Regularizing Properties for Transition Semigroups and Semilinear Parabolic Equations in Banach Spaces

Stochastic Optimal Control for the Stochastic Heat Equation with Exponentially Growing Coefficients and with Control and Noise on a Subdomain

“…Applying the results in Section 4, we can also solve the associated Hamilton Jacobi Bellman equation. Hamilton Jacobi Bellman equations in Banach spaces can be also solved by a fixed point argument, as in [18]. Unlike in the present article, in [18], the transition semigroup related to Equation (2.2) is asked to be regularizing in a suitable sense.…”

“…Hamilton Jacobi Bellman equations in Banach spaces can be also solved by a fixed point argument, as in [18]. Unlike in the present article, in [18], the transition semigroup related to Equation (2.2) is asked to be regularizing in a suitable sense. The transition semigroup related to a linear heat equation with noise on a subdomain does not possess such regularizing properties.…”

“…In [12] and [17] existence and uniqueness of a mild solution (see, e.g., [7,9] and references therein for the definition of mild solution) is proved, via the BSDE approach, for a Hamilton Jacobi Bellman equation respectively on a infinite dimensional Hilbert space and on a Banach space. In [18], Hamilton Jacobi Bellman equations on a Banach space are studied, with Hamiltonian and final datum only lipschitz continuous, but with some regularizing properties for the transition semigroup (see also Section 4 for further discussions). Also viscosity solutions of partial differential equations of parabolic type on an infinite dimensional Hilbert space have been studied by many authors, we refer to [4] as a general reference.…”

“…The transition semigroup related to a linear heat equation with noise on a subdomain does not possess such regularizing properties. Indeed, see, for example, [7], Section 9.4.1, [16], Section 3.1 and [18], Section 2, in the case of the heat equation with noise on a subdomain, such regularizing properties are related to null controllability of a linear heat equation with control on a subinterval and to the behaviour of the minimal energy steering to 0 an initial condition. Deterministic linear heat equations with control on a subinterval are null controllable, see, for example, [14] and [22], but the minimal energy blows up too fast, see, for example, [10] and [21].…”

“…Deterministic linear heat equations with control on a subinterval are null controllable, see, for example, [14] and [22], but the minimal energy blows up too fast, see, for example, [10] and [21]. So with the techniques of [18], the Hamilton Jacobi Bellman equation related to the stochastic heat equation with control and noise on a subdomain can not be solved; in the present article, see Theorem 5.5 below, existence and uniqueness of a mild solution of this Hamilton Jacobi Bellman equation are stated.…”