On mild solutions of gradient systems in Hilbert spaces

We consider the Cauchy problem for an infinite-dimensional Ornstein-Uhlenbeck equation perturbed by gradient of a potential. We prove some results on existence and uniqueness of mild solutions of the problem. We also provide stochastic representation of mild solutions in terms of linear backward stochastic differential equations determined by the Ornstein-Uhlenbeck operator and the potential.     

Lp solutions of reflected BSDEs under monotonicity condition

We prove existence and uniqueness of ${\mathbb{L}}^p$ solutions, $p\in [1,2]$, of reflected backward stochastic differential equations with $p$-integrable data and generators satisfying the monotonicity condition. We also show that the solution may be approximated by the penalization method. Our results are new even in the classical case $p=2$.     

Stochastic Representation of Weak Solutions of Viscous Conservation Laws: A BSDE Approach

We consider the Cauchy problem for systems of viscous conservation laws. We obtain three different but related stochastic representations of weak solutions of the problem: in terms of solutions to systems of usual backward stochastic differential equations, in terms of solutions to some stochastic backward systems, and in terms of solutions to some forward-backward stochastic differential equations.     

On Dirichlet Processes Associated with Second Order Divergence Form Operators

Let be a d - dimensional Markov family corresponding to a uniformly elliptic second order divergence form operator. We show that for any quasi continuous in the Sobolev space the process (X) admits under P ^x a decomposition into a martingale additive functional (AF) M and a continuous AF A of zero quadratic variation for almost every starting point x if q=2, for quasi every x if q>2 and for every if is continuous, d=1 and or d>1 and q>d. Our decomposition enables us to show that in the case of symmetric operator the energy of A equals zero if q=2 and that the decomposition of (X) into the martingale AF M and the AF of zero energy A is strict if for some q>d. Moreover, our decomposition provides a probabilistic representation of A .     

Large Time Behavior of Solutions to Parabolic Equations with Dirichlet Operators and Nonlinear Dependence on Measure Data

Potential Analysis - We study large time behavior of renormalized solutions of the Cauchy problem for equations of the form ?tu ? Lu + λu = f(x, u) + g(x, u) ? μ, where...     

On the structure of diffuse measures for parabolic capacities

Let $Q=(0,T)\times \mathrm{\Omega }$, where Ω is a bounded open subset of ${\mathbb{R}}^d$. We consider the parabolic p-capacity on Q naturally associated with the usual p-Laplacian. Droniou, Porretta, and Prignet have shown that if a bounded Radon measure μ on Q is diffuse, i.e. charges no set of zero p-capacity, $p>1$, then it is of the form $\mu =f+\text{div}(G)+g_t$ for some $f\in L^1(Q)$, $G\in {(L^{p^{\prime }}(Q))}^d$ and $g\in L^p(0,T;W_0^{1,p}(\mathrm{\Omega })\cap L^2(\mathrm{\Omega }))$. We show the converse of this result: if $p>1$, then each bounded Radon measure μ on Q admitting such a decomposition is diffuse.