相关论文: A parabolic PDE-based approach to Borell--Brascamp…

Using purely probabilistic methods, we prove the existence and the uniqueness of solutions fora system of coupled forward-backward stochastic differential equations (FBSDEs) with measurable, possibly discontinuous coefficients. As a…

The Brascamp-Lieb inequality in harmonic analysis was proved by Brascamp and Lieb in the rank one case in 1976, and by Lieb in 1990. It says that in a certain inequality, the optimal constant can be determined by checking the inequality for…

We prove the Lp,q-solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the…

We show that the Harnack inequality for a class of degenerate parabolic quasilinear PDE $$\p_t u=-X_i^* A_i(x,t,u,Xu)+ B(x,t,u,Xu),$$ associated to a system of Lipschitz continuous vector fields $X=(X_1,...,X_m)$ in in $\Om\times (0,T)$…

We extend the results of the FBSDE theory in order to construct a probabilistic representation of a viscosity solution to the Cauchy problem for a system of quasilinear parabolic equations. We derive a BSDE associated with a class of…

We prove new versions of the isomorphic Busemann-Petty problem for two different measures and show how these results can be used to recover slicing and distance inequalities. We also prove a sharp upper estimate for the outer volume ratio…

The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage…

We prove Aleksandrov-Bakelman-Pucci estimates and Harnack inequalities for viscosity solutions of a class of degenerate fully nonlinear pseudo-$p$-Laplacian equations in nondivergence form. Our main approach is an adaptation of the sliding…

We consider degenerate fully nonlinear parabolic equations, which generalize the p-parabolic equation with $p>2$ to nondivergence form operators. We prove an intrinsic Harnack inequality for nonnegative solutions and a weak Harnack…

Consider the Leibenson equation \begin{equation*} \partial_t u = \Delta_p u^q, \end{equation*} where $\Delta_p f = div(|\nabla f|^{p-2}\nabla f)$ for $p>1$ and $q>0$, which is a simultaneous generalization of the porous media and the…

We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influence of a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as an energy-amplifying supercritical…

We prove both necessary and sufficient conditions for $L^p$-bound\-ed\-ness of certain multilinear generalized Radon transforms that arise as Heisenberg group analogues of the Brascamp--Lieb inequalities on Euclidean space. The necessary…

We develop a relativistic lattice Boltzmann (LB) model, providing a more accurate description of dissipative phenomena in relativistic hydrodynamics than previously available with existing LB schemes. The procedure applies to the…

We establish a stable form of the general Euclidean Brascamp-Lieb inequality in all cases in which the Lebesgue exponents are strictly between 1 and 2, asserting that all near-extremizers are nearly Gaussian.

This paper deals with the problem of existence and uniqueness of a solution for a backward stochastic differential equation (BSDE for short) with one reflecting barrier in the case when the terminal value, the generator and the obstacle…

The dimension-free Harnack inequality and uniform heat kernel upper/lower bounds are derived for a class of infinite-dimensional GEM processes, which was introduced in \cite{FW} to simulate the two-parameter GEM distributions. In…

The maximal inequalities for diffusion processes have drawn increasing attention in recent years. However, the existing proof of the $L^p$ maximum inequalities for the Ornstein-Uhlenbeck process was dubious. Here we give a rigorous proof of…

We discuss interplays between log-concave functions and log-concave sequences. We prove a Bernstein-type theorem, which characterizes the Laplace transform of log-concave measures on the half-line in terms of log-concavity of the…

Following a recently introduced approach to approximate Lie symmetries of differential equations which is consistent with the principles of perturbative analysis of differential equations containing small terms, we analyze the case of…

We establish Zaremba problem for Laplacian and $p$-Laplacian with degenerate weights when the Dirichlet condition is only imposed in a set of positive weighted capacity. We prove weighted Sobolev-Poincar\'{e} inequality with sharp…