A Semi-Linear Backward Parabolic cauchy Problem with Unbounded Coefficients of Hamilton-Jacobi-Bellman Type and Applications to optimal control

We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton-Jacobi-Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the Value Function of the controlled equation and that the feedback law is verified

A nonlinear Bismut-Elworthy formula for HJB equations with quadratic Hamiltonian in Banach spaces

We consider a Backward Stochastic Differential Equation (BSDE for short) in a Markovian framework for the pair of processes $(Y,Z)$, with generator with quadratic growth with respect to $Z$. The forward equation is an evolution equation in an abstract Banach space. We prove an analogue of the Bismut-Elworty formula when the diffusion operator has a pseudo-inverse not necessarily bounded and when the generator has quadratic growth with respect to $Z$. In particular, our model covers the case of the heat equation in space dimension greater than or equal to 2. We apply these results to solve semilinear Kolmogorov equations for the unknown $v$, with nonlinear term with quadratic growth with respect to $\nabla v$ and final condition only bounded and continuous, and to solve stochastic optimal control problems with quadratic growth

Invariant measures for systems of Kolmogorov equations

In this paper we provide sufficient conditions which guarantee the existence of a system of invariant measures for semigroups associated to systems of parabolic differential equations with unbounded coefficients. We prove that these measures are absolutely continuous with respect to the Lebesgue measure and study some of their main properties. Finally, we show that they characterize the asymptotic behaviour of the semigroup at infinity

Characterizations of Sobolev spaces on sublevel sets in abstract Wiener spaces

In this paper we consider an abstract Wiener space $(X,\gamma,H)$ and an open subset $O\subseteq X$ which satisfies suitable assumptions. For every $p\in(1,+\infty)$ we define the Sobolev space $W_{0}^{1,p}(O,\gamma)$ as the closure of Lipschitz continuous functions which support with positive distance from $\partial O$ with respect to the natural Sobolev norm, and we show that under the assumptions on $O$ the space $W_{0}^{1,p}(O,\gamma)$ can be characterized as the space of functions in $W^{1,p}(O,\gamma)$ which have null trace at the boundary $\partial O$, or, equivalently, as the space of functions defined on $O$ whose trivial extension belongs to $W^{1,p}(X,\gamma)$

ON WEAKLY COUPLED SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS WITH DIFFERENT DIFFUSION TERMS

We prove maximal Schauder regularity for solutions to elliptic systems and Cauchy problems, in the space Cb(Rd;Rm) of bounded and continuous functions, associated to a class of nonautonomous weakly coupled secondorder elliptic operators A, with possibly unbounded coefficients and diffusion and drift terms which vary from equation to equation. We also provide estimates of the spatial derivatives up to the third-order and continuity properties both of the evolution operator G(t, s) associated to the Cauchy problem Dtu = A(t)u in Cb(Rd;Rm), and, for fixed t, of the semigroup Tt(τ) associated to the autonomous Cauchy problem Dτu = A(t)u in Cb(Rd;Rm). These results allow us to deal with elliptic problems whose coefficients also depend on time

Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems

We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over ℝd{\mathbb{R}^{d}} and in Lp{L^{p}}-spaces with respect to tight evolution systems of measures. Here, the linear part of the equation is a nonautonomous second-order elliptic operator with unbounded coefficients defined in I×ℝd{I\times\mathbb{R}^{d}}, (I being a right-halfline). To the above Cauchy problem we associate a nonlinear evolution operator, which we study in detail, proving some summability improving properties. We also study the stability of the null solution to the Cauchy problem