arXiv research

Study improves understanding of solutions to complex equations in geometry.

In this paper, we bring in General Landau-Lifshitz-Bloch equation and prove that it admits a local strong solution.

Study on interest rate model with jumps, proving strong convergence in simulations.

In this note, we reveal that our solution of Demailly's strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Kollár and Jonsson-Mustată implies the truth of twisted versions of the strong openness conjecture; our optimal $L^{2}$ extension implies Berndtsson…

New varifold solutions for mean curvature flow converge and are unique.

The paper studies curves in Riemannian manifolds using total variation flow.

Existence of strong randomized equilibria in mean-field games with common noise.

Novel weak solutions for volume-preserving mean curvature flow established.

Strong geodesic convex function and strong monotone vector field of order $m$ on Riemannian manifolds have been established. A characterization of strong geodesic convex function of order $m$ for the continuously differentiable functions has been discussed. The relation between the solution of a new variational inequal…

Study on G2 structures with torsion and existence of solutions.

The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for $C^0$ spacelike hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped product splittin…

Proves strong solutions for graphical Brakke flows with $L^2$ normal velocity.

The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.

We extend Bony's propagation of support argument \cite{Bony} to $C^1$ solutions of the non-homogeneous sub-elliptic $p-$Laplacian associated to a system of smooth vector fields satisfying Hörmander's finite rank condition. As a consequence we prove a strong maximum principle and strong comparison principle that general…

Study compares nodal sets of solutions to the Allen-Cahn equation.

Study on continuity of solutions for complex Monge-Ampère equations with movable singularities.

Study proves stability of big bang singularity in complex system.

Study proves stability and uniqueness for a specific type of flow.

Study develops numerical schemes for non-Markovian volatility models with memory.

Global solutions found for a wave-Klein-Gordon system with strong couplings in divergence form.

The study proves strong cosmic censorship violation for spherically symmetric dust clouds.

Study on non-negative solutions for stochastic Volterra equations with jumps.

In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional $κ$-solutions. In this paper, we present an alternative proof for this fact and show that compact $κ$-solutions are rotational symmetric. Our proof arose from independent work relating to our Strong Stability Theorem for singular…

The paper is concerned with the problem of existence of solutions for the Heath-Jarrow-Morton equation with linear volatility. Necessary conditions and sufficient conditions for the existence of weak solutions and strong solutions are provided. It is shown that the key role is played by the logarithmic growth condition…

In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let $g(t)$ be a smooth complete solution to the Ricci flow on $\mathbb{R}^{3}$, with the canonical Euclidean metric $E$ as initial data, then $g(t)$ is trivial, i.e. $g(t)\equiv E$.

We study two-dimensional stochastic differential equations (SDEs) of McKean--Vlasov type in which the conditional distribution of the second component of the solution given the first enters the equation for the first component of the solution. Such SDEs arise when one tries to invert the Markovian projection developed …

Paper shows how solutions to Allen-Cahn converge to multiphase mean curvature flow.

Study of stochastic differential equations on non-compact manifolds, solving open problem on strong completeness.

This paper considers binomial approximation of continuous time stochastic processes. It is shown that, under some mild integrability conditions, a process can be approximated in mean square sense and in other strong metrics by binomial processes, i.e., by processes with fixed size binary increments at sampling points. …

We generalize the Zermelo navigation problem and its solution on Riemannian manifolds $(M, h)$ admitting a space dependence of a ship's speed $0<|u(x)|_h\leq1$ in the presence of a perturbation $\tilde{W}$ determined by a strong velocity vector field satisfying $|\tilde{W}(x)|_h=|u(x)|_h$, with application of Finsler m…

The paper explores strong G2-structures with torsion and their geometric properties.

A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is closed. We classify the invariant SKT structures on four-dimensional solvable Lie groups. The classification includes solutions on groups that do not admit compact four-dimensional quotients. It also shows that there are solvable g…

Study bounds derivatives of solutions to a specific equation on domains.

We apply results of Malliavin-Thalmaier-Watanabe for strong and weak Taylor expansions of solutions of perturbed stochastic differential equations (SDEs). In particular, we work out weight expressions for the Taylor coefficients of the expansion. The results are applied to LIBOR market models in order to deal with the …

We study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is $C^{1,1}$. We obtain as a consequence a Liouville theorem for entire solutio…

Using the classical approach we show the existence of disc type solutions to the asymptotic Plateau problem in certain Hadamard manifolds which may have arbitrarily strong curvature and volume growth.

This paper shows how to learn variational inequalities fast with strong monotonicity.

We study convergence properties of the full truncation Euler scheme for the Cox-Ingersoll-Ross process in the regime where the boundary point zero is inaccessible. Under some conditions on the model parameters (precisely, when the Feller ratio is greater than three), we establish the strong order 1/2 convergence in $L^…

A new adaptive splitting method improves accuracy for Cox-Ingersoll-Ross model.

Optimal dividend strategy with ratcheting and capital injection under Cramér-Lundberg model.

The variational calculus for the Faddeev-Hopf model on a general Riemannian domain, with general Kaehler target space, is studied in the strong coupling limit. In this limit, the model has key similarities with pure Yang-Mills theory, namely conformal invariance in dimension 4 and an infinite dimensional symmetry group…

We prove existence, regularity and a Feynman-Kač representation formula of the strong solution to the free boundary problem arising in the financial problem of the pricing of the American Asian option with arithmetic average.

Any surface can be foliated into equipotential hypersurfaces of the level sets. A current result is that the contours are the progressing wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, as well as with a corresponding eikonal equation. The lev…

We construct solutions to the constraint equations in general relativity using the limit equation criterion introduced by Dahl, Humbert and the first author. We focus on solutions over compact 3-manifolds admitting a $\bS^1$-symmetry group. When the quotient manifold has genus greater than 2, we obtain strong far from …

The paper solves the Cauchy problem for Friedrichs systems on specific spacetime manifolds.

We consider geometric flows of hypersurfaces expanding by a function of the extrinsic curvature and we show that the homothethic sphere is the unique solution of the flow which converges to a point at the initial time. The result does not require assumptions on the speed other than positivity and monotonicity and it is…

The paper classifies solutions to semilinear equations on curved spaces.

The paper studies CR Yamabe solutions on Sasakian manifolds with nonnegative curvature.