Related papers: A note on stochastic integrals as $L^2$-curves
This paper begins by giving an historical context to fractional Brownian Motion and its development. Section 2 then introduces the fractional calculus, from the Riemann-Liouville perspective. In Section 3, we introduce Brownian motion and…
We show that an interesting class of functionals of stochastic differential equations can be approximated by a Chen-Fliess series of iterated stochastic integrals and give a L^{2} error estimate, thus generalizing the standard stochastic…
In the theory of differential geometry curves, a curve is said to be of constant-ratio if the ratio of the length of the tangential and normal components of its position vector function is constant. In this paper, we study and characterize…
We consider additive functionals of stationary Markov processes and show that under Kipnis-Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Levy area that can be…
This paper initiates a study of Hodge integrals on moduli spaces of pseudostable curves. We prove an explicit comparison formula that allows one to effectively compute any pseudostable Hodge integral in terms of intersection numbers on…
We introduce a general approach to contour integrals. It covers usual Abelian integrals, the higher order Melnikov integrals and the generalized Abelian integrals. We prove that the generating function always satisfies a linear differential…
In this paper we consider Lipschitz graphs of functions which are stationary points of strictly polyconvex energies. Such graphs can be thought as integral currents, resp. varifolds, which are stationary for some elliptic integrands. The…
We give a natural cohomological interpretation of Letzter-Makar-Limanov invariants for rings of differential operators on algebraic curves.
The goal of this paper is to encode equivalently the fractional Lagrange dynamics as a nonholonomic almost Kahler geometry. We use the fractional Caputo derivative generalized for nontrivial nonlinear connections (N-connections) originally…
In this paper, for general curves $(t,\gamma(t))$ satisfying some suitable curvature conditions, we obtain some $L^p(\mathbb{R})\times L^q(\mathbb{R}) \rightarrow L^r(\mathbb{R})$ estimates for the bilinear fractional integrals…
This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. The main result is to derive stochastic governing equations for such systems from a critical point of a stochastic action.…
We study the continuity property of multiple Q-adapted quantum stochastic integrals with respect to noncommuting integrands given by the non-adapted multiple integral kernels in Fock scale. The noncommutative algebra of relatively…
Using the spectral theory of weakly convergent sequences of finite graphs, we prove the uniform existence of the integrated density of states for a large class of infinite graphs.
This is the author's 2008 thesis from the University of Chicago. We generalize the notion of the Clifford index to an arbitrary very ample line bundle and show how it determines when a curve and its various secant varieties have…
In this work, we introduce a theory of stochastic integration with respect to symmetric $\alpha$-stable cylindrical L\'evy processes. Since $\alpha$-stable cylindrical L\'evy processes do not enjoy a semi-martingale decomposition, our…
The purpose of this note is to prove that there is an algebraic stack U parameterizing all curves. The curves that appear in the algebraic stack U are allowed to be arbitrarily singular, non-reduced, disconnected, and reducible. We also…
The article is devoted to the expansions of iterated Stratonovich stochastic integrals on the basis of the method of generalized multiple Fourier series that converge in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k\in\mathbb{N}.$…
We to define a Path Integral in Lorentzian time by restricting the relevant domain of integration on $C([0,1],M)$ over a Riemannian configuration manifold $(M,g)$ and considering the dynamics of a particle evolving between to fixed…
A theory of integration for anticommuting paths is described. This is combined with standard It\^o calculus to give a geometric theory of Brownian paths on curved supermanifolds. (Invited lecture given at meeting on `Espaces de Lacets',…
The paper considers the integration theory for $G$-L\'evy processes with finite activity. We introduce the It\^o-L\'evy integrals, give the It\^o formula for them and establish SDE's, BSDE's and decoupled FBSDE's driven by $G$-L\'evy…