Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A stochastic approach to a new type of parabolic variational
  inequalities   We study the following quasilinear partial differential equation with two
subdifferential operators:
  $${\frac{\partial u}{\partial s}(s,x)} + (\mathcal{L}u)(s,x,u(s,x),(\nabla
u(s,x))^\ast\sigma(s,x,u(s,x))) + f(s,x,u(s,x),(\nabla
u(s,x))^\ast\sigma(s,x,u(s,x))) \in \partial\varphi(u(s,x)) +
<\partial\psi(x),\nabla u(s,x)>,
  (s,x) \in[0,T]\times Dom\psi, u(T,x) =g(x),\quad x\in Dom\psi.$$ where for
$u\in C^{1,2}\big([0,T]\times Dom\psi\big)$ and $(s,x,y,z)\in [0,T]\times
Dom\psi\times Dom\varphi\times\mathbb{R}^{1\times d}$,
  $$(\mathcal{L}u)(s,x,y,z) := 1/2\sum_{i,j=1}^n
(\sigma\sigma^\ast)_{i,j}(s,x,y)\frac{\partial^2u}{\partial x_{i}\partial
x_{j}}(s,x) +\sum_{i=1}^n b_i(s,x,y,z)\frac{\partial u}{\partial x_i}(s,x). $$
  The operator $\partial\psi$ (resp. $\partial\varphi$) is the subdifferential
of the convex lower semicontinuous function $\psi:\mathbb{R}^{n}\to
(-\infty,+\infty]$ (resp. $\varphi:\mathbb{R}\to(-\infty,+\infty]$).
  We define the viscosity solution for such kind of partial differential
equations and prove the uniqueness of the viscosity solutions when $\sigma$
does not depend on $y$. To prove the existence of a viscosity solution, a
stochastic representation formula of Feymann-Kac type will be developed. For
this end, we investigate a fully coupled forward-backward stochastic
variational inequality.

26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics