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Let be a one dimensional positive recurrent diffusion continuously observed on [0,] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when → ∞. The main point of our work is that we do not suppose the process to be in stationary...

Let be a one-dimensional positive recurrent diffusion with initial distribution and invariant probability . Suppose that for some >1, ∈ℝ such that ∀∈ℝ, and , where is the hitting time of . For such a diffusion, we derive non-asymptotic deviation bounds of the form ℙ(|(1/)0 ( ) d−()|≥)≤()(1/ /2)(1/ )(). Here bounded or bounded and compactly supported and ()=‖‖∞ when is bounded and ()=(||) when is bounded and...

Let be a regular continuous positively recurrent Markov process with state space ℝ, scale function and speed measure . For ∈ℝ denote +=sup≥ (], +∞[)(()−()), −=sup≤ (]−∞; [)(()−()). It is well known that the finiteness of ± is equivalent to the existence of spectral gaps of generators associated with . We show how these quantities appear independently in the study of the exponential moments of hitting times of . Then...

Let $X$ be a $\mu$-symmetric Hunt process on a LCCB space $𝙴$. For an open set $𝙶\subseteq 𝙴$, let ${\tau }_{𝙶}$ be the exit time of $X$ from $𝙶$ and $A^{𝙶}$ be the generator of the process killed when it leaves $𝙶$. Let $r:[0,\infty [\to [0,\infty [$ and $R\left(t\right)={\int }_0^{t}r\left(s\right)\mathrm{d}s$. We give necessary and sufficient conditions for ${𝔼}_{\mu }R\left({\tau }_{𝙶}\right)lt;\infty$ in terms of the behavior near the origin of the spectral measure of $-A^{𝙶}$. When $r\left(t\right)=t^l$, $l\ge 0$, by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order $l+1$ for ${\tau }_{𝙶}$ implies the...

Let be a one dimensional positive recurrent diffusion continuously observed on [0,] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when → ∞. The main point of our work is that we do not suppose the process to be in stationary...