We consider diffusion processes in media with pockets of large diffusivity. The asymptotic behavior of such processes is described when the diffusion coefficients in the pockets tend to infinity. The limiting process is identified as a diffusion on the space where each of the pockets is treated as a single point, and certain conditions on the behavior of the process on the boundary of the pockets are imposed. Calculation of various probabilities and expectations related to the limiting process leads to new initial-boundary (and boundary) problems for the corresponding parabolic (and elliptic) PDEs.Our goal is to show that X x,ε t converge, on an appropriate state space, to a limiting family of processes Y x t when ε ↓ 0. First, let us give an intuitive description of the behavior of X x,ε t for small ε. For a small δ > 0, letAssume that x ∈ U, in which case X x,ε t moves as a Brownian motion until it reaches ∂D k for some k. Once the process reaches ∂D k , it will reach D −δ k very soon, and will then move very fast in the interior of D k due to the large parameter ε −1 at L. Next, we need to understand how the process exits D +δ k (if ε is small, the process will reach ∂D k and then go back to D −δ k many times prior to exiting D +δ k ). We will argue that the distribution of the exit point from D +δ k is nearly uniform with respect to the (d − 1)-dimensional volume measure (i.e., the measure corresponding to the volume form on ∂D +δ k ) if δ and ε are small (a small δ is fixed first, and then ε is taken to zero). Thus, if we disregard the time spent inside D k , the process gets almost immediately re-distributed along ∂D k (or rather ∂D +δ k ) upon reaching ∂D k . Processes with somewhat similar behavior were considered in [6]. Instead of a selfadjoint operator multiplied by a large parameter, which amounts to fast motion inside D k , the re-distribution along the boundary in [6] was due to a trapping mechanism: the motion inside D k was assumed to be nearly deterministic with a large vector field pointing inside the domain. The exit times were exponentially long as ε ↓ 0, and the exit was due to large deviations, while now the exit times will tend to zero.Due to fast mixing inside D k , it is impossible to distinguish between different points of D k without time re-scaling when studying the limiting behavior of X x,ε t . Let U ′ be the metric space obtained from U by identifying all points of ∂D k , turning every ∂D k , k = 1, ..., n, into one point d k . The family of limiting processes Y x t , x ∈ U ′ , will be defined in terms of its generator. Since we expect Y x t to coincide with a Wiener process inside U, the generator coincides with 1 2 ∆ on a certain class of functions. The domain of the generator of the limiting process, however, should be restricted by certain boundary conditions to account for non-trivial behavior of Y x t on the boundary of U and for the delay at the points d k .The generator of the limiting process will be carefully defined in the Section 2, where we also formulate the main resul...