We consider a real-valued diffusion process with a linear jump term driven by a Poisson point process and we assume that the jump amplitudes have a centered density with finite moments. We show upper and lower estimates for the density of the solutio
n in the case that the jump amplitudes follow a Gaussian or Laplacian law. The proof of the lower bound uses a general expression for the density of the solution in terms of the convolution of the density of the continuous part and the jump amplitude density. The upper bound uses an upper tail estimate in terms of the jump amplitude distribution and techniques of the Malliavin calculus in order to bound the density by the tails of the solution. We also extend the lower bounds to the multidimensional case.
We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for $d=1$ and $d=2$. We consider a class of jump diffusion processes whose invariant density belongs to some Holder spac
e. Firstly, in dimension one, we show that the kernel density estimator achieves the convergence rate $frac{1}{T}$, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in [Amorino, Gloter (2021)], which depends on the Blumenthal-Getoor index for $d=1$ and is equal to $frac{log T}{T}$ for $d=2$. Secondly, we show that is not possible to find an estimator with faster rates of estimation. Indeed, we get some lower bounds with the same rates ${frac{1}{T},frac{log T}{T}}$ in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case.
The purpose of this paper is extend recent results of Bonder-Groisman and Foondun-Nualart to the stochastic wave equation. In particular, a suitable integrability condition for non-existence of global solutions is derived.
We study the following equation begin{equation*} frac{partial u(t,,x)}{partial t}= Delta u(t,,x)+b(u(t,,x))+sigma dot{W}(t,,x),quad t>0, end{equation*} where $sigma$ is a positive constant and $dot{W}$ is a space-time white noise. The initial conditi
on $u(0,x)=u_0(x)$ is assumed to be a nonnegative and continuous function. We first study the problem on $[0,,1]$ with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together with a theorem of Bonder and Groisman, our first result shows that the solution blows up in finite time if and only if begin{equation*} int_{cdot}^inftyfrac{1}{b(s)},d s<infty, end{equation*} which is the well-known Osgood condition. We also consider the same equation on the whole line and show that the above condition is sufficient for the nonexistence of global solutions. Various other extensions are provided; we look at equations with fractional Laplacian and spatial colored noise in $mathbb{R}^d$.
Consider the following stochastic heat equation, begin{align*} frac{partial u_t(x)}{partial t}=- u(-Delta)^{alpha/2} u_t(x)+sigma(u_t(x))dot{F}(t,,x), quad t>0, ; x in R^d. end{align*} Here $- u(-Delta)^{alpha/2}$ is the fractional Laplacian with $ u
>0$ and $alpha in (0,2]$, $sigma: Rrightarrow R$ is a globally Lipschitz function, and $dot{F}(t,,x)$ is a Gaussian noise which is white in time and colored in space. Under some suitable additional conditions, we prove a strong comparison theorem and explore the effect of the initial data on the spatial asymptotic properties of the solution. This constitutes an important extension over a series of recent works.
In this paper, we obtain upper and lower bounds for the moments of the solution to a class of fractional stochastic heat equations on the ball driven by a Gaussian noise which is white in time, and with a spatial correlation in space of Riesz kernel
type. We also consider the space-time white noise case on an interval.
We consider a stochastic differential equation with additive fractional noise with Hurst parameter $H>1/2$, and a non-linear drift depending on an unknown parameter. We show the Local Asymptotic Normality property (LAN) of this parametric model with
rate $sqrt{tau}$ as $taurightarrow infty$, when the solution is observed continuously on the time interval $[0,tau]$. The proof uses ergodic properties of the equation and a Girsanov-type transform. We analyse the particular case of the fractional Ornstein-Uhlenbeck process and show that the Maximum Likelihood Estimator is asymptotically efficient in the sense of the Minimax Theorem.
In this paper, we consider a multidimensional ergodic diffusion with jumps driven by a Brownian motion and a Poisson random measure associated with a pure-jump Levy process with finite Levy measure, whose drift coefficient depends on an unknown param
eter. Considering the process discretely observed at high frequency, we derive the local asymptotic normality (LAN) property.
Consider the following equation $$partial_t u_t(x)=frac{1}{2}partial _{xx}u_t(x)+lambda sigma(u_t(x))dot{W}(t,,x)$$ on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentiall
y fast if $lambda$ is large enough. But if $lambda$ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what $lambda$ is. We also provide various extensions.
In this paper we prove that the spatially homogeneous Landau equation for Maxwellian molecules can be represented through the product of two elementary processes. The first one is the Brownian motion on the group of rotations. The second one is, cond
itionally on the first one, a Gaussian process. Using this representation, we establish sharp multi-scale upper and lower bounds for the transition density of the Landau equation, the multi-scale structure depending on the shape of the support of the initial condition.
