Related papers: On Loewner chains driven by semimartingales and co…

We give a geometric derivation of SLE($\kappa,\rho$) in terms of conformally invariant random growing subsets of polygons. We relate the parameters $\rho_j$ to the exterior angles of the polygons. We also show that SLE($\kappa,\rho$) can be…

We analyse analytic properties of nonlocal transition semigroups associated with a class of stochastic differential equations (SDEs) in $\mathbb{R}^d$ driven by pure jump--type L\'evy processes. First, we will show under which conditions…

Our aim in this paper is to establish some strong stability properties of a solution of a stochastic differential equation driven by a fractional Brownian motion for which the pathwise uniqueness holds. The results are obtained using…

Existence, uniqueness, and $L_p$-approximation results are presented for scalar stochastic differential equations (SDEs) by considering the case where, the drift coefficient has finitely many spatial discontinuities while both coefficients…

In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [arXiv:0807.1594], of the radial and chordal variant of the Loewner differential equation, which is of…

In this paper we solve a L\'evy driven linear stochastic first order partial differential equation (transport equation) understood in the canonical (Marcus) form. The solution can be obtained with the help of the method of stochastic…

We consider collections of $N$ chordal random curves obtained from a critical lattice model on a planar graph, in the limit when a fine-mesh graph approximates a simply-connected domain. We define and study candidates for such limits in…

The paper considers a coupled system of linear Boltzmann transport equations (BTE), and its Continuous Slowing Down Approximation (CSDA). This system can be used to model the relevant transport of particles used e.g. in dose calculation in…

We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear…

This paper proves conjectures originating in the physics literature regarding the intersection exponents of Brownian motion in a half-plane. For instance, suppose that B and B' are two independent planar Brownian motions started from…

We present basic properties of Dipolar SLEs, a new version of stochastic Loewner evolutions (SLE) in which the critical interfaces end randomly on an interval of the boundary of a planar domain. We present a general argument explaining why…

Loewner Theory, based on dynamical viewpoint, is a powerful tool in Complex Analysis, which plays a crucial role in such important achievements as the proof of famous Bieberbach's conjecture and well-celebrated Schramm's Stochastic Loewner…

We revisit the Bieberbach conjecture in the framework of SLE processes and, more generally, L\'evy processes. The study of their unbounded whole-plane versions leads to a discrete series of exact results for the expectations of coefficients…

Semilinear, $N-$dimensional stochastic differential equations (SDEs) driven by additive L\'evy noise are investigated. Specifically, given $\alpha\in\left(\frac{1}{2},1\right)$, the interest is on SDEs driven by $2\alpha-$stable,…

SLE($\kappa,\rho$) is a variant of the Schramm-Loewner Evolution which describes the curves which are not conformal invariant, but are self-similar due to the presence of some other preferred points on the boundary. In this paper we study…

We establish well-posedness results for multidimensional non degenerate $\alpha$-stable driven SDEs with time inhomogeneous singular drifts in $\mathbb{L}^r-{\mathbb B}_{p,q}^{-1+\gamma}$ with $\gamma<1$ and $\alpha$ in $(1,2]$, where…

We show that any stochastic differential equation (SDE) driven by Brownian motion with drift satisfying the Krylov-R\"ockner condition has exactly one solution in an ordinary sense for almost every trajectory of the Brownian motion.…

In this work, we will show the existence and uniqueness of the solution to the semi linear stochastic differential equations driven by weighted fractional Brownian motion with delay. We also prove smoothness of the density of the solution…

We consider the Hastings--Levitov HL(0) model in the small particle scaling limit and prove a large deviation principle. The rate function is given by the relative entropy of the driving measure $\rho$ for the Loewner--Kufarev equation: \[…

Consider the following stochastic differential equation (SDE) $$dX_t = b(t,X_{t-}) \, dt+ dL_t, \quad X_0 = x,$$ driven by a $d$-dimensional L\'evy process $(L_t)_{t \geq 0}$. We establish conditions on the L\'evy process and the drift…