Exit-Problem for a Class of Non-Markov Processes With Path Dependency: Proofs of Auxiliary Lemmas

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Abstract and 1 Introduction

1.1 State of the art

1.2 Some remarks on dynamics and initial condition

1.3 Outline of the paper

1.4 List of notations

2 Large Deviation Principle

2.1 Establishing the LDP for the SID

2.2 Results related to the LDP

2.3 Compactness results

3 Exit-time

3.1 Auxiliary results

3.2 Proof of the main theorem

3.3 Proofs of auxiliary lemmas

4 Generalization and References

3.3 Proofs of auxiliary lemmas

3.3.1 Proof of Lemma 3.1: Initial descent to the point of attraction

3.3.2 Proof of Lemma 3.2: Convergence of deterministic process towards a

Then, by Grönwall’s inequality:

That guarantees exponentially fast convergence towards 0:

.3.3.3 Proof of Lemma 3.3: Attraction of stochastic process towards a

And this concludes the proof of the lemma.

3.3.4 Proof of Lemma 3.4: Behaviour in the annulus between Bρ(a) and ∂G

3.3.5 Proof of Lemma 3.5: Stabilization of the occupation measure

Proof. The proof of the following lemma follow the same logic as the one of Lemma 3.3.

We have to show now that

This completes the proof.

Now we are ready to prove Lemma 3.5 itself.

That gives us:

We recall that Lemma 3.9 establishes the following asymptotic behaviour for small σ:

Let us first deal with the term B. By (3.22), we have

Consider now A. Separate each probability inside the sum the following way

3.3.6 Proof of Lemma 3.6: Exit before nearing a

First, we show that it is possible to establish a lower bound for

By Lemma 2.5,

which proves the lemma by taking ρ → 0.

3.3.7 Proof of Lemma 3.7: Control of dynamics for small time intervals

Use the Jensen’s inequality, integrate the second part of the last integral over s, and get:

We apply Grönwall’s inequality and get:

Thus, it follows that, for the absolute value of the Brownian motion itself,

Taking into account these remarks, we can now use the Schilder theorem [DZ10, Lemma 5.2.1] that provides the LDP for the path of the Brownian motion. Hence, for some fixed ϵ and 0 < T < 1:

where d is the dimension of the space.

Note that for any ϵ > 0 the following limit holds:

3.3.8 Proof of Lemma 3.8: Uniform lower bound for probability of exit from G