Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On the well-posedness of a Hamilton-Jacobi-Bellman equation with transport noise (2209.06660v3)

Published 14 Sep 2022 in math.AP and math.OC

Abstract: In this paper we consider the following non-linear stochastic partial differential equation (SPDE): \begin{align*} \begin{cases} \mathrm{d}u(s,x)=\sumn_{i=1} \mathscr{L}_i u(s,x)\circ \mathrm{d}W_i(s)+\left(V(x)+\mu\Delta u(s,x)-\frac{1}{2}\vert\nabla u(s,x)\vert2\right)\mathrm{d}s, \quad &\text{in } (0,T)\times \mathbb{T}n, u(0,x)=u_0(x), & \text{on } \mathbb{T}n, \end{cases} \end{align*} where $\mathbb{T}n$ is the $n$-dimensional torus, the functions $u_0, V: \mathbb{T}n \to \mathbb{R}$ are given and ${\mathscr{L}_i}_i$ is a collection of first order linear operators. This can be seen as a Cauchy problem for a Hamilton-Jacobi-BeLLMan equation with transport noise in any space dimension. We introduce the concept of a strong solution from the realm of PDEs and establish the existence and uniqueness of maximal solutions (strong solutions upto a stopping time). Moreover, for a particular class of ${\mathscr{L}_i}_i$ we establish global well-posedness of strong solutions. The proof relies on studying an associated truncated version of the original SPDE and showing its global well-posedness in the class of strong solutions.

Summary

We haven't generated a summary for this paper yet.