The paper explores symmetry in solutions of semilinear PDEs on Riemannian domains.
problem Symmetry phenomena in solutions of semilinear PDEs on Riemannian domains.
method General framework for formulating the symmetry problem; evidence from stable solutions; consideration of manifolds with density.
result Evidence that the framework is natural, with results for stable solutions.
Paper introduces a new method to solve complex PDEs efficiently.
problem Solving high-dimensional semilinear PDEs and BSDEs.
method Decomposes PDEs into linear and nonlinear parts, uses Deep BSDE solver with control variate method.
result Errors of the new method are much smaller than those of the original Deep BSDE solver.
Deep neural nets solve complex insurance math equations.
problem Optimal control problems in insurance math.
method Deep neural network algorithm for elliptic PDEs.
result Solves high-dimensional semilinear elliptic PDEs.
The aim of this paper is to suggest a new viewpoint to study qualitative properties of solutions of semilinear elliptic PDE's defined outside a compact set. The relevant tools come from spectral theory and from a combination of stochastic properties of the relevant differential operators. Possible links between spectra…
We extend the viscosity solution characterization proved in [5] for call/put American option prices to the case of a general payoff function in a multi-dimensional setting: the price satisfies a semilinear re-action/diffusion type equation. Based on this, we propose two new numerical schemes inspired by the branching p…
We study the semilinear partial differential equation (PDE) associated with the non-linear BSDE characterizing buyer's and seller's XVA in a framework that allows for asymmetries in funding, repo and collateral rates, as well as for early contract termination due to counterparty credit risk. We show the existence of a …
New learning scheme solves high-dimensional semi-linear PDEs using sparse grids and Picard approximations.
problem Solving high-dimensional semi-linear parabolic PDEs.
method Probabilistic learning scheme based on Picard iteration with SGD, employing sparse grid approximation.
result Convergence proof and polynomial complexity in ε−1 for high-dimensional PDEs. New approach simplifies proof of wave equations on black holes.
problem Global existence and decay for semilinear wave equations on extremal Reissner-Nordström black holes.
method Develops a new approach based on weaker estimates, avoiding near-horizon sharp estimates.
result Simpler and more streamlined proof without requiring near-horizon sharp estimates.
We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Ito and point processes, such PDEs are shown to provide a natural description for the solution of hedging and valuatio…
We study conditions for existence, uniqueness and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al (2011). These equations take the form of semilinear PDEs and Forward-Backward Stochastic Differential Equations (FBSDEs). After summarizing the cash flows definitions all…
Paper tackles DOCTR-L with SciPhy RL, solving neural PDEs from data.
problem High-dimensional optimal control with stochastic policies.
method Soft HJB equation, Neural PDEs, Physics-Informed Neural Networks.
result Reduces DOCTR-L to solving neural PDEs from data.
We take the holistic approach of computing an OTC claim value that incorporates credit and funding liquidity risks and their interplays, instead of forcing individual price adjustments: CVA, DVA, FVA, KVA. The resulting nonlinear mathematical problem features semilinear PDEs and FBSDEs. We show that for the benchmark v…
Study phase transitions with prescribed mean curvature in Riemannian manifolds.
problem Understanding phase transitions with prescribed mean curvature in geometric settings.
method Analyzing solutions to inhomogeneous semilinear elliptic PDEs, establishing bounds and asymptotics.
result Established upper and lower bounds for eigenvalues of phase transition problems.
Extends XVA valuation under stochastic volatility, characterizing value processes via mild solutions.
problem Valuation of contingent claims in presence of default, collateral, and funding under stochastic volatility.
method Characterizes pre-default value processes via mild solutions to parabolic semilinear PDEs under stochastic volatility.
result Characterizes pre-default value processes via mild solutions to parabolic semilinear PDEs under stochastic volatility, providing sufficient conditions for existence and uniqueness.
The article derives gradient estimations for semilinear equations on geometric flows.
problem Gradient estimation for semilinear equations on geometric flows.
method Derives both Hamilton and Souplet-Zhang type gradient estimations.
result Gradient estimations for semilinear equations on geometric flows.
Researchers solve an inverse problem for a semilinear elliptic equation on complex manifolds.
problem Determining an unknown function in a semilinear elliptic equation on complex manifolds.
method Analyzing higher order linearizations and interactions of Gaussian quasimode solutions.
result An unknown smooth function can be uniquely determined from the Dirichlet-to-Neumann map.
NO approximates non-Markovian BSDEs with polynomial scaling in 1/ε.
problem Complexity of NO approximations for structured families of BSDEs.
method Identifying structured families of non-Markovian BSDEs, informing NO's inductive bias.
result Polynomial scaling in 1/ε for NO approximations of BSDE solution operators.
Derives matrix Harnack inequalities for semilinear heat equations on manifolds.
problem Bounding solutions of semilinear heat equations on manifolds with geometric constraints.
method Applies Li-Yau estimates to derive Harnack inequalities for positive solutions.
result Derives matrix Harnack inequalities for positive solutions of semilinear heat equations.
The paper proves Liouville theorems and nonexistence results for semilinear equations on pseudo-Hermitian manifolds.
problem Analyzing semilinear elliptic equations and inequalities on pseudo-Hermitian manifolds.
method Using a generalized Jerison-Lee's formula and volume estimates.
result Established Liouville theorems and nonexistence results for specific equations and inequalities.
Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.
problem Proving a Liouville theorem for a generalized elliptic equation on H-type groups.
method Proof based on an a priori integral estimate and a generalized differential identity.
result Obtained a Liouville type theorem for the semilinear subcritical elliptic equation on H-type groups.
The paper classifies solutions to semilinear equations on curved spaces.
problem Classifying solutions to semilinear equations on manifolds with nonnegative Ricci curvature.
method Proving classification results for subcritical and critical semilinear elliptic equations.
result Strong rigidity results for nontrivial solutions in the critical case.
Study on blow-up solutions for semilinear wave equations on specific manifolds.
problem Investigate blow-up and lifespan estimates for semilinear wave equations on asymptotically Euclidean manifolds.
method Use of exponential perturbation metric and construction of entire solutions for a related equation.
result Sharp upper bound estimates for the lifespan of solutions.
On an affine flat manifold with coordinates x^j and convex local potential function f, we call the affine Kahler metric f_{ij} dx^i dx^j semi-flat Calabi-Yau if it satisfies det f_{ij} = 1. Recently Gross-Wilson have constructed many such metrics on S^2 minus 24 singularities, as degenerate limits of Calabi-Yau metrics…
Derives Li & Yau estimates for heat equations on manifolds.
problem Analyzing positive solutions of semilinear heat equations on manifolds.
method Adapts Li & Yau estimates to derive new inequalities.
result Derives Harnack inequality and discusses monotonicity, convexity, decay estimates.
Trivial solution proof for heat equation on certain manifolds.
problem Proving trivial solutions for semilinear heat equations on specific manifolds.
method Analyzing pointwise monotonicity and boundedness over time.
result Trivial solutions exist only for certain values of p.
Sharp conditions found for solving heat equation on Riemannian manifolds.
problem Solving semilinear heat equation on Riemannian manifolds.
method Sharp conditions derived for local-in-time solvability.
result Sharp conditions on solvability given for complete and connected manifolds.
New Kelvin transform for anisotropic elliptic problems.
problem Semilinear and quasilinear anisotropic elliptic problems.
method Introducing a new Kelvin-type transform in the anisotropic setting.
result New insights into anisotropic elliptic problems.
Paper proves no nontrivial solutions to certain elliptic equations on graphs.
problem Proving nonexistence of solutions to semilinear elliptic equations on metric graphs.
method Constructed a modified distance function and introduced test functions to show nonexistence under volume growth conditions.
result No nontrivial solutions exist for the equations under suitable conditions.
Develops comparison methods for semilinear elliptic problems on Riemannian manifolds with Ricci lower bound.
problem Analyzing positive solutions of semilinear Dirichlet problems on Riemannian manifolds with Ricci lower bound.
method Sharp pointwise gradient comparison method, derived from admissibility and structural conditions on f.
result Explicit isoperimetric-type inequality and quantitative hot-spot localization estimate.
We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.
Classifies self-similar solutions for heat equations with positive speed.
problem Classifying self-similar solutions for semilinear heat equations.
method Analyzes the semilinear heat equation ut=Δu+∣u∣p−1u for p>1. result Finite time blowing up solutions converge to a positive constant after rescaling.
Measuring wave sources uniquely identifies manifold properties.
problem Determining Riemannian manifold structure from wave observations.
method Semilinear wave equation measurements at a single point.
result Topological, differential, and geometric structure can be inferred.
The paper classifies solutions to a specific elliptic equation in the Heisenberg group.
problem Classifying positive solutions to a critical semilinear elliptic equation in the Heisenberg group.
method Proof based on Jerison-Lee's differential identity and pointwise/integral estimates.
result The solutions are the Jerison-Lee's bubbles in the Heisenberg group.
Nonexistence results for semilinear parabolic and hyperbolic inequalities on metric graphs
problem Nonexistence of solutions to semilinear parabolic and hyperbolic inequalities on metric graphs
method Construction of a new pseudo-metric and space-time test functions
result All solutions must be identically zero
We consider the Dirichlet problem for semilinear elliptic equations on a bounded domain which is diffeomorphic to a ball and investigate bifurcation from a given (trivial) branch of solutions, where the radius of the ball serves as bifurcation parameter. Our methods are based on well known results from variational bifu…
We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian manifolds. As a byproduct of the proofs we also obtain some rigidity, or partial symmetry…
Study semilinear equations on weighted manifolds to prove rigidity.
problem Prove rigidity of weighted manifolds via classification of semilinear equations.
method Classify positive solutions at the Sobolev-critical exponent, proving rigidity and weight triviality.
result Existence of positive solutions implies rigidity and weight triviality under certain curvature conditions.
We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on no-arbitrage arguments, we derive backward stochastic differential equations (BSDEs) associated with the replicating portfolios of long an…
Solves a 30-year-old problem on singular solutions for Yamabe equations.
problem Constructing singular solutions to semilinear equations without phase-plane analysis.
method Careful gluing in weighted L∞ spaces, exploiting semilinearity and stability of linearized operator. result Provides an alternative construction for the Yamabe problem with maximal singular set.
Assume that f(s)=F′(s) where F is a double-well potential. Under certain conditions on the Lipschitz constant of f on [−1,1], we prove that arbitrary bounded global solutions of the semilinear equation Δu=f(u) on hyperbolic space $\HH^n$ must reduce to functions of one variable provided they admit asympto…
The paper extends Liouville theorems to sub-Riemannian manifolds.
problem Generalizing Liouville theorems to sub-Riemannian manifolds.
method Constructing 'good' cut-off functions and applying a nonnegative generalized curvature-dimension inequality.
result The Liouville theorems are extended to sub-Riemannian manifolds.
We consider a semilinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity which is related to a stochastic control problem with fuel constraint. The fuel constraint translates into a singular initial condition for the HJB equation. We first propose a transformation based on a change of vari…
Study on ground states of semilinear elliptic equations with various potential wells.
problem Characterizing ground states of semilinear elliptic equations with arbitrary potential wells.
method Analyzing solutions in convex domains and manifolds with non-negative Ricci curvature, using Morse theory and min-max methods.
result Ground states are mountain-pass type with Morse index 1 in convex domains and manifolds with non-negative Ricci curvature.
We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not…
Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth (n−2)-dimensional subm…
In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the γk functions on the space of its hermitian metrics.
The classical Pohozaev identity constrains potential solutions of certain semilinear PDE boundary value problems. The Kazdan-Warner identity is a similar necessary condition important for the Nirenberg problem of conformally prescribing scalar curvature on the sphere. For dimensions n≥3 both identities are captur…
Study on price formation in financial markets with a single default event.
problem Equilibrium price formation in financial markets with a single default risk.
method Characterized optimal strategies using quadratic-growth BSDEs, derived market-clearing condition, and established mean-field BSDE solvability.
result Characterized equilibrium risk premium and its dependence on default risk factors.