laplace, the first rigorous formalization of Large Deviations has been done by Varadhan in 1960s. Nowadays, this is one of the main subject of study in Probability, expecially in the theory of random processes, because, unlike classical discrete probability, it is often very difficult to obtain simple exact formulas in problems connected with them. In this work we study Large Deviations for Stochastic Differential Equations, namely we are interested in giving asymptotical estimates for a family of random processes Xε
satisfying dXε
t = b(Xtε) dt + ε dWt, where W is a standard Brownian motion. These
estimates are stated in terms of the drift b and the intensity of the noise ε and are asymptotical in the limit ε → 0. The paradigm of this type of investigations is the following result due to Friedlin and Wentzell:
ε2log P {Xε∼ ϕ} → 1 2
Z T
0
k ˙ϕt− b(ϕt)k2dt,
where ϕ ∈ C([0, T ], Rd) is absolutely continuous and ϕ(0) = 0. The first chapter of this
work deals with Large Deviations in abstract metric spaces and is preparatory to the following. It contains all the definitions and first results necessary to get started and it is deliberately shallow, as this work is not about Large Deviations on their own. In the second chapter, we develop the theory of Large Deviations for SDEs. Sections 1 and 2 of this chapter are taken from the book of Freidlin and Wentzell [9]. The main results of these sections are Schilder’s Theorem (1966) and Freidlin-Wentzell Theorem (1969), which concern respectively the case with null drift and bounded and Lipschitz continuous drift. Section 3 applies some ideas from [10] to generalize Freidlin-Wentzell Theorem to a bounded and continuous drift. The unbounded case is treated in Section 4 and a counterexample for a bounded and measurable drift is given in Section 5. As far as we know, the results of Sections 4 and 5 are original. In the third chapter, Large Deviations in presence of a Peano phenomenon are investigated. In particular, we study the equation dXtε = b(Xε
t) dt+ε dWtwith b : Rd→ Rdgiven by b(x) := xkxkγ−1, γ ∈ (0, 1). In addition
to the Large Deviations Principle given by results of Chapter 2, here another LDP holds, in correspondence with bifurcation of solutions to the unperturbed equation:
Chapter 0. Introduction
Section 2 of this chapter is the d-dimensional generalization of [10], where the case d = 1 is considered. The main tool to perform such a generalization is Lemma 3.4, which is a novelty. Some notions about one-parameter semigroups and viscosity solutions are given in the appendix.
Finally, I would like to thank all the people with whom I studied mathematics in these years and from whom I learned a great part of what I know. A special thanks goes to Prof. Flandoli, who has carefully guided me over the last few years: his concern and dedication are the greatest thing a student can ask.
Chapter 1
Large Deviations
1.1
Basic definitions
Here we give the first definitions involving Large Deviations. We follow mainly the approach of [9], but [7] is a classical reference as well. Let (X , ρ) be a metric space. On the σ-field of its Borelian let:
µε be a family of probability measures depending on a parameter ε > 0;
λ : R+ → R+ be a function going to +∞ as its argument goes to zero;
S : X → [0, +∞] be a function such that the set Φ(s) := {x ∈ X : S(x) ≤ s} is compact for every s ≥ 0.
We call such λ a rate function and such S a renormalized action function. We say that S is trivial if S takes only values 0 and +∞. We always assume S to be non-trivial. The first topological property of a renormalized action function is stated in the following: Proposition 1.1. A renormalized action function S always attains its infimum on every nonempty closed set A.
Proof. Let xn ∈ A be a sequence such that S(xn) is decreasing to infx∈AS(x); obviously
we have Φ(S(xn+1)) ∩ A ⊆ Φ(S(xn)) ∩ A, hence by Cantor’s intersection Theorem there
exists a x ∈T
k∈NΦ(S(xk)) ∩ A, at which clairly S attains its infimum over A.
Definition 1.1. We say that the triple (λ, S, µε) satifies the Freidlin-Wentzell conditions
if:
(FW1) for any δ > 0 , γ > 0 , x ∈ X there exists an ε0 >0 such that for every ε ≤ ε0
µε{y ∈ X : ρ(x, y) < δ} ≥ exp(−λ(ε)[S(x) + γ]); (1.1) (FW2) for any δ > 0 , γ > 0 , s > 0 there exists an ε0 >0 such that for every ε ≤ ε0
Chapter 1. Large Deviations
Remark 1.1. Notice that condition (FW1) implies that for any δ > 0,γ > 0 and any s0 > 0 there exists an ε0 > 0 such that the inequality (1.1) holds for every ε < ε0 and
x ∈ Φ(s0): indeed, by lower semicontinuity of S, for every x ∈ Φ(s0) there exists a
positive δ = δ(x) such that S(x) ≤ S(y) + γ/2 for every y ∈ Bδ(x), thus by compactness
of Φ(s0) there exist xi, i = 1, . . . , N and δ > 0 such that Φ(s0) ⊆ ∪iBδ(xi), hence given
any x ∈ Φ(s0) there exist xk such that
µε{y ∈ X : ρ(x, y) < 2δ} ≥ µε{y ∈ X : ρ(x
k, y) < δ}
≥ exp(−λ(ε)[S(xk) + γ/2])
≥ exp(−λ(ε)[S(x) + γ])
for every ε ≤ ε0 with ε0 sufficiently small but independent on x, since it can be chosen
such that (1.1) holds only for xi, i = 1, . . . , N . Similarly, condition (FW2) implies that
for any δ > 0,γ > 0 and any s0 >0 there exists an ε0 >0 such that the inequality (1.2)
holds for every ε < ε0 and s ≤ s0: indeed, define si := iγ/2, i = 0, . . . , d2s0/γe and take
ε0 such that (1.2) holds for every si with γ = γ/2. Therefore for every s ≤ s0 there exists
sk ∈ [s − γ/2, s) for which for every ε ≤ ε0 we have:
µε{y ∈ X : ρ(Φ(s), y) ≥ δ} ≤ µε{y ∈ X : ρ(Φ(s
k), y) ≥ δ}
≤ exp(−λ(ε)[sk− γ/2])
≤ exp(−λ(ε)[s − γ]).
The opposite implications are quite obvious, so the conditions above together are in fact equivalent to the Freidlin-Wentzell conditions. We refer to them respectively as conditions (FW1) and (FW2) as well.
Definition 1.2. We say that the triple (λ, S, µε) satifies the Varadhan conditions if:
(V1) for any open A ⊆ X we have lim inf
ε→0+ λ(ε)
−1
log µε(A) ≥ − inf
x∈AS(x);
(V2) for any closed A ⊆ X we have lim sup
ε→0+
λ(ε)−1log µε(A) ≤ − inf x∈AS(x).
Definition 1.3. We say that the triple (λ, S, µε) satifies the Borovkov condition if:
(B1) for any Borel A ⊆ X such that infx∈AS(x) = infx∈ ˚AS(x) we have lim
ε→0+λ(ε)
−1
log µε(A) = − inf x∈AS(x).
In the following we say that a Borelian A ⊆ X is regular with respect to S if infx∈AS(x) = infx∈ ˚AS(x) holds.
Proposition 1.2. The Freidlin-Wentzell conditions, the Varadhan conditions and the Borovkov condition are all equivalent.
1.1. Basic definitions
Proof. We prove first the equivalence between the Freidlin-Wentzell conditions and the Varadhan conditions, then the equivalence between the Varadhan conditions and the Borovkov condition. Let divide the proof in single steps.
(FW1) ⇒ (V1): take x ∈ A such that S(x) ≤ infx∈AS(x) + γ; since A is open there
exists a δ > 0 such that lim inf
ε→0+ λ(ε)
−1
log µε(A) ≥ lim inf ε→0+ λ(ε)
−1
log µε{y ∈ X : ρ(x, y) < δ}
≥ −[S(x) + γ] ≥ − inf
x∈AS(x) − 2γ.
Since γ is arbitrary condition (V1) holds.
(FW2) ⇒ (V2): take s = infx∈AS(x) − γ. The closed set A does not intersect the
compact set Φ(s), therefore δ = ρ(A, Φ(s)) > 0 is such that µε(A) ≤ µε{y ∈ X : ρ(Φ(s), y) ≥ δ}
≤ exp(−λ(ε)[s − γ]) = exp(−λ(ε)[ inf
x∈AS(x) − 2γ])
for ε sufficiently small. Taking logarithms, dividing by λ(ε) and taking the limit for ε → 0+ we obtain (V2) by the arbitrariness of γ.
(V1) ⇒ (FW1): since the set {y ∈ X : ρ(x, y) < δ} = Bδ(x) is open we have
µε{y ∈ X : ρ(x, y) < δ} ≥ exp(−λ(ε) inf
y∈Bδ(x)
S(y)) ≥ exp(−λ(ε)[S(x) + γ]) for every ε small enough and γ > 0.
(V2) ⇒ (FW2): the set A = {y ∈ X : ρ(Φ(s), y) ≥ δ} is closed and S > s in it, therefore infx∈AS(x) ≥ s. By hypothesis for every γ > 0 there exists ε0 > 0 such
that for every ε ≤ ε0 we have λ(ε)−1log µε(A) ≤ −s + γ, that is (FW2).
(V1) + (V2) ⇒ (B1): we have the following chain of inequalities: lim inf
ε→0+ λ(ε)
−1
log µε(A) ≥ lim inf ε→0+ λ(ε) −1 log µε( ˚A) ≥ − inf x∈ ˚A S(x) = − inf x∈A S(x) ≥ lim sup ε→0+ λ(ε)−1log µε(A) ≥ lim sup ε→0+ λ(ε)−1log µε(A), then the limit exists and it coincides with − infx∈AS(x).
(B1) ⇒ (V1): for any positive δ define A−δ := {x ∈ X : ρ(Ac, x) > δ} and s(−δ) :=
infx∈A−δS(x), the infimum over the empty set being +∞. The function s is defined
for every negative value of the argument and it is nonincreasing, then it is continuous except at a countable number of points. Observe that continuity of s at −δ is equiv-alent to regularity of the set A−δ, in the sense that infx∈A−δS(x) = infx∈ ˚A−δS(x).
Chapter 1. Large Deviations
Now choose any γ > 0 and take δ such that s(−δ) < infx∈AS(x) + γ and s is
continuous at −δ: this can be done since if lim−δ→0−s(−δ) > infx∈AS(x) then any
x ∈ A such that S(x) < infx∈AS(x) + γ must be arbitrarily close to Ac, absurd
since A is open. We have lim inf
ε→0+ λ(ε)
−1
log µε(A) ≥ lim inf ε→0+ λ(ε) −1 log µε(A −δ) > −inf x∈AS(x) − γ.
Since γ is arbitrary condition (V1) is proved.
(B1) ⇒ (V2): for any positive real δ define A+δ := {x ∈ X : ρ(A, x) < δ} and
s(+δ) := infx∈A+δS(x). The function s is defined for every positive value of the
argument and it is nonincreasing, then it is continuous except at a countable number of points. Observe that continuity of s at +δ is equivalent to regularity of the set A+δ, in the sense that infx∈A+δ S(x) = infx∈ ˚A+δS(x). Now choose any γ > 0 and
take δ such that s(+δ) > infx∈AS(x)−γ and s is continuous at +δ: this can be done
since every sequence (xn)n∈Nsuch that xn ∈ A+δn, δn → 0
+ as n → ∞ and S(x n) →
limδ→0+s(δ) has its tail in the compact set Φ(s(1)), so there is a subsequence (xn k)
converging to x∞∈ A (since A is closed) and S is lower semicontinuous. We have
lim sup
ε→0+
λ(ε)−1log µε(A) ≤ lim sup ε→0+
λ(ε)−1log µε(A +δ)
< −inf
x∈AS(x) + γ.
Since γ is arbitrary condition (V2) is proved.
We say that λ(ε)S is an action function for µε if one of the conditions above is satisfied.
In a more informal way we say that a Large Deviations Principle holds.
1.2
First results
In order to establish a Large Deviations Principle (in the following LDP) for a familiy of probability measures µεwith action function λ(ε)S some techniques have been developed
(see [7], Chapter 4, for a more comprehensive discussion of the subject). First of all, under certain assumption we can check Varadhan conditions locally.
Proposition 1.3. Suppose that the triple (λ, S, µε) satifies the following condition:
(V1−) for any x ∈ X and δ > 0 we have
lim inf
ε→0+ λ(ε)
−1
log µε(Bδ(x)) ≥ −S(x) − o(1)δ→0+.
1.2. First results
Proof. Take A ⊆ X open and x ∈ A, choose a δ > 0 such that Bδ(x) ⊆ A. We get
lim inf
ε→0+ λ(ε)
−1
log µε(A) ≥ lim inf ε→0+ λ(ε)
−1
log µε(B
δ(x)) ≥ −S(x) − o(1)δ→0+.
The thesis follows by taking δ → 0+ and the supremum in x ∈ A.
Definition 1.4. We say that a family of probability measures µε on X is exponentially
tight with rate λ if for every M ∈ N there exists a compact set KM ⊆ X such that
lim sup
ε→0+
λ(ε)−1log µε(Kc
M) < −M. (1.3)
Lemma 1.4. Suppose that µε is exponentially tight with rate λ and the triple (λ, S, µε)
satifies the following condition:
(V2 weak) for any compact A ⊆ X we have lim sup
ε→0+
λ(ε)−1log µε(A) ≤ − inf x∈AS(x).
Then condition (V2) holds for the triple (λ, S, µε).
Proof. By Proposition 1.1 the minimum of S on A is attained at a certain x0 and without
loss of generality we can suppose KM ⊆ KM +1 and x0 ∈ KM for every M . Thus
lim sup
ε→0+
λ(ε)−1log µε(A) ≤ lim sup
ε→0+ λ(ε)−1log (µε(A ∩ KM) + µε(KMc )) ≤ lim sup ε→0+ λ(ε)−1log (2 max {µε(A ∩ K M), µε(KMc )}) ≤ − min {S(x0), M } .
Taking M → ∞ we have the thesis.
Proposition 1.5. Suppose that µε is exponentially tight with rate λ and the triple
(λ, S, µε) satifies the following condition:
(V2−) for any x ∈ X and δ > 0 we have
lim sup ε→0+ λ(ε)−1log µεB δ(x) ≤ −S(x) + o(1)δ→0+,
where o(1) is uniform in x ∈ X . Then condition (V2) holds for the triple (λ, S, µε).
Proof. By Lemma 1.4 we can suppose A compact. Given any δ > 0 by compactness there exist x1, . . . , xNδ in A such that A ⊆ ∪
Nδ
k=1Bδ(xk). With the same calculations of
the lemma above we obtain lim sup
ε→0+
λ(ε)−1log µε(A) ≤ lim sup ε→0+ λ(ε)−1logNδmax k n µεBδ(xk) o ≤ − min k (S(xk) − o(1)δ→0 +) .
Taking δ → 0+ we finally obtain
lim sup
ε→0+
λ(ε)−1log µε(A) ≤ − min
Chapter 1. Large Deviations
Remark 1.2. Conditions (V1−) and (V2−) give a useful interpretation of the role of S in
Large Deviations: in fact, at least formally we can write µε(Bδ(x)) ∼ exp (−λ(ε)S(x)) .
Since λ is stricly positive, the probability µε(B
δ(x)) goes to zero exponentially fast as
ε → 0+ for every x such that S(x) > 0, and the convergence is faster the higher the
value of S(x). If S(x) = 0 then no information about the convergence to zero or not of µε(B
δ(x)) can be deduced by the validity of a LDP; the only exception to this fact
is given by the case when x is the only point where S vanishes, which forces µε * δ x.
This is true for example in Schilder’s Theorem and Freidlin-Wentzell Theorem of Chapter 2. In the example of Chapter 3, instead, the action function vanishes on a rich set and further investigation are needed.
X S
x0
+∞
Fig.1 The typical shape of the function S. In this example, µε* δ
x0 since x0 is the only zero of S.
The following result is probably the most important theorem in Large Deviations: it allows to push forward a LDP from one metric space to another via continuous maps. Theorem 1.6 (Contraction Principle). Let λ(ε)Sµ be an action function for the family
of probabilities µε on a metric space (X , ρ
X) and let f be a continuous mapping from
(X , ρX) to another metric space (Y, ρY). Define νε := f∗(µε): then λ(ε)Sν is an action
function for νε, where Sν(y) := inf
x∈f−1(y)Sµ(x).
Proof. By Proposition 1.1 the infimum that defines Sν is actually a minimum whenever
the set f−1(y) is nonempty, hence given any s > 0 we have the equality Φν(s) = f (Φµ(s)),
where Φµ(s) := {x ∈ X : Sµ(x) ≤ s} and Φν(s) := {y ∈ Y : Sν(y) ≤ s} and compactness
of Φν(s) follows. We now prove that the Freidlin-Wentzell conditions are satisfied:
(FW1): fix δ > 0,γ > 0 and y ∈ Y. If Sν(y) = +∞ there is nothing to prove; if
not, take x ∈ f−1(y) such that Sµ(x) = Sν(y). By continuity of f there exists a
r= r(δ) such that f (BX r (x)) ⊆ B Y δ(y), hence νε(BδY(y)) ≥ µε(BX r (x)) ≥ exp(−λ(ε)[Sµ(x) + γ]) = exp(−λ(ε)[Sν(y) + γ])
1.2. First results (FW2): the preimage of the set {y ∈ Y : ρY(Φν(s), y) ≥ δ} is closed and disjoint
from Φµ(s), thus there exists a positive δ0 such that
νε{y ∈ Y : ρY(Φν(s), y) ≥ δ} ≤ µε{x ∈ X : ρX(Φµ(s), x) ≥ δ0}
≤ exp(−λ(ε)[s − γ]) for every ε sufficiently small.
If ζε is a family of random elements of X defined on the probability spaces (Ωε, Fε, Pε),
then every action function for the family of the laws of ζεis said to be an action function
for the family ζε. In the following we investigate mostly the special case (Ωε, Fε, Pε) =
(Ω, F, P) and ζε a random process with takes values in the space of continuous functions
from [0, T ] to Rd; in this setting the parameter ε is understood as the intensity of a noise
perturbing the process ζε. When the action function is defined on the space of continuous
Chapter 2
Large Deviations for SDEs
In this chapter we prove a LDP for certain families of random processes. The first example, due to Schilder ([13]), states a LDP for standard Brownian Motion in Rd, d ≥ 1
and it has been generalized to Itˆo diffusions with Lipschitz coefficients by Freidlin and Wentzell (see [9]). Further generalizations are discussed in Sections 3 and 4.
2.1
Schilder’s Theorem for Brownian Motion
Let X = C0([0, T ], Rd) be the space of continuous functions from [0, T ] to Rd vanishing
at zero, T < +∞ and d ≥ 1, endowed with the sup norm ρ(ϕ, ψ) := supt∈[0,T ]kϕt− ψtk.
A well known result is that the Borel σ-field Bρ associated with the distance ρ coincides
with the σ-field generated by the projection maps πt(ϕ) = ϕt, i.e. Bρ = σ{πt, t ≤ T }
(see for instance [14], Section 1.3). In the following we work with (C0([0, T ], Rd), Bρ) as
a measurable space. On C0([0, T ], Rd) we define the following functional:
S0T(ϕ) := 1 2 Z T 0 k ˙ϕtk2 dt
if ϕ is absoluteley continuous, and S0T(ϕ) := +∞ otherwise.
Proposition 2.1. The functional S0T is lower semicontinuous with respect to the uniform
convergence in C0([0, T ], Rd).
Proof. We use the following characterization: a function ϕ ∈ C0([0, T ], Rd) is absolutely
continuous and its derivative is square integrable if and only if sup 0≤t1≤···≤tN≤T N X i=1 ϕti − ϕti−1 2 ti− ti−1 <+∞
and in that case the supremum equals RT
0 k ˙ϕtk 2
dt. Now take ϕn → ϕ uniformly as
n → ∞: we get sup 0≤t1≤···≤tN≤T N X i=1 ϕti − ϕti−1 2 ti− ti−1 = sup 0≤t1≤···≤tN≤T lim n→∞ N X i=1 ϕn,ti− ϕn,ti−1 2 ti− ti−1 .
Chapter 2. Large Deviations for SDEs
Since the last sum is smaller thanRT
0 k ˙ϕn,tk 2
dt by the characterization above, the previ-ous equality becomes
S0T(ϕ) = 1 2 Z T 0 k ˙ϕtk 2 dt ≤ lim inf n→∞ 1 2 Z T 0 k ˙ϕn,tk 2 dt = lim inf n→∞ S0T(ϕn),
that is exactly lower semicontinuity of the functional S0T.
Proposition 2.2. The sublevels Φ(s) :=ϕ ∈ C0([0, T ], Rd) : S0T(ϕ) ≤ s are compact
for every s ≥ 0.
Proof. Closedness of Φ(s) is a consequence of Proposition 2.1, so it suffices to prove that Φ(s) is relatively compact. Take ϕ ∈ Φ(s): we have
kϕtk = Z t 0 ˙ ϕsds ≤ Z t 0 k ˙ϕsk ds ≤ s t Z t 0 k ˙ϕsk 2 ds ≤√2T s; kϕt+h− ϕtk ≤ Z t+h t k ˙ϕsk ds ≤ s h Z t+h t k ˙ϕsk2 ds ≤ √ 2hs.
Hence the family of functions Φ(s) is equibounded and equicontinuous, then relatively compact by Ascoli-Arzel`a Theorem.
Let (Ω, F, P), (Ft)t∈[0,T ] be a filtered probability space which supports a d-dimensional
standard Brownian motion W : Ω → C0([0, T ], Rd) and suppose that (Ft) is complete
and right continuous.
Lemma 2.3. Let µε be the law of the process εW on C
0([0, T ], Rd) and let νε be the
law of the process εW − ϕ on C0([0, T ], Rd), with S0T(ϕ) < +∞. Then νε is absolutely
continuous with respect to µε and the density is given by
dνε dµε(x) = exp −ε−2 Z T 0 ˙ ϕs· dxs− ε−2 2 Z T 0 k ˙ϕsk2 ds .
Proof. For every bounded measurable f : C0([0, T ], Rd) → R we have the equality
Z C0([0,T ],Rd) f(x) µε(dx) = Z C0([0,T ],Rd) f(x + ϕ) νε(dx). (2.1)
Define the random process Xt := −
Rt 0ϕ˙sd(ε
−1W
s): by Girsanov Theorem and Levy’s
characterization of Brownian motion the law of εW − [εW, X] = εW + ϕ is µε under
the probability on Ω with density with respect to P given by E(X)T, where E(X) is the
Dol´eans-Dade exponential of X. Hence for every bounded measurable f : C0([0, T ], Rd) →
R we have Z C0([0,T ],Rd) f(x) µε(dx) = Z Ω f(εW (ω) + ϕ)E(X)T(ω) P(dω). (2.2)
2.1. Schilder’s Theorem for Brownian Motion
Since the integrand in the definition of X is determinisitic the Dol´eans-Dade exponential of X is actually equal to E(X)T(ω) = exp −ε−1 Z T 0 ˙ ϕs· dWs(ω) − ε−2 2 Z T 0 k ˙ϕsk2 ds , therefore the quantity in (2.2) can be rewritten as
Z C0([0,T ],Rd) f(x + ϕ) exp −ε−2 Z T 0 ˙ ϕs· dxs− ε−2 2 Z T 0 k ˙ϕsk2 ds µε(dx), that together with the equation (2.1) gives
dνε dµε(x) = exp −ε−2 Z T 0 ˙ ϕs· dxs− ε−2 2 Z T 0 k ˙ϕsk 2 ds .
Theorem 2.4 (Schilder’s). The functional ε−2S
0T is an action functional for the family
of processes Xε= εW .
Proof. Compactness of the sublevels of S0T is given by Proposition 2.2, so we only need
to prove asymptotic estimates (FW1) and (FW2), namely:
(FW1) for any δ > 0, γ > 0 and ϕ ∈ C0([0, T ], Rd) there exists an ε0 >0 such that for
any ε ≤ ε0
P {ρ(εW, ϕ) < δ} ≥ exp −ε−2[S0T(ϕ) + γ] ;
(FW2) for any δ > 0,γ > 0 and any s > 0 there exists an ε0 > 0 such that for any
ε ≤ ε0
P {ρ(εW, Φ(s)) ≥ δ} ≤ exp(−ε−2[s − γ]).
We start from condition (FW1). Take δ > 0, γ > 0 and ϕ ∈ C0([0, T ], Rd). If S0T(ϕ) =
+∞ there is nothing to prove; if not, we are interested in P {ρ(εW, ϕ) < δ} = P {ρ(εW − ϕ, 0) < δ}
= νεx ∈ C
0([0, T ], Rd) : ρ(x, 0) < δ .
Since S0T(ϕ) < +∞ by Lemma 2.3 the quantity above is actually equal to
Z {ρ(x,0)<δ} exp −ε−2 Z T 0 ˙ ϕs· dxs− ε−2 2 Z T 0 k ˙ϕsk2 ds µε(dx).
Now observe that µεis the law of a Brownian motion multiplied by ε, therefore the
quan-tity µε{ρ(x, 0) < δ} converge to 1 as ε → 0+; in particular, we have that µε{ρ(x, 0) < δ}
is greater than 3/4 for every ε sufficiently small. By Chebyshev inequality and Itˆo isom-etry we have also the following estimate:
µε −ε−2 Z T 0 ˙ ϕs· dxs < −ε−1p8S0T(ϕ) ≤ ε−4E RT 0 ϕ˙s· dxs 2 ε−28S 0T(ϕ) = 1 4.
Chapter 2. Large Deviations for SDEs
where the expected value is taken with respect to the measure µε, so that the ε actually
simplifies in the ratio above. Now define: A:= {ρ(x, 0) < δ} ∩ −ε−2 Z T 0 ˙ ϕs· dxs≥ −ε−1p8S0T(ϕ)
and notice that by the previous remarks we have that µε(A) ≥ 1
2 for every ε sufficiently
small. This leads to the following estimate: Z {ρ(x,0)<δ} exp −ε−2 Z T 0 ˙ ϕs· dxs− ε−2 2 Z T 0 k ˙ϕsk 2 ds µε(dx) ≥ exp −ε−2S0T(ϕ) Z A exp −ε−2 Z T 0 ˙ ϕs· dxs µε(dx) ≥1 2exp −ε−2S0T(ϕ) − ε−1p8S0T(ϕ) ≥ exp −ε−2[S0T(ϕ) + γ] ,
where the last inequality holds for every ε sufficiently small: condition (FW1) is proved. We prove now condition (FW2). Denote lε the random process given by the linear
interpolation of points 0, εW∆, εW2∆, . . . , εWT, where ∆ is such that T /∆ is integer. The
event {ρ(εW, Φ(s)) ≥ δ} may occur in two ways: either ρ(εW, lε) < δ or ρ(εW, lε) ≥ δ.
In the first case necessarily S0T(lε) > s, therefore we have
P {ρ(εW, Φ(s)) ≥ δ} ≤ P {S0T(lε) > s} + P {ρ(εW, lε) ≥ δ} .
We study the two addend separately. For the first one, we have the following identity: S0T(lε) = 1 2 Z T 0 ˙l ε s 2 ds = ε 2 2 T /∆ X k=1 Wk∆− W(k−1)∆ 2 ∆ = ε2 2 dT /∆ X k=1 Zk2,
where Zk are i.i.d. standard gaussian. We recall that Zk2 has exponential moments of any
order stricly less than 1/2, then by Chebyshev inequality we get
P {S0T(lε) > s} = P 1 − γ/2s 2 dT /∆ X k=1 Zk2 > ε−2[s − γ/2] = P exp 1 − γ/2s 2 dT /∆ X k=1 Zk2 >exp ε −2 [s − γ/2] ≤ CdT /∆1−γ/2s 2 exp −ε−2[s − γ/2] ≤ 1 2exp(−ε −2 [s − γ])
for every ε sufficiently small. Consider now the quantity P {ρ(εW, lε) ≥ δ}. It is easy to
verify the inclusions of events giving the following estimate: P {ρ(εW, lε) ≥ δ} ≤ T ∆P max t∈[0,∆]kεWt− l ε tk ≥ δ ≤ T ∆P max t∈[0,∆]kεWtk ≥ δ/2 .
2.2. Freidlin-Wentzell Theorem for SDEs with regular drift
Now notice that in order to the maxt∈[0,∆]kεWtk to be greater than or equal to δ/2 at
least one of its components must be greater than or equal to δ/2d, hence P {ρ(εW, lε) ≥ δ} ≤ T d ∆P max t∈[0,∆]|εZt| ≥ δ/2d ,
where Z is an unidimensional standard Brownian motion. By the reflection princi-ple of the Brownian motion and the classical estimate on Normal cumulative function P {Z1 ≥ a} ≤ (2π)−1/2a−1exp(−a2/2) we have T d ∆P max t∈[0,∆]|εZt| ≥ δ/2d ≤ 4T d ∆ P {Z∆ ≥ δ/2εd} ≤ 4T d ∆ 1 √ 2π 2εd√∆ δ exp − δ 2 8ε2d2∆ . Now it suffices to take ∆ < δ2/8d2s and ε sufficiently small to obtain the estimate
P {ρ(εW, lε) ≥ δ} ≤ 1
2exp(−ε
−2[s − γ]),
which together with the previous one finally gives
P {ρ(εW, Φ(s)) ≥ δ} ≤ exp(−ε−2[s − γ]). This proves condition (FW2) and conclude the proof.
2.2
Freidlin-Wentzell Theorem for SDEs with
regu-lar drift
In this section we consider random perturbations of the dynamical system ˙xt = b(xt),
x0 = x ∈ Rd. We assume that the drift b : Rd→ Rdis bounded and Lipschitz continuous.
Define Xε
t to be the solution of the stochastic differential equation
Xtε= x + Z t
0
b(Xsε) ds + εWt, t ∈[0, T ] (2.3)
where W is a d-dimensional standard Brownian motion. Classical theory of stochastic differential equations ensures strong existence and uniqueness for every fixed ε > 0 (see for instance [15]) and convergence of Xε
t to the solution of the unperturbed dynamical system
xt in probability as ε → 0 (a more general setting is studied in [3]). Notice also that
weak existence and uniqueness would suffice to ask about Large Deviations for the family Xε, since only the law of Xε matters. Consider the operator B : Rd× C
0([0, T ], Rd) →
C([0, T ], Rd) that maps the couple (x, ψ) into the solution v of the integral equation vt = x +
Z t
0
b(vs) ds + ψt, t ∈[0, T ].
For every fixed x ∈ Rd the map B
x := B(x, ·) actually takes values in the space
Cx([0, T ], Rd) :=ϕ ∈ C([0, T ], Rd) : ϕ0 = x and is invertible, with
Bx−1(v)t = vt− x −
Z t
0
Chapter 2. Large Deviations for SDEs
Lemma 2.5. The operator B : Rd×C
0([0, T ], Rd) → C([0, T ], Rd) is Lipschitz continuous.
Proof. As usually, denote by ρ the sup norm on C0([0, T ], Rd) and C([0, T ], Rd). For every
t ∈[0, T ] by the triangular inequality we have
kB(x, ψ)t− B(y, ϕ)tk ≤ kB(x, ψ)t− B(y, ψ)tk + kB(y, ψ)t− B(y, ϕ)tk
Since the field b is Lipschitz continuous with Lipschitz constant equal to K we have the following estimates: kB(x, ψ)t− B(y, ψ)tk = x+ Z t 0 b(B(x, ψ)s) ds − y − Z t 0 b(B(y, ψ)s) ds ≤ kx − yk + K Z T 0 kB(x, ψ)s− B(y, ψ)sk ds; kB(y, ψ)t− B(y, ϕ)tk = Z t 0 b(B(y, ψ)s) ds + ψt− Z t 0 b(B(y, ϕ)s) ds − ϕt ≤ kψt− ϕtk + K Z T 0 kB(y, ψ)s− B(y, ϕ)sk ds,
therefore by Gr¨onwall inequality we obtain
ρ(B(x, ψ), B(y, ψ)) ≤ eKT kx − yk , ρ(B(y, ψ), B(y, ϕ)) ≤ eKTρ(ψ, ϕ).
By summing up the previous inequalities we finally get the result ρ(B(x, ψ), B(y, ϕ)) ≤ eKT(kx − yk + ρ(ψ, ϕ)) .
Theorem 2.6 (Freidlin–Wentzell). For every fixed x ∈ X the functional ε−2S
0T ◦ Bx−1 :
Cx([0, T ], Rd) → [0, +∞] is an action functional for the family of processes Xε defined in
eq. (2.3).
Proof. Since the map Bx : C0([0, T ], Rd) → Cx([0, T ], Rd) is (Lipschitz) continuous and
bijective, the result is a direct consequence of Schilder’s Theorem and the Contracion Principle.
Remark 2.1. Notice that the action functional in the theorem above can be rewritten in the more explicit form
ε−2S0T ◦ Bx−1(ϕ) = ε−2 2 Z T 0 k ˙ϕt− b(ϕt)k 2 dt
if ϕ is absoluteley continuous and ϕ0 = x, and +∞ otherwise. With this formulation it
is clear that S0T ◦ Bx−1(ϕ) = 0 if and only if ϕ is the unique solution of the unperturbed
dynamical system ˙xt= b(xt), x0 = x.
Let us now discuss peculiarities arising in considering the point x ∈ Rd, from which the
trajectory of the perturbed system is issued, as a parameter. We remark that, since the inclusion map ι : Cx([0, T ], Rd) → C([0, T ], Rd) is continuous, by Contraction Principle
2.2. Freidlin-Wentzell Theorem for SDEs with regular drift
can follow two routes: we either consider a whole family of functionals depending on the parameter x (the functional corresponding to x is equal to +∞ for all function ϕ such that ϕ0 6= x) or we introduce a new definition of the action functional suitable for the
situation: here we give the idea. Denote by Bx(εW ) the solution of eq. (2.3), to stress
also the dependence of x ∈ Rd. Hence ϕ 7→ ε−2S
0T ◦ Bϕ−10(ϕ) is an action functional for
the family (Bx(εW ))x,ε uniformly in x, in the sense that the action functional is lower
semicontinuous on C([0, T ], Rd) and:
(C) for every s ≥ 0 and A ⊆ Rdcompact, the A-sublevel Φ
A(s) := ∪x∈AΦx(s) is compact,
where Φx(s) is the set of functions ϕ such that ϕ0 = x and S0T ◦ B−1x (ϕ) ≤ s;
(FW1+) for any δ > 0,γ > 0 and s
0 >0 there is an ε0 such that for every ε ≤ ε0, x ∈ Rd
and ϕ ∈ Φx(s0)
P {ρ(Bx(εW ), ϕ) < δ} ≥ exp −ε−2[S0T(Bx−1(ϕ)) + γ] ;
(FW2+) for any δ > 0,γ > 0 and s
0 >0 there is an ε0 such that for every ε ≤ ε0, s ≤ s0
and x ∈ Rd
P {ρ(Bx(εW ), Φx(s)) ≥ δ} ≤ exp −ε−2[s − γ] .
Theorem 2.7. The map C([0, T ], Rd) 3 ϕ 7→ ε−2S
0T◦ Bϕ−10(ϕ) is an action functional for
the family (Bx(εW ))x,ε uniformly in x.
Proof. Since B : Rd× C0([0, T ], Rd) → C([0, T ], Rd) is continuous we have that for every
compact A ⊆ Rd the set B(A × Φ
0(s)) = ∪x∈AΦx(s) is compact in C([0, T ], Rd). The
semicontinuity of the action functional can be deduced by closedness of A-sublevels notic-ing that a convergnotic-ing sequence in C([0, T ], Rd) is bounded. Condition (C) is proved. Let
us now prove condition (FW1+) and (FW2+). Since B
x is Lipschitz with constant eKT
independent on x ∈ Rd we have the following set inclusions
ρ(εW, B−1
x (ϕ)) < e −KT
δ ⊆ {ρ(Bx(εW ), ϕ) < δ} ,
{ρ(Bx(εW ), Φx(s)) ≥ δ} ⊆ρ(εW, Φ(s)) ≥ e−KTδ ,
hence by the Remark 1.1 and the fact that Bx(Φ(s0)) = Φx(s0) we have that for any
δ >0,γ > 0 and s0 >0 there is an ε0 such that for every ε ≤ ε0, x ∈ Rd and ϕ ∈ Φx(s0)
P {ρ(Bx(εW ), ϕ) < δ} ≥ Pρ(εW, Bx−1(ϕ)) < e −KT
δ ≥ exp −ε−2[S0T(Bx−1(ϕ)) + γ] ,
and similarly for any δ > 0,γ > 0 and s0 >0 there is an ε0 such that for every ε ≤ ε0,
s ≤ s0 and x ∈ Rd
P {ρ(Bx(εW ), Φx(s)) ≥ δ} ≤ Pρ(εW, Φ(s)) ≥ e−KTδ ≤ exp −ε−2[s − γ] .
Even though taking the initial condition as a parameter might lead to more general results, we will not do that in the following. The result above suffices to see the point, and it is better to keep the discussion as simple as possible fixing x once and for all.
Chapter 2. Large Deviations for SDEs
2.3
Freidlin-Wentzell Theorem for bounded
contin-uous drift
The argument presented in the previous section relies on the continuity of the operator B. If the drift b is not Lipschitz continuous, then not only B might be discontinuous, but even the integral equation defining it might lack of existence or uniqueness, so that the definition of B given above does not make any sense. In the following we extend the Freidlin-Wentzell Theorem for a bounded and continuous drift b. Even though the dependence of the initial condition x is easy to treat, we prefer to assume x = 0 in the following to avoid cumbersome notation: we simply denote by S the functional on C0([0, T ], Rd) S(ϕ) = 1 2 Z T 0 k ˙ϕt− b(ϕt)k 2 dt
if ϕ is absolutely continuous, and S(ϕ) := +∞ otherwise. Quite considerably we recover the same action functional of the Lipschitz case. Our result is based on a local approxima-tion of b by bounded and Lipschitz continuous funcapproxima-tions bn, for which Freidlin-Wentzell
Theorem applies; in a second moment we transfer estimates given by (FW1) and (FW2) for the SDE driven by bn to the SDE driven by b. Let us specify what we mean by
approximation of b: we would like that the functions bn approximate b uniformly on a
sequence of sets which covers the whole of Rd. Since b is continuous, by Heine-Cantor
Theorem we can take δn >0 such that for every x, y ∈ Bn+1(0) with kx − yk < δnit holds
kb(x) − b(y)k < 1/n. Now take a smooth mollyfier ηn with support in B
δn∧1(0) and a
smooth cutoff function ψn identically equal to 1 on B
n(0) and equal to 0 out of Bn+1(0).
Define bn = (b ∗ ηn) ψn. Then bn ∈ C∞
c (Rd) and thus it is bounded and Lipschitz
con-tinuous. Moreover, kbn(x) − b(x)k < 1/n for kxk < n and kbnk
∞ < kbk∞ for every n.
We shall use these bn in the following and we denote by Sn the anologous of S for a drift
bn. Notice that by Contraction Principle every Sn has compact sublevels, so it is a good
functional for the LDP setup. Now we prove goodness of the functional S as well. Proposition 2.8. The sublevels Φ(s) :=ϕ ∈ C0([0, T ], Rd) : S(ϕ) ≤ s are compact for
every s ≥ 0.
Proof. Take ϕn → ϕ∞uniformly with respect to the sup norm on [0, T ]; since a converging
sequence is bounded, for every ϕ = ϕn or ϕ = ϕ∞, N sufficiently large and every c > 0
Young inequality implies S(ϕ) = 1 2 Z T 0 ϕ˙s− bN(ϕs) + bN(ϕs) − b(ϕs) 2 ds ≤ 1 + c 2 2 Z T 0 ϕ˙s− bN(ϕs) 2 ds+1 + c −2 2 Z T 0 bN(ϕs) − b(ϕs) 2 ds ≤ (1 + c2)SN(ϕ) + 1 + c −2 2 T N2,
and with the same calculations
SN(ϕ) ≤ (1 + c2)S(ϕ) + 1 + c
−2
2 T N2.
2.3. Freidlin-Wentzell Theorem for bounded continuous drift
Since bN is bounded and Lipschitz continuous, SN has compact sublevels, which implies
that SN is lower semicontinuous. Hence
S(ϕ∞) ≤ (1 + c2)SN(ϕ∞ ) + 1 + c −2 2 T N2 ≤ lim inf n→∞ (1 + c 2)SN(ϕn) + 1 + c −2 2 T N2 ≤ lim inf n→∞ (1 + c 2)2S(ϕn) + (2 + c2)1 + c −2 2 T N2,
and by taking the limits first for N → ∞ and then for c → 0 we obtain that S is lower semicontinuous, and therefore the sublevels Φ(s) are closed. For compactness we reason like in Proposition 2.2: kϕtk = Z t 0 ˙ ϕsds = Z t 0 ( ˙ϕs− b(ϕs) + b(ϕs)) ds ≤ Z t 0 ( ˙ϕs− b(ϕs)) ds + Z t 0 b(ϕs) ds ≤ Z t 0 k ˙ϕs− b(ϕs)k ds + Z t 0 kb(ϕs)k ds ≤√2T s +pT kbk2 ∞, kϕt+h− ϕtk ≤ √ 2hs +phkbk2 ∞.
Hence the family of functions Φ(s) is equibounded and equicontinuous, then relatively compact by Ascoli-Arzel`a Theorem.
The proposition above states the goodness of the functional S in the LDP setup. As usual, denote by Xεthe strong solution of the stochastic differential equation in dimension d ≥ 1:
dXε
t = b(Xtε)dt + εdWt, with zero initial condition (existence and uniqueness of a strong
solution for bounded and measurable drift is ensured by a result due to Veretennikov). As we mentioned before, we are interested in proving the following
Theorem 2.9. The functional ε−2S : C
0([0, T ], Rd) → [0, +∞] is an action functional
for the family of processes Xε.
We separate the proof into two parts: the lower bound (FW1) and the upper bound (FW2). Before proving Theorem 2.9 we show the following lemma, very usefull to obtain sharp exponential estimates for stochastic integrals.
Lemma 2.10 (Exponential trick). Let Y ∈ L∞(Ω, L∞
([0, T ]; Rd)) be a progressively
measurable process. Then for every c > 0 we have P Z T 0 Ys· dWs > c ≤ exp − c 2 2kY k2 ∞T .
Proof. Take λ a real number and rewrite the event we are interested in as exp λ Z T 0 Ys· dWs− λ2 2 Z T 0 kYsk2ds >exp λc −λ 2 2 Z T 0 kYsk2ds ,
Chapter 2. Large Deviations for SDEs
so that the desired probability is less or equal to the probability P exp λ Z T 0 Ys· dWs− λ2 2 Z T 0 kYsk2ds >exp λc −λ 2 2 kY k 2 ∞T .
Since λY is bounded it satisfies the Novikov’s condition, therefore the LHS of the expres-sion above is the value of a martingale at time T and thus its expected value is equal to 1 (we refer to [2], Theorem 12.2, for a proof of this fact). Markov inequality gives
P Z T 0 Ys· dWs > c ≤ exp −λc + λ 2 2 kY k 2 ∞T , and choosing λ = c/kY k2
∞T the thesis follows.
We are now ready to prove Theorem 2.9. Let us first establish the lower bound (FW1) and then the upper bound (FW2). To ease the notation, we allow some flexibility when we take γ and δ of the Freidlin-Wentzell conditions.
Proof of condition (FW1). Let Xε be as usual define Xε,n in a similar way, just taking
bn instead of b. Define the process Y
t = −ε−1
Rt
0 b(X ε
s) · dWs: since Y satifies Novikov’s
condition, by Girsanov Theorem we have that Xε has the law of a brownian motion
multiplied by ε under the probability Q, which density with respect to P is given by: dQ dP(ω) = exp −ε−1 Z T 0 b(Xsε) · dWs− ε−2 2 Z T 0 kb(Xsε)k 2 ds (ω). Using the SDE solved by Xε we can rewrite the density as
dQ dP(ω) = exp −ε−2 Z T 0 b(Xε s) · dXs+ ε−2 2 Z T 0 kb(Xε s)k 2 ds (ω),
hence given any ϕ ∈ C0([0, T ], Rd) absolutely continuous (if not, there is nothing to prove)
and δ > 0: P {ρ(Xε, ϕ) < δ} = EP1{ρ(Xε,ϕ)<δ} = EQ 1{ρ(Xε,ϕ)<δ}dP dQ = EQ h 1{ρ(Xε,ϕ)<δ}eε −2RT 0 b(X ε s)·dXs−ε−22 R0Tkb(Xsε)k 2 dsi = EP h 1{ρ(εW,ϕ)<δ}eε −1RT 0 b(εWs)·dWs− ε−2 2 RT 0 kb(εWs)k 2dsi , by the very definition of the measure Q. To lighten the notation, define
ξT = ε−1 Z T 0 b(εWs) · dWs− ε−2 2 Z T 0 kb(εWs)k2ds, ξTn = ε−1 Z T 0 bn(εWs) · dWs− ε−2 2 Z T 0 kbn(εWs)k2ds,
so that the previous formula and its analogous for Xε,n read
P {ρ(Xε, ϕ) < δ} = EP1{ρ(εW,ϕ)<δ}e ξT ,
P {ρ(Xε,n, ϕ) < δ} = EP1{ρ(εW,ϕ)<δ}e ξn
2.3. Freidlin-Wentzell Theorem for bounded continuous drift
Now consider the auxiliary events Eα := {|ξT − ξTn| < α}, defined for any α > 0, and
Fn :=supt∈[0,T ]kεWtk ≤ n . We have the following estimate:
P {ρ(Xε, ϕ) < δ} ≥ EP1{ρ(εW,ϕ)<δ}1Eαe ξT ≥ E P1{ρ(εW,ϕ)<δ}1Eαe ξn T−α = EP1{ρ(εW,ϕ)<δ}e ξn T e−α− E P1{ρ(εW,ϕ)<δ}1Eαce ξn T−α ≥ P {ρ(Xε,n, ϕ) < δ} e−α − EP1Eαce ξn T ≥ P {ρ(Xε,n, ϕ) < δ} e−α− P (Eαc) 1 2 EPe 2ξn T 1 2 ,
where the last line follows from H¨older inequality. Now take α = ε−2γ in the equation
above: the next step is to prove that, at least for n sufficiently large, the quantity P(Eεc−2γ) 1 2E Pe 2ξn T 1
2 does not affect the behaviour of the exponential for small ε, namely
it can be absorbed into the γ. Let us first provide the following bound on EPe 2ξn T 1 2: EPe 2ξn T 1 2 = EP h e2ε−1R0Tbn(εWs)·dWs−ε−2 RT 0 kbn(εWs)k2ds i12 = EP h e2ε−1 RT 0 b n(εW s)·dWs−4ε−22 RT 0 kb n(εW s)k2ds+ε−2 RT 0 kb n(εW s)k2ds i12 ≤ EP h e2ε−1R0Tbn(εWs)·dWs− 4ε−2 2 RT 0 kbn(εWs)k 2dsi12 eε−22 kbk2∞T = eε−22 kbk 2 ∞T < ∞,
uniformly in n. On the other hand, by concavity: P Eεc−2γ 12 ≤ P Eεc−2γ∩ Fn 12 + P Eεc−2γ∩ Fnc 12 . (2.4)
Using the estimate, holding in Fn,
ε−2 2 Z T 0 kb(εWs)k2− kbn(εWs)k2 ds ≤ ε −2T kbk ∞ n ,
the first term on the RHS of eq. (2.4) can be controlled by the quantity
P ( Z T 0 (b(εWs) − bn(εWs)) · dWs > ε −1γ 2 ,t∈[0,T ]sup kεWtk ≤ n )12 ,
at least for n = nγ sufficiently large. Finally, by the esponential trick we bound the
quantity above asymptotically in n by 2 exp (−ε−2γ2n2/16T ); the second term on the
RHS of eq. (2.4) can be controlled simply by P (Fc n)
1
2, which by classical estimates, similar
to those used in the proof of Schilder’s Theorem, is bounded by CT ,dexp (−ε−2n2/2d2T),
uniformly in ε ∈ (0, 1). Putting all together we obtain for n sufficiently large: P {ρ(Xε, ϕ) < δ} ≥ P {ρ(Xε,n, ϕ) < δ} e−ε −2γ − P Ec ε−2γ 12 EPe 2ξn T 1 2 ≥ P {ρ(Xε,n, ϕ) < δ} e−ε−2γ − 2 exp −ε −2γ2n2 16T + ε−2 2 kbk 2 ∞T − CT ,dexp −ε −2n2 2d2T + ε−2 2 kbk 2 ∞T .
Chapter 2. Large Deviations for SDEs
Now we fix n = nγ,ϕ large enough such that the following conditions hold:
T kbk∞ n < γ 2, γ2n2 16T − 1 2kbk 2 ∞T > S(ϕ) + 5γ, Sn(ϕ) < S(ϕ) + γ, n 2 2d2T − 1 2kbk 2 ∞T > S(ϕ) + 5γ.
Thanks to these inequalities for n = nγ,ϕ we obtain the estimate
P {ρ(Xε, ϕ) < δ} ≥ P {ρ(Xε,n, ϕ) < δ} e−ε
−2γ
− (2 + CT ,d) exp −ε−2[S(ϕ) + 5γ] ,
uniformly in ε ∈ (0, 1). Notice that in the inequality above n = nγ,ϕ is fixed. Since bn is
bounded and Lipschitz continuous condition (FW1) holds for the law of Xε,n with action
functional ε−2Sn, therefore we obtain for small ε:
P {ρ(Xε, ϕ) < δ} ≥ exp −ε−2[Sn(ϕ) + 2γ] − (2 + CT,d) exp −ε−2[S(ϕ) + 5γ]
≥ exp −ε−2[S(ϕ) + 3γ] − (2 + CT ,d) exp −ε−2[S(ϕ) + 5γ] ,
that for ε sufficiently small gives the desired condition (FW1): P {ρ(Xε, ϕ) < δ} ≥ exp −ε−2[S(ϕ) + 4γ] .
Proof of condition (FW2). Let us keep the same notation as above. Take s positive and define Φn(s) := ϕ ∈ C
0([0, T ], Rd) : Sn(ϕ) ≤ s . Given any δ > 0 by the same
calculation performed for the lower bound we obtain P {ρ(Xε,Φn(s)) ≥ δ} = EP1{ρ(εW,Φn(s))≥δ}e ξT = EP1{ρ(εW,Φn(s))≥δ}1Eαe ξT + E P1{ρ(εW,Φn(s))≥δ}1Ecαe ξT ≤ EP1{ρ(εW,Φn(s))≥δ}eξ n T eα+ E P1Ec αe ξT ≤ P {ρ(Xε,n,Φn(s)) ≥ δ} eα + P (Ec α) 1 2 EPe 2ξTn12 .
Now take α = ε−2γ. With computations similar to those performed before we get the
following inequality for n = nγ sufficiently large:
P {ρ(Xε,Φn(s)) ≥ δ} ≤ P {ρ(Xε,n,Φn(s)) ≥ δ} eε −2γ + 2 exp −ε −2γ2n2 16T + ε−2 2 kbk 2 ∞T + CT,dexp −ε −2n2 2d2T + ε−2 2 kbk 2 ∞T .
In the same spirit of the lower bound estimate, we fix n = nγ,s large enough such that
for every n > nγ,s the following conditions hold:
T kbk∞ n < γ 2, γ2n2 16T − 1 2kbk 2 ∞T > s, n2 2d2T − 1 2kbk 2 ∞T > s.
2.3. Freidlin-Wentzell Theorem for bounded continuous drift
Thanks to these inequalities we obtain the estimate for any n > nγ,s:
P {ρ(Xε,Φn(s)) ≥ δ} ≤ P {ρ(Xε,n,Φn(s)) ≥ δ} eε
−2γ
+ (2 + CT,d) exp −ε−2s ,
uniformly in ε ∈ (0, 1). Now we claim that there exists n > nγ,s such that for every
x ∈ Φn(s) there exists y ∈ Φ(s) such that ρ(x, y) < δ; if the claim is true, we have the
following set inclusion:
{ρ(Xε,Φ(s)) ≥ 2δ} ⊆ {ρ(Xε,Φn(s)) ≥ δ} ,
so that we obtain
P {ρ(Xε,Φ(s)) ≥ 2δ} ≤ P {ρ(Xε,Φn(s)) ≥ δ}
≤ P {ρ(Xε,n,Φn(s)) ≥ δ} eε−2γ+ (2 + C
T ,d) exp −ε−2s .
Since bn is bounded and Lipschitz continuous condition (FW2) holds for the law of Xε,n
with action functional ε−2Sn, therefore we obtain the following estimate for ε sufficiently
small:
P {ρ(Xε,Φ(s)) ≥ 2δ} ≤ exp −ε−2[s − 2γ] + (2 + CT,d) exp −ε−2s
≤ exp −ε−2[s − 3γ] ,
that is exactly the condition (FW2). Let us prove the claim. Suppose per absurdum that the claim is false and take a sequence of functions ϕn∈ Φn(s) such that ρ(ϕn,Φ(s)) ≥ δ
for every n > nγ,s. Consider the following estimates:
kϕn tk = Z t 0 ˙ ϕn sds = Z t 0 ˙ ϕn s − b n(ϕn s) + b n(ϕn s) ds ≤ Z t 0 ˙ ϕn s − b n(ϕn s) ds + Z t 0 bn(ϕn s) ds ≤ Z t 0 ˙ϕns − bn(ϕns) ds+ Z t 0 kbn(ϕn s)k ds ≤√2T s +pT kbk2 ∞, kϕn t+h− ϕ n tk ≤ √ 2hs +phkbk2 ∞,
which imply that ϕn converges (up to a subsequence ϕnk) uniformly to a certain function
ϕ∞ by Ascoli-Arzel`a Theorem. In particular, the sequence (ϕnk) is bounded in the sup
norm. Notice also that by continuity ρ(ϕ∞,Φ(s)) ≥ δ. Since S is lower semicontinuous
and by Young inequality we get S(ϕ∞) ≤ lim inf k→∞ S(ϕ nk) = lim inf k→∞ 1 2 Z T 0 k ˙ϕnk s − bnk(ϕnsk) + bnk(ϕnsk) − b(ϕnsk)k2ds ≤ lim inf k→∞ 1 + c2 Snk(ϕnk) + 1 + c −2 2 Z T 0 kbnk(ϕnk s ) − b(ϕnsk)k2ds ≤ lim inf k→∞ 1 + c2 Snk(ϕnk) + 1 + c −2 2 T nk2 ≤ 1 + c2 s,
so that taking c → 0 we finally obtain S(ϕ∞) ≤ s, in contradiction with the fact that
Chapter 2. Large Deviations for SDEs
2.4
Freidlin-Wentzell Theorem for unbounded drift
Go back to eq. (2.3): in the previous section we established a LDP for Xε under the
hypothesis of b being bounded and continuous. Here we try to further generalize the result in the case of unbounded drift. First of all, throughout this section we suppose that the local strong solution of eq. (2.3) does not explode before time T > 0 with probability one. Some limitations appear: in fact, we shall restrict the time interval in order to establish the lower bound given by condition (FW1), while for the upper bound stronger assumptions are needed. The idea of this section is to approximate b with a sequence of bounded continuous drifts bR given by
bR(x) = (bR 1(x), . . . b R d(x)), b R i (x) := (bi(x) ∧ R) ∨ −R, i= 1, . . . , d.
In this section S denotes the classical Freidlin-Wentzell functional and SR its analogous
for the drift bR. Notice that SR is a good functional in the LDP setup for every R > 0,
while we need a specific proof of that for S. Theorem 2.11. Assume that E −ε−1R·
0b(X ε
s)dWs is a martingale on [0, T ] for every
ε >0 and for some T < d−1/2. Then Freidlin-Wentzell condition (FW1) holds for the law
of Xε on C
0([0, T ], Rd). The rate function is given by ε−2 and the functional is given by
S.
Remark 2.2. Notice that we did not call S action functional since a priori its goodness for the LDP setup has to be proved yet. Nevertheless, checking condition (FW1) does not requires any topological property of S.
Proof. The proof essentially replicates that of Theorem 2.9. Let Xε be as usual define
Xε,R in a similar way, just taking bR instead of b. The same calculations of Theorem 2.9
give (here we need the hypothesis of E −ε−1R·
0b(X ε s)dWs being a martingale) P {ρ(Xε, ϕ) < δ} = EP1{ρ(εW,ϕ)<δ}e ξT , Pρ(Xε,R, ϕ) < δ = EP h 1{ρ(εW,ϕ)<δ}eξ R T i ,
for any given ϕ ∈ C0([0, T ], Rd) absolutely continuous and δ > 0, where ξT and ξTR are
given by ξT = ε−1 Z T 0 b(εWs) · dWs− ε−2 2 Z T 0 kb(εWs)k 2 ds, ξTR = ε−1 Z T 0 bn(εWs) · dWs− ε−2 2 Z T 0 bR(εWs) 2 ds.
Now consider the auxiliary events Eα := |ξT − ξTR| < α , defined for any α > 0, and
FR :=maxi=1,...,dsupt∈[0,T ]|εWti| ≤ R . We have the following estimate:
P {ρ(Xε, ϕ) < δ} ≥ Pρ(Xε,R, ϕ) < δ e−α− P (Eαc) 1 2 EP h e2ξRT i12 .
Now take α = ε−2γ in the equation above: as in the proof of Theorem 2.9, the next step
is to prove that for R sufficiently large, the quantity P(Ec ε−2γ) 1 2E P h e2ξR T i12
2.4. Freidlin-Wentzell Theorem for unbounded drift
the behaviour of the exponential for small ε. Here it will arise the condition on T . Usual calculations give EP h e2ξTR i12 ≤ exp ε −2R2dT 2 . On the other hand, Ec
ε−2γ ⊆ FRc, hence for R sufficiently large we have P(Eεc−2γ) 1 2 ≤
CT ,dexp (−ε−2R2/2T ), uniformly in ε ∈ (0, 1). Putting all together we obtain for R
sufficiently large: P {ρ(Xε, ϕ) < δ} ≥ Pρ(Xε,R, ϕ) < δ e−ε −2γ − CT ,dexp −ε −2R2 2T + ε−2R2dT 2 . If T < d−1/2 then the argument of the exponential above is strictly negative; now we fix
R= Rγ,ϕ large enough such that the following conditions hold:
SR(ϕ) < S(ϕ) + γ, R
2
2T − R2dT
2 > S(ϕ) + 4γ. Thanks to these inequalities for R = Rγ,ϕ we obtain the estimate
P {ρ(Xε, ϕ) < δ} ≥ Pρ(Xε,R, ϕ) < δ e−ε
−2γ
− CT,dexp −ε−2[S(ϕ) + 4γ] ,
uniformly in ε ∈ (0, 1), and since bR is bounded and continuous we obtain for small ε:
P {ρ(Xε, ϕ) < δ} ≥ exp −ε−2SR(ϕ) + 2γ − CT ,dexp −ε−2[S(ϕ) + 4γ]
≥ exp −ε−2[S(ϕ) + 3γ] .
Proposition 2.12. Suppose that the same hypotheses of Theorem 2.11 hold, and assume that Xε is a family exponentially tight with rate ε−2: then S has compact sublevels.
Proof. Closedness of Φ(s) is easy. Take ϕn → ϕ∞uniformly with respect to the sup norm
on [0, T ]; since a converging sequence is bounded, for every ϕ = ϕn or ϕ = ϕ∞ and R
sufficiently large: S(ϕ) = 1 2 Z T 0 k ˙ϕs− b(ϕs)k 2 ds = 1 2 Z T 0 ϕ˙s− bR(ϕs) 2 ds = SR(ϕ).
Since bR is bounded and continuous, SR has compact sublevels, which implies that SR is
lower semicontinuous. Hence S(ϕ∞) = SR(ϕ∞ ) ≤ lim inf n→∞ S R(ϕn) = lim inf n→∞ S(ϕ n).
This means that S is lower semicontinuous, and therefore the sublevels Φ(s) are closed. For the compactness we consider KM given by the definition of exponential tightness; if
ϕ ∈ Kc
M, then by Theorem 2.11
−S(ϕ) − γ ≤ lim inf
ε→0+ ε
2
log P {Xε ∈ Bδ(ϕ)} ≤ lim sup ε→0+
ε2log P {Xε∈ KMc } < −M, that implies S(ϕ) ≥ M by arbitrarity of γ. This means that Φ(s) ⊆ KM for every M > s:
Chapter 2. Large Deviations for SDEs
Theorem 2.13. Suppose that Xε is a family exponentially tight with rate ε−2. Then
Varadhan condition (V2) holds for the law of Xε on C
0([0, T ], Rd). The rate function is
given by ε−2 and the renormalized action functional is given by S.
Proof. By exponential tightness and Lemma 1.4 we just need to prove condition (V2 weak). With obvious notations, we have
P {Xε ∈ A} = P {Xε ∈ A, Xε∈ KM} + P {Xε∈ A, Xε ∈ KMc } .
Let BR :=ϕ ∈ C0([0, T ], Rd) : maxi=1,...,dsupt≤T |ϕit| ≤ R denote the square ball of
ra-dius R centered in the origin. Since KM is compact, for every M there exists R = RM
such that KM ⊆ BR. Therefore Xε= Xε,R on KM and we obtain
P {Xε ∈ A} ≤ PXε,R ∈ A + P {Xε ∈ KMc } .
Since A is compact we can choose R large enough such that A ⊆ BR; notice that S = SR
on BR, hence lim supε→0+ε2log P {Xε ∈ A} is less or equal to
max
lim sup
ε→0+
ε2log PXε,R ∈ A , lim sup
ε→0+ ε2log P {Xε∈ Kc M} = max − inf x∈AS R (x), −M = max − inf x∈AS(x), −M = − inf x∈AS(x),
taking M sufficiently large. Condition (V2 weak) is proved. We sum up the results of this section in the following
Theorem 2.14. Consider eq. (2.3), with b : Rd → Rd continuous. Suppose that Xε is a
family exponentially tight with rate ε−2 and that E −ε−1R·
0b(X ε
s) · dWs is a martingale
on [0, T ] for every ε > 0 and for some T < d−1/2 (the process Xε is supposed to be
defined until time T , as stated in the beginning of this section). Then ε−2S is an action
functional for the family of processes Xε on C
0([0, T ], Rd).
The hypotheses of the theorem above might seem rather artificial, but they are actually satisfied by a large number of SDEs of interest, for example by the equation studied in Chapter 3 (and in fact one could try to further generalize that example, since there the hypotheses of Theorem 2.14 hold easily). The power of Theorem 2.14 is that it is plenty of tools to check its assumptions: exponential tightness can be usually proved with compactness theorems and classical estimates, while the martingale property often comes from Novikov’s or Kazamaki’s condition.
2.5
A counterexample for the case b ∈
L
∞The next natural step in our work would be that of extending the Freidlin-Wentzell Theorem to a discontinuous drift b. The following negative example shows that this is not so straightforward. Consider indeed d = 1 and b given by
b(x) := −1, if x ≥ 0, +1, if x < 0.
2.5. A counterexample for the case b ∈L∞
The key point in this drift is that no continuous solution to ˙xt = b(xt), x0 = 0 exists.
Moreover, the infimum of S on C0([0, T ], Rd) is stricly positive, since
S(ϕ) = 1 2 Z T 0 | ˙ϕs− b(ϕs)|2ds= 1 2 Z T 0 | ˙ϕs|2− 2 ˙ϕsb(ϕs) + 1 ds = 1 2 Z T 0 | ˙ϕs|2ds+ |ϕT| + T 2 ≥ T 2,
therefore if per absurdum a Freidlin-Wentzell LDP holds for the family of processes Xε
with action functional ε−2S, by condition (V2) we would have
lim sup ε→0+ ε2log PXε ∈ C0([0, T ], Rd) ≤ − inf ϕ∈C0([0,T ],Rd) S(ϕ) = −T 2 <0, in contradiction with PXε∈ C 0([0, T ], Rd) ≡ 1 for every ε > 0.
Remark 2.3. In the previous counterexample we only proved that a Freidlin-Wentzell LDP can not hold for the family of processes Xε. We know nothing about the possibility
that others LDPs hold, with action functionals different from that of Freidlin-Wentzell. In particular, by comparison with the case b = 0, it looks reasonable that in the zero-noise limit the constant function ϕt ≡ 0 is selected, but it is not clear if this convergence is
Chapter 3
Large Deviations for Peano
Phenomenon
In this chapter we consider the following stochastic differential equation Xtε= Z t 0 b(Xε s) ds + εWt, t ∈[0, T ], (3.1) where b : Rd
→ Rd is given by b(x) := xkxkγ−1, γ ∈ (0, 1) and W is a d-dimensional
Brownian motion. Here b basically generalizes the drift b(x) = sgn(x)|x|γ to
dimen-sion greater than one. The equation above admits an unique strong solution such that P n RT 0 kX ε sk2γds < ∞ o
= 1 (see for instance [8]). The unperturbed system: xt=
Z t
0
b(xs) ds, t ∈[0, T ], (3.2)
admits an infinite family of solutions of the form xt = c (1 − γ) (t − t0)+
1−γ1
, c ∈ Sd−1, t0 ∈ [0, T ].
This lack of uniqueness is called Peano phenomenon. The solutions starting immediately from zero (namely for which t0 = 0) are called extremal and their norm is given by
x+(t) = ((1 − γ) t)1−γ1 . We study large deviations for the family Xε in C
0([0, T ], Rd),
the space of continuous functions ϕ : [0, T ] → Rd such that ϕ
0 = 0. Two LDPs hold:
a first-order LDP, with rate ε−2, according to the theory of Freidlin-Wentzell and the
generalizations of the previous chapter, which selects among all the continuous functions the solutions to eq. (3.2), and a second-order LDP, with rate ε−21−γ1+γ, which selects among
all the solutions to eq. (3.2) the extremal ones. Before diving into the proofs, let us first clarify this coexistence of LDPs with the following example. With the same notations of Chapter 1, let X := {x0, x1, x2} be a set of three points, equipped with the discrete
metric ρ. On (X , ρ) define the following family of probabilities µε, for ε ∈ (0, 1/2):
µε({x2}) = exp(−ε−2),
µε({x1}) = exp(−ε−1),
Chapter 3. Large Deviations for Peano Phenomenon
Clearly weak convergence of µε to δ
x0 occurs. Moreover, two different non trivial LDPs
hold for the family µε: indeed, at rate ε−2 one has:
lim ε→0+ε 2 log µε({x2}) = −1, lim ε→0+ε 2 log µε({x1}) = 0, lim ε→0+ε 2 log µε({x0}) = 0,
hence conditions (V1−) and (V2−) are satisfied (notice that µε is exponentially tight by
compactness of X ) and a non trivial LDP holds for µεat rate ε−2with renormalized action
function S(x2) = 1 and S(x1) = S(x0) = 0: as pointed out in Remark 1.2, no information
about the convergence µε* δ
x0 can be deduced by this LDP, being the points x1 and x0
equivalent for the action functional. However, at rate ε−1 one has instead:
lim ε→0+εlog µ ε ({x2}) = −∞, lim ε→0+εlog µ ε ({x1}) = −1, lim ε→0+εlog µ ε ({x0}) = 0,
thus a non trivial LDP holds for µεat rate ε−1 with renormalized action function ˜S(x 2) =
∞, ˜S(x1) = 1 and ˜S(x0) = 0. It is also clear that in this example the latter principle is
more accurate than the former, being able to distinguish all points from each other. A similar phenomenon arises in eq. (3.1): at a first rate, the action function vanishes on the whole set of solutions to eq. (3.2), and further selection occurs only at a slower rate. Remark 3.1. With the previous example, as much as in the following of the chapter, we show that coexistence of LDPs is possible. However, some constraints appear when two different LDPs simultaneously hold on the same space. Indeed, suppose that (λ, S) and (˜λ, ˜S) are two couples rate function - renormalized action function: from (V1), (V2) and lower semicontinuity of S follows
S(x) = − lim δ→0+lim infε→0+ λ(ε) −1 log µε(B δ(x)) = − lim δ→0+lim sup ε→0+ λ(ε)−1log µε(B δ(x)),
therefore if λ = ˜λ then necessarily S = ˜S, namely the renormalized action function is uniquely determined by the rate function. Moreover, if for example ˜λ(ε)−1λ(ε) → +∞
and S(x) > 0, then ˜ S(x) = − lim δ→0+ε→0lim+ ˜ λ(ε)−1λ(ε) lim inf ε→0+ λ(ε) −1 log µε(Bδ(x)) = +∞;
instead, in the case ˜λ(ε)−1λ(ε) → 0 and S(x) < +∞, the same calculations give ˜S(x) = 0.
3.1
First order LDP
In this section we establish a classical Freidlin-Wentzell LDP for the family Xε. We want
to apply Theorem 2.14 from the previous chapter, hence here we only need to check the validity of its hypotheses, namely the exponential tightness of the family of processes Xε
with rate ε−2 and the martingale property of the process E −ε−1R·
0b(X ε
s) · dWs.
Proposition 3.1. The family of processes Xε is exponentially tight with rate ε−2.
Proof. The proof of this fact is based on Ascoli-Arzel`a Theorem. Thanks to the inequality aγ ≤ (1 + a)γ ≤ 1 + a, holding for every a ≥ 0, we get
kXtεk ≤ Z t 0 kXsεkγds+ kεWtk ≤ Z t 0 kXsεk ds + T + sup s∈[0,T ] kεWsk.
3.1. First order LDP
By Gr¨onwall inequality there exists a constant C = CT < ∞such that
sup s∈[0,T ] kXε sk ≤ C 1 + sup s∈[0,T ] kεWsk ! , (3.3)
hence by classical estimates involving Brownian motion we obtain
P ( sup s∈[0,T ] kXε sk > M C + C ) ≤ P ( sup s∈[0,T ] kεWsk > M ) ≤ √1 2π εd√T M exp − M 2 2ε2d2T .
Define EM := ϕ ∈ C0([0, T ], Rd) : sups∈[0,T ]kϕsk ≤ M C + C . Let k · kα stand for the
α H¨older seminorm on [0, T ], α ∈ (0, 1/2). On EM the following estimate holds
kXεk α := sup 0≤s<t≤T kXε t − Xsεk (t − s)α ≤ sup 0≤s<t≤T Rt s kX ε rkγdr+ εkWt− Wsk (t − s)α ≤ sup 0≤s<t≤T (t − s)(M C + C)γ+ εkW t− Wsk (t − s)α ≤ T 1−α(M C + C)γ+ εkW k α.
Denote Rα the random variable kW kα. Classical arguments for Brownian motion give
the bound kRαkLp ≤ C
√
p for p sufficiently large, with C = Cα,d,T < ∞ (this result can
be proved, for example, mimicking the proof of Kolmogorov’s criterion of continuity for Brownian motion). This implies that there exists a constant c = cα,d,T > 0 such that
E [exp (cRα2)] < ∞, which by Markov inequality gives
P {εRα > M }= P exp cR2α > exp cM2 ε2 ≤ Eexp cR2 α exp −cM 2 ε2 . Define FM := ϕ ∈ C0([0, T ], Rd) : kϕkα ≤ M + T1−α(M C + C)γ . By Ascoli-Arzel`a
Theorem, KM := EM ∩ FM turns out to be compact in C0([0, T ], Rd), and thanks to
the calculations performed above we have lim sup
ε→0+
ε2log P {Xε 6∈ KM} ≤ lim sup ε→0+ ε2log (P {Xε 6∈ EM} + P {Xε∈ EM, Xε 6∈ FM}) = max −M 2 2T , −cM 2 → −∞ as M → ∞.
Proposition 3.2. The process E −ε−1R·
0b(X ε
s) · dWs is a martingale on [0, T ] for every
ε >0 and T < ∞.
Proof. We check that Novikov’s condition holds, namely
E exp ε −2 2 Z T 0 kXε sk2γds < ∞.
Chapter 3. Large Deviations for Peano Phenomenon
Recall that this would imply that E −ε−1R·
0b(X ε
s) · dWs
is a (uniformly integrable) martingale on [0, T ]. By eq. (3.3) and the inequality (1 + a)2γ ≤ C
γ(1 + a2γ), holding for
some Cγ < ∞and a ≥ 0, we get
E exp ε −2 2 Z T 0 kXε sk 2γds ≤ E " exp Cγ,T ε−2 2 1 + sups∈[0,T ] kεWsk2γ !!# . Since γ < 1, by reflection principle and exponential integrability of gaussian r.v. we get the desired result.
Theorem 3.3. The family of processes Xεgiven by eq. (3.1) satisfies a LDP on the space
C0([0, T ], Rd), T < d−1/2 with rate ε−2 and action functional given by
S(ϕ) = 1 2 Z T 0 k ˙ϕt− b(ϕt)k 2 dt if ϕ is absolutely continuous, and S(ϕ) := +∞ otherwise.
Proof. By Proposition 3.1 the family Xε is exponentially tight and by Proposition 3.2
the process E −ε−1R·
0b(X ε
s) · dWs is a martingale. By Theorem 2.14 the thesis follows
for every T < d−1/2.
t ϕt
S(ϕ) = 0
S(ϕ) > 0
Fig.2 In blue, some solutions of eq. (3.2) in dimension d = 1. Since the red curve is not a solution of eq. (3.2), the probability that Xε is close to this curve is exponentially small with rate ε−2.
3.2
Second order LDP
In the previous section a first order LDP has been established. Since S is equal to zero on the set of solutions to eq. (3.2), no information about the convergence of Xε beyond
rate ε−2 can be deduced by Theorem 3.3 (cfr Remark 1.2). In this section we prove a
second order LDP for the family Xε which selects among all solutions to eq. (3.2) the
extremal ones. The results in this section are basically aimed to generalize those of [10], where the one-dimensional case is considered. The approach is the following: first we show a representation formula for the density of Xε
t on Rd which allows to prove a LDP
for the density, then we pass from the density to the process itself by the argument of Theorem 3.11. We start with the following:
