This is an advanced text on ordinary differential equations (ODEs) in Banach and more
general locally convex spaces, most notably ODEs on measures and various function …

Stochastic gradient descent (SGD) has been widely used in machine learning due to its
computational efficiency and favorable generalization properties. Recently, it has been …

Consider the following stochastic differential equation (SDE) d X t= b (t, X t−) d t+ d L t, X 0=
x, driven by a d-dimensional Lévy process (L t) t≥ 0. We establish conditions on the Lévy …

Let α∈(0, 2) and d∈ N. Consider the following stochastic differential equation (SDE) in Rd:
dXt= b (t, Xt) dt+ a (t, Xt−) dL(α) t, X0= x, where L (α) is a d-dimensional rotationally invariant …

We consider the stochastic differential equation dXt= b (Xt) dt+ dLt, where the drift b is a
generalized function and L is a symmetric one dimensional α-stable Lévy processes, α∈(1 …

We study the strong rate of convergence of the Euler--Maruyama scheme for a
multidimensional stochastic differential equation (SDE) $$ dX_t= b (X_t) dt+ dL_t, $$ with …

Heat Kernel of Anisotropic Nonlocal Operators Page 1 Documenta Math. 1 Heat Kernel of
Anisotropic Nonlocal Operators Krzysztof Bogdan, Pawe l Sztonyk, and Victoria Knopova …

We construct the fundamental solution (the heat kernel) p κ to the equation∂ t= L κ, where
under certain assumptions the operator L κ takes one of the following forms, L κ f (x):=∫ R d …

In this paper, we study the following time-dependent stochastic differential equation (SDE) in
$\mathbb {R}^ d $:\begin {equation*}\mathrm {d} X_ {t}=\sigma (t, X_ {t-})\mathrm {d} Z_t+ b (t …

A stable-like process is a Feller process $(X_t) _ {t\geq 0} $ taking values in $\mathbb {R}^ d
$ and whose generator behaves, locally, like an $\alpha $-stable L\'evy process, but the …