We consider a class of space-time coupled evolution equations (CEEs), obtained by a subordination of the heat operator. Our CEEs reformulate and extend known governing equations of non-Markovian processes arising as scaling limits of continuous time random walks, with widespread applications. In particular we allow for initial conditions imposed on the past, general spatial operators on Euclidean domains and a forcing term. We prove existence, uniqueness and stochastic representation for solutions.

We look at estimates for the Green's function of time-fractional evolution equations of the form $D^{\nu}_{0+*} u = Lu$, where $D^{\nu}_{0+*}$ is a Caputo-type time-fractional derivative, depending on a L\'evy kernel $\nu$ with variable coefficients, which is comparable to $y^{-1-\beta}$ for $\beta \in (0, 1)$, and $L$ is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green's function of $D^{\beta}_0 u = Lu$ in the case that $L$ is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green's function of $D^{\beta}_0 u=\Psi(-i\nabla)u$ where $\Psi$ is a pseudo-differential operator with constant coefficients that is homogeneous of order $\alpha$. Thirdly, we obtain local two-sided estimates for the Green's function of $D^{\beta}_0 u = Lu$ where $L$ is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green's functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green's functions associated with $L$ and $\Psi$, as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form $D^{(\nu, t)}_0 u = Lu$, where $D^{(\nu, t)}$ is a Caputo-type operator with variable coefficients.

In this article we obtain two-sided estimates for the Greens function of fractional boundary value problems on $\mathbb R_+ \times \mathbb R_+ \times \mathbb R^d$ of the form \[(-{}_{t_1}D^\beta_{0+*} - {}_{t_2}D^\gamma_{0+*})u(t_1, t_2, x) = L_{x}u(t_1, t_2, x),\] with some prescribed boundary functions on the boundaries $\{0\} \times \mathbb R_+ \times \mathbb R^d$ and $\mathbb R_+ \times\{0\}\times \mathbb R^d$. The operators ${}_{t_1}D^\beta$ and ${}_{t_1}D^\gamma$ are Caputo fractional derivatives of order $\beta, \gamma \in (0, 1)$ and $L_{x}$ is the generator of a diffusion semigroup: $L_x= \nabla \cdot(a(x) \nabla)$ for some nice function $a(x)$. The Greens function of such boundary value problems are decomposed into its components along each boundary, giving rise to a natural extension to the case involving $k \geq 2$ number of fractional derivatives on the left hand side.

Sebastian Andres (University of Cambridge) Green kernel asymptotics for two-dimensional random walks under random conductances The random conductance model is a well-established model for a random walk in random environment. In recent years the behaviour of the associated heat kernel and Green function has been intensively studied, and in dimension d ≥ 3 the asymptotics of the Green kernel are meanwhile quite well-understood. In this talk we present precise asymptotics of the potential kernel and the Green function of the walk killed upon exiting balls in dimension d = 2. This result holds, for instance, in the case of strictly elliptic conductances, random walks on supercritical percolation clusters or ergodic degenerate conductances satisfying a moment condition. This talk is based on a joint work with Jean-Dominique Deuschel and Martin Slowik (TU Berlin). George Andriopoulos (University of Warwick) Invariance principles for random walks in random environment on trees Consider the nearest-neighbor random walk in random environment (RWRE) on a (locally nite) rooted ordered tree. For a xed environment, it is a crucial fact that this model is reversible, and therefore it can be described as an electical network with conductances that are given in terms of the potential of the RWRE. In Sinai's model, in which the potential converges to a Brownian motion, the study of the potential is of particular importance since it determines the behavior of the walk. Under an assumption that is reminiscent to Sinai's regime, we suppose that the collection of rooted plane trees equipped with the unique invariant measure of the RWRE, and its potential, converges with respect to the spatial Gromov-Hausdor -vague topology, and moreover that a certain condition for the non-explosion of the resistances, rst introduced by Croydon (2017), is satis ed. Proving that these two conditions are valid, and using recent results of Croydon's on the convergence of processes associated with resistance forms, we are able to deduce scaling limits for the RWRE in various settings. Our rst application gives us as a corollary Seignourel's result on the convergence of a random walk on a random environment with di usive time scaling to the Brox di usion. Our second example includes a scaling limit for the biased random walk on the range of large critical branching random walk in high dimensions. Noam Berger (Technical University of Munich) A probabilistic approach to quantitative homogenization In this talk I'll present an approach for quantitative homogenization which is based on direct random walk calculations. This approach yields results in balanced (or non-divergence form) cases, often without ellipticity assumptions. Based on joint work with D. Criens and J.-D. Deuschel