We introduce a notion of viscosity solution for Hamilton--Jacobi--Bellman (HJB) equations with distributional drift, based on paracontrolled test functions and related through a Zvonkin transformation to classical viscosity theory. The equations considered are of the form \[ \left(\partial_t+\frac12\Delta+b\cdot\nabla\right)h(t,x) =-H(t,x,h(t,x),\nabla h(t,x)), \] where $b$ is singular in the sense of \cite{paradistrib} and has regularity $\mathcal C^{-\alpha}$ for $\alpha\in(1/2,2/3)$. Using doubling-of-variables arguments, we derive a priori gradient estimates that also cover Hamiltonians with slightly superquadratic growth in $\nabla h$. We also obtain probabilistic representations through singular control problems for convex $H$ and weak singular forward--backward SDEs for possibly nonconvex Hamiltonians with at most quadratic growth.
We investigate the regularity of viscosity solutions to a class of nonlocal Hamilton-Jacobi equations driven by x-dependent integro-differential operators and coercive superlinear Hamiltonians. We first establish H{\"o}lder regularity for bounded viscosity solutions under general structural and continuity assumptions o...
A. Ciomaga, T. M. Lê, Olivier Ley et al.· 0 citations
We develop a Banach-valued lateral Stieltjes calculus adapted to linear evolution equations on a Gelfand triple \[ V\hookrightarrow H\simeq H^*\hookrightarrow V^*. \] The evolution problem is formulated with the posterior state, \[ D_g u(t)+A(t) S_g u(t)=f(t),\qquad u(a)=u_0. \] This choice is decisive at atoms of the...
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
We study instantaneous shrinking of supports for nonnegative solutions of the stochastic partial differential equation \[ \partial_t u=a(t,x)\,\partial_x^2 u + b(t,x)\,\partial_x u + c(t,x)\,u +\sigma(u)\,\xi(t,x), \qquad (t,x)\in(0,\infty)\times\mathbb R, \] where $\xi$ is space-time white noise, the coefficients $a$,...
We study quasilinear symmetrizable partially dissipative hyperbolic systems with non-autonomous relaxation coefficients in $\mathbb{R}^d$ ($d\geq1$). The existence of global strong solutions is established in a critical regularity setting for systems satisfying the so-called Shizuta-Kawashima (SK) and entropy condition...
We establish scale-invariant interior $C^{1,\alpha}$ estimates for bounded viscosity solutions of $(-\Delta)^su+b\cdot\nabla u=f$ for $s\in[1/2,1)$ with locally H\"older $b$ and $f$. The critical case uses Silvestre's parabolic theorem; the subcritical case uses Schauder estimates and interpolation. Applying this visco...
Qi Xue, Chao Zhang· 0 citations
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