Abstract. In this paper we study the parabolic Anderson equation ∂u(x, t)/∂t = κΔu(x, t) + ξ(x, t)u(x, t), x ∈ Z d, t ≥ 0, where the u-field and the ξ -field are R-valued, κ ∈ [0, ∞) is the diffusion constant, and Δ is the discrete Laplacian. The ξ -field plays the role of a dynamic random environment that drives the equation. The initial condition u(x, 0) = u 0 (x), x ∈ Z d , is taken to be nonnegative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2dκ, split into two at rate ξ ∨ 0, and die at rate (−ξ) ∨ 0. Our goal is to prove a number of basic properties of the solution u under assumptions on ξ that are as weak as possible. These properties will serve as a jump board for later refinements.Throughout the paper we assume that ξ is stationary and ergodic under translations in space and time, is not constant and satisfies E(|ξ(0, 0)|) < ∞, where E denotes expectation w.r.t. ξ . Under a mild assumption on the tails of the distribution of ξ , we show that the solution to the parabolic Anderson equation exists and is unique for all κ ∈ [0, ∞). Our main object of interest is the quenched Lyapunov exponent λ 0 (κ) = lim t→∞ 1 t log u(0, t). It was shown in Gärtner, den Hollander and Maillard (In Probability in Complex Physical Systems. In Honour of Erwin Bolthausen and Jürgen Gärtner (2012) 159-193 Springer) that this exponent exists and is constant ξ -a.s., satisfies λ 0 (0) = E(ξ(0, 0)) and λ 0 (κ) > E(ξ(0, 0)) for κ ∈ (0, ∞), and is such that κ → λ 0 (κ) is globally Lipschitz on (0, ∞) outside any neighborhood of 0 where it is finite. Under certain weak space-time mixing assumptions on ξ , we show the following properties: (1) λ 0 (κ) does not depend on the initial condition u 0 ; (2) Bolthausen and Jürgen Gärtner (2012) 159-193 Springer) properties (1), (2) and (4) were not addressed, while property (3) was shown under much more restrictive assumptions on ξ .) Finally, we prove that our weak space-time mixing conditions on ξ are satisfied for several classes of interacting particle systems. (4) n'ont pas été abordées, tandis que la propriété (3) a été prouvée sous des hypothèses beaucoup plus restrictives sur ξ .) Finalement, nous prouvons que nos conditions faibles de mélange en espace-temps sur ξ sont satisfaites par plusieurs systèmes de particules en interaction.MSC: Primary 60H25; 82C44; secondary 60F10; 35B40