Let $α\in(0,2)$ and $d\in{\mathbb N}$. Consider the following SDE in ${\mathbb R}^d$:$${\rm d}X_t=b(t,X_t){\rm d} t+a(t,X_{t-}){\rm d} L^{(α)}_t,\ \ X_0=x,$$where $L^{(α)}$ is a $d$-dimensional rotationally invariant $α$-stable process, $b:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ and $a:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ are H{ö}lder continuous functions in space, with respective order $β,γ\in (0,1)$ such that $(β\wedge γ)+α>1$, uniformly in $t$. Here $b$ may be unbounded.When $a$ is bounded and uniformly elliptic, we show that the unique solution $X_t(x)$ of the above SDE admits a continuous density, which enjoys sharp two-sided estimates. We also establish sharp upper-bound for the logarithmic derivative. In particular, we cover the whole supercritical range $α\in (0,1) $.Our proof is based on ad hoc parametrix expansions and probabilistic techniques.
We consider non degenerate Brownian SDEs with H{ö}lder continuous in space diffusion coefficient and unbounded drift with linear growth. We derive two sided bounds for the associated density and pointwise controls of its derivatives up to order two under some additional spatial H{ö}lder continuity assumptions on the drift. Importantly, the estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized, flow.
Consider the following McKean-Vlasov SDE: $$ d X_t=\sqrt{2}d W_t+\int_{{\mathbb R}^d}K(t,X_t-y)μ_{X_t}(dy)d t,\ \ X_0=x, $$ where $μ_{X_t}$ stands for the distribution of $X_t$ and $K(t,x): {\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ is a time-dependent divergence free vector field. Under the assumption $K\in L^q_t(\widetilde L_x^p)$ with $\frac dp+\frac2q<2$, where $\widetilde L^p_x$ stands for the localized $L^p$-space, we show the existence of weak solutions to the above SDE. As an application, we provide a new proof for the existence of weak solutions to 2D-Navier-Stokes equations wi…
Consider the following time-dependent stable-like operator with drift $$ \mathscr{L}_tφ(x)=\int_{\mathbb{R}^d}\big[φ(x+z)-φ(x)-z^{(α)}\cdot\nablaφ(x)\big]σ(t,x,z)ν_α(d z)+b(t,x)\cdot\nabla φ(x), $$ where $d\geq 1$, $ν_α$ is an $α$-stable type Lévy measure with $α\in(0,1]$ and $z^{(α)}=1_{α=1}1_{|z|\leq1}z$, $σ$ is a real-valued Borel function on $\mathbb{R}_+\times\mathbb{R}^d\times\mathbb{R}^d$ and $b$ is an $\mathbb{R}^d$-valued Borel function on $\mathbb{R}_+\times\mathbb{R}^d$. By using the Littlewood-Paley theory, we establish the well-posedness for the martingale problem associated with …
We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric $α$-stable Lévy processes with values in $R^d$ under some mild H{ö}lder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index $α\in (0, 1)$. New strong well-posedness results are also derived from the previous analysis.
We consider the impact of additive Gaussian white noise on a supercritical pitchfork bifurcation in an unbounded domain. As an example we focus on the stochastic Swift-Hohenberg equation with polynomial nonlinearity. Here we identify the order where small noise first impacts the bifurcation. Using an approximation via modulation equations, we provide a tool to analyse how the noise influences the dynamics close to a change of stability.
We justify the WKB analysis for the semiclassical nonlinear Schrödinger equation with a subquadratic potential. This concerns subcritical, critical, and supercritical cases as far as the geometrical optics method is concerned. In the supercritical case, this extends a previous result by E. Grenier; we also have to restrict to nonlinearities which are defocusing and cubic at the origin, but besides subquadratic potentials, we consider initial phases which may be unbounded. For this, we construct solutions for some compressible Euler equations with unbounded source term and unbounded initial vel…
We study the validity of geometric optics in $L^\infty$ for nonlinear wave equations in three space dimensions whose solutions, pulse like, focus at a point. If the amplitude of the initial data is subcritical, then no nonlinear effect occurs at leading order. If the amplitude of the initial data is sufficiently big, strong nonlinear effects occur; we study the cases where the equation is either dissipative or accretive. When the equation is dissipative, pulses are absorbed before reaching the focal point. When the equation is accretive, the family of pulses becomes unbounded.
Optimal control of stochastic Volterra integral equations with completely monotone kernels and stochastic differential equations on Hilbert spaces with unbounded control and diffusion operators
The paper investigates the existence of global attractors and their upper semicontinuity for a structural damped wave equation on $\mathbb{R}^{N}: u_{tt}-Δu+(-Δ)^αu_{t}+u_{t}+u+g(u)=f(x)$, where $α\in (1/2, 1)$ is called a dissipative index. We propose a new method based on the harmonic analysis technique and the commutator estimate to exploit the dissipative effect of the structural damping $(-Δ)^αu_{t}$ and to overcome the essential difficulty: "both the unbounded domain $\mathbb{R}^N$ and the supercritical nonlinearity cause that the Sobolev embedding loses its compactness"; Meanwhile we sh…
In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar…
We consider the Cauchy-problem for a parabolic equation of the following type: \begin{equation*} \frac{\partial u}{\partial t}= Δu+ f(u,|x|), \end{equation*} where $f=f(u,|x|)$ is supercritical. We supply this equation by the initial condition $u(x,0)=ϕ$, and we allow $ϕ$ to be either bounded or unbounded in the origin but smaller than stationary singular solutions. We discuss local existence and long time behaviour for the solutions $u(t,x;ϕ)$ for a wide class of non-homogeneous non-linearities $f$. We show that in the supercritical case, Ground States with slow decay lie on the threshold bet…
We consider continuous-time random walks on a random locally finite subset of $\mathbb{R}^d$ with random symmetric jump probability rates. The jump range can be unbounded. We assume some second--moment conditions and that the above randomness is left invariant by the action of the group $\mathbb{G}=\mathbb{R}^d$ or $\mathbb{G}=\mathbb{Z}^d$. We then add a site-exclusion interaction, thus making the particle system a simple exclusion process. We show that, for almost all environments, under diffusive space-time rescaling the system exhibits a hydrodynamic limit in path space. The hydrodynamic e…
