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We consider a class of stochastic dynamical systems, called piecewise deterministic Markov processes, with states $(x, \s)\in \O\times \G$, $\O$ being a region in $\bbR^d$ or the $d$--dimensional torus, $\G$ being a finite set. The…
A new class of fractional-order stochastic evolution equations of the form $(\partial_t + A)^\gamma X(t) = \dot{W}^Q(t)$, $t\in[0,T]$, $\gamma \in (0,\infty)$, is introduced, where $-A$ generates a $C_0$-semigroup on a separable Hilbert…
We prove that the mild solution to a semilinear stochastic evolution equation on a Hilbert space, driven by either a square integrable martingale or a Poisson random measure, is (jointly) continuous, in a suitable topology, with respect to…
We consider the evolution of a quantum simple harmonic oscillator in a general Gaussian state under simultaneous time-continuous weak position and momentum measurements. We deduce the stochastic evolution equations for position and momentum…
The planar symmetric Markov random flight $\bold X(t), \; t>0,$ is represented by the stochastic motion of a particle moving with constant finite speed $c>0$ in the Euclidean plane $\Bbb R^2$ and taking on its initial and each new…
This paper investigates a non-autonomous slow-fast system, which is generalized by stochastic differential equations (SDEs) with locally Lipschitz coefficients, subjected to standard Brownian motion (Bm) and fractional Brownian motion (fBm)…
We solve the time-dependent Schr\"odinger equation by learning the score function, the gradient of the log-probability density, on Bohmian trajectories. In Bohm's formulation of quantum mechanics, particles follow deterministic paths under…
Stochastic homogenization is achieved for a class of elliptic and parabolic equations describing the lifetime, in large domains, of stationary diffusion processes in random environment which are small, statistically isotropic perturbations…
We consider a stochastic version of the Wilson-Cowan model which accommodates for discrete populations of excitatory and inhibitory neurons. The model assumes a finite carrying capacity with the two populations being constant in size. The…
In a series of recent papers and in a book, this author put forward a mathematical model capable of embracing the search for extra-terrestrial intelligence (SETI), Darwinian Evolution and Human History into a single, unified statistical…
We study the cohomological equation $Xu=f$ for smooth locally Hamiltonian flows on compact surfaces. The main novelty of the proposed approach is that it is used to study the regularity of the solution $u$ when the flow has saddle loops,…
In this paper we consider the Stochastic isothermal, nonlinear, incompressible bipolar viscous fluids driven by a genuine cylindrical fractional Bronwnian motion with Hurst parameter $H \in (1/4,1/2)$ under Dirichlet boundary condition on…
We construct a Brownian motion on complex partial flag manifolds with blocks of equal size as a matrix-valued diffusion from a Brownian motion on the unitary group. This construction leads to an explicit expression for the characteristic…
A large class of Jordan curves on the Riemann sphere can be encoded by circle homeomorphisms via conformal welding, among which we consider the welding homeomorphism of the random SLE loops and the Weil-Petersson class of quasicircles. It…
Recently, it has been shown in [Jentzen, A., M\"uller-Gronbach, T., and Yaroslavtseva, L., Commun. Math. Sci., 14, 2016] that there exists a system of autonomous stochastic differential equations (SDE) on the time interval $[0,T]$ with…
Standard stochastic Loewner evolution (SLE) is driven by a continuous Brownian motion, which then produces a continuous fractal trace. If jumps are added to the driving function, the trace branches. We consider a generalized SLE driven by a…
Continuing the research initiated in \cite{Fr-Ki2}, we study the existence of solutions and their regularity for the cohomological equations $X u=f$ for locally Hamiltonian flows (determined by the vector field $X$) on a compact surface $M$…
In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify these as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general…
The signature of a sample path is a formal series of iterated integrals along the path. The expected signature of a stochastic process gives a summary of the process that is especially useful for studying stochastic differential equations…
In this work, closure of the Boltzmann--BGK moment hierarchy is accomplished via projection of the distribution function $f$ onto a space $\mathbb{H}^{N}$ spanned by $N$-order Hermite polynomials. While successive order approximations…