Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise

Abstract

We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L2(G), where G is an open bounded domain in $\mathbb {R}^{d}$ ℝ d with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

On pointwise Malliavin differentiability of solutions to semilinear parabolic SPDEs

“…admits a unique mild solution u λ ∈ L p (Ω; C([0, T ]; C(G))) for every p > 0, because u 0 ∈ C(G) is non-random. Therefore, as proved in [8], u λ (t, x) ∈ D 1,p for every (t, x) ∈ [0, T ] × G and every p > 0, with Du λ ∈ L ∞ (0, T ; L p (Ω; L ∞ (G; H))), and…”

“…It was shown in [8] that u(t, x) ∈ D 1,p loc for every (t, x) ∈ [0, T ] × G, and that, for every x ∈ G, Du(•, x) = Du n (•, x) on [[0, T n ]] for every n ∈ N, where is defined T n is the first time t when u(t) C(G) reaches n, and u n is the unique mild solution in L p (Ω; C([0, T ]; C(G))) to the equation…”

“…Our goal is to fill at least in part the gap between the results of [8] and those of [7], showing that u(t, x) belongs to D 1,p for all p 1. To this purpose, we develop two different approaches: one uses stochastic calculus in vector-valued L q spaces and monotonicity, and another one a comparison principle for mild solutions to stochastic evolution equations.…”

Absolute continuity of the solution to stochastic generalized Burgers-Huxley equation

“…The stochastic calculus of variations (or Malliavin calculus, [29]) has been used extensively to study the existence of density for solutions of stochastic partial differential equations (SPDEs), see for example [8,10,20,21,24,25,32,35], etc. The authors in [32] established the absolute continuity of the solution of a parabolic SPDE, where the drift and diffusion functions are assumed to be measurable and locally bounded with locally bounded derivatives.…”

“…For a class of SPDEs like the stochastic wave and heat equation with multiplicative noise, Lipschitz coefficients, the authors in [35] proved the existence of density of the law of the solution in some Besov space. Absolute continuity of the law of solution to a parabolic dissipative SPDE of reaction-diffusion type perturbed by multiplicative Wiener noise in an open bounded domain in R d with smooth boundary is established in [24]. The existence of densities for stochastic differential equations (SDEs) perturbed by a stable-like Lévy process under some non-degeneracy condition with Hölder continuous coefficients in some Besov spaces has been discussed in [7].…”