Work

On the well-posedness of a Hamilton–Jacobi–Bellman equation with transport noise

Year: 2025

Type: article

Abstract: In this paper we consider the following non-linear stochastic partial differential equation (SPDE): $$\begin{aligned} {\left\{ \begin{array}{ll} \textrm{d}u(s,x)=\sum ^n_{i=1} \mathscr {L}_i u(s,x)\circ \textrm{d}W_i(s)\\ \qquad \qquad +\left( V(x)+\mu \Delta u(s,x)-\frac{1}{2}\vert \nabla u(s,x)\vert ^2\right) \textrm{d}s, \quad & \text {in } (0,T)\times {\mathbb {T}}^n,\\ u(0,x)=u_0(x),\quad \text {on } {\mathbb {T}}^n, \end{array}\right. } \end{aligned}$$ d u ( s , x ) = ∑ i = 1 n L i u ( s , x ) ∘ d W i ( s ) + V ( x ) + μ Δ u ( s , x ) - 1 2 | ∇ u ( s , x ) | 2 d s , in ( 0 , T ) × T n , u ( 0 , x ) = u 0 ( x ) , on T n ,

Source: Nonlinear Differential Equations and Applications NoDEA

DOI: 10.1007/s00030-025-01046-6

Authors: Neeraj Bhauryal, Ana Bela Cruzeiro, Carlos Oliveira