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Isaac Ohavi

Publications and source records attributed to Isaac Ohavi.

10 recordsLinked to original sources
arXiv2026

Strong comparison principle for fully nonlinear partial differential equations with Hamiltonians discontinuous in all variables

We establish a strong comparison principle for viscosity solutions of fully nonlinear elliptic equations driven by second-order Hamiltonians that may be discontinuous with respect to all variables. The analysis relies only on a controlled superlinear growth in the gradient variable together with a structural monotonicity in the unknown, and requires neither ellipticity nor continuity of the coefficients. The proof is based on the construction of tailored viscosity test functions obtained as solutions of auxiliary eikonal-type equations. These functions compensate for the lack of regularity and allow for a fully local comparison argument despite the complete discontinuity of the Hamiltonian. This yields a robust comparison principle in open subsets of $\mathbb{R}^N$. As a consequence, we prove the existence of continuous viscosity solutions via a Perron's method adapted to discontinuous frameworks, combined with Ishii's semicontinuous envelope technique. The class of Hamiltonians covered includes linear and quasilinear equations with merely Borel measurable coefficients, as well as Hamilton-Jacobi-Bellman equations arising in stochastic control and differential games.

math.AP↗
arXiv2025

Viscosity solutions posed on star-shaped network with Kirchhoff's boundary condition: Well-posedness

The aim of this work is to establish the well-posedness of fully nonlinear partial differential equations (PDE) posed on a star-shaped network, having nonlinear Kirchhoff's boundary condition at the vertex, and possibly degenerate. We obtain a comparison theorem, for discontinuous viscosity solutions, following the recent ideas obtained by Ohavi for second order problems, building test functions at the vertex solutions of Eikonal equations with well-designed coefficients. Another strong result obtained in this contribution is to show that any generalized Kirchhoff's viscosity solution introduced by Lions-Souganidis, is indeed a Kirchhoff's viscosity solution. In other terms, the values of the Hamiltonians are not required at the vertex in the analysis of these types of PDE systems.

math.AP↗
arXiv2025

On spider diffusions having a spinning measure selected from their own local time

The aim of this article is to give several results related to Walsh's spider diffusions living on a star-shaped network that have a spinning measure selected from the own local time of the motion at the vertex (cf.[17]). We prove the corresponding It\^o's formula and give some global trajectory properties such as $L^1$-approximation of the local time and the Markov property. Regarding the behavior of the process at the vertex, we show that that the distribution of the process is non atomic at the junction point and we characterize the instantaneous scattering distribution along some ray with the aid of the probability coefficients of diffraction. We obtain also a Feynmann-Kac representation for linear parabolic systems posed on star-shaped networks that where introduced in [18] possessing a so-called local-time Kirchhoff's boundary condition.

math.PR↗
arXiv2025

Stochastic scattering control of spider diffusion governed by an optimal diffraction probability measure selected from its own local-time

The purpose of this article is to study a new problem of stochastic control, related to Walsh's spider diffusion, named: stochastic optimal scattering control. The optimal scattering control of the spider diffusion at the junction point is governed by an appropriate and highly non-trivial condition of the Kirchhoff Law type, involving an optimal diffraction probability measure selected from the own local time of the spider process at the vertex. In this work, we prove first the weak dynamic programming principle in the spirit of [32], adapted to the new class of spider diffusion introduced recently in [37]-[38]. Thereafter, we show that the value function of the problem is characterized uniquely in terms of a Hamilton Jacobi Bellman (HJB) system posed on a star-shaped network, having a new boundary condition at the vertex called : non linear local-time Kirchhoff's transmission. The key main point is to use the recent comparison theorem obtained in [40], that has significantly unlocked the study of this type of problem. We conclude by discussing the formulation of stochastic scattering control problems, where there is no dependency w.r.t. the local-time variable, for which their well-posedness appear as a simpler consequence of the results of this work and the advances contained in [40].

math.AP↗
arXiv2023

Comparison principle for Walsh's spider HJB equations with non linear local time Kirchhoff's boundary transmission

The main purpose of this work is to obtain a comparison principle for viscosity solutions of a system of elliptic Walsh's spider Hamilton-Jacobi-Bellman equations, possessing a new boundary condition called non linear local-time Kirchhoff's transmission. The main idea is to build test functions at the neighborhood of the vertex solutions of ODE, with well-designed coefficients. The key point is to impose a 'local-time' derivative at the vertex absorbing the error term induced by - what we decide to call here - the Kirchhoff's speed of the Hamiltonians.

math.AP↗
arXiv2023

Martingale problem for a Walsh spider process with spinning measure selected from its own local time

The objective of this article is to prove existence and weak uniqueness of a Walsh spider diffusion process, whose spinning measure and coefficients are allowed to depend on the local time spent at the junction vertex. The methodology is to show carefully that an effectively designed martingale problem is well-posed. Exploiting fully the results coming from the pioneering work of [16], the construction of the solution is performed using a concatenation procedure, as introduced in the seminal reference [26]. Uniqueness is shown by making use of the recent results obtained in [25] for the solution of the corresponding parabolic PDE that involves a new class of transmission condition called local time Kirchhoff 's transmission condition. As a byproduct of our main result, we manage to compute the explicit law of the diffusion when it behaves as a standard Brownian motion on each branch. The case I = 2 permits us to derive also that there is existence and uniqueness for solutions of generalized SDE on the real line that involve the local time of the unknown process in all its coefficients.

math.PR↗
arXiv2023

Well posedness of linear parabolic partial differential equations posed on a star-shaped network with local time Kirchhoff's boundary condition at the vertex

The main purpose of this work is to provide an existence and uniqueness result for the solution of a linear parabolic system posed on a star-shaped network, which presents a new type of Kirchhoff's boundary transmission condition at the junction. This new type of Kirchhoff's condition-that we decide to call here local-time Kirchhoff 's condition-induces a dynamical behavior with respect to an external variable that may be interpreted as a local time parameter, designed to drive the system only at the singular point of the network. The seeds of this study point towards a forthcoming theoretical inquiry of a particular generalization of Walsh's random spider motions, whose spinning measures would select the available directions according to the local time of the motion at the junction of the network.

math.AP↗
arXiv2020

Stochastic control on networks: weak DPP, and verification theorem

The purpose of this article is to study a stochastic control problem on a junction, with control at the junction point. The problem of control is formulated in the weak sense, using a relaxed control, namely a control which takes values in the space of probability measures on a compact set. We prove first the compactness of the admissible rules and the dynamic programming principle (DPP). We complete this article by giving a verification Theorem for the value function of the problem, using some recent results on quasi linear non degenerate PDE posed on a junction, with non linear Neumann boundary condition at the junction point. An example is given, where the optimal control at the junction point is solution of a convex quadratic optimization problem with linear constraints.

math.OC↗
arXiv2019

Non Homogeneous Stochastic Diffusion on a Junction

The purpose of this article is to give another proof on the existence of a diffusion on a junction, which has been already done by M.Freidlin and S-J.Sheu, in Diffusion processes on graphs, (2000). We generalize the result to time dependent and borel coefficients. Such a process can be seen as a couple (x, i) with x a one dimensional continuous diffusion whose coefficients depends on the edge i where it is located. We then provide an It{\^o}'s formula for this process. Finally, we give an estimate of the local time of the process at the junction point.

math.PR↗