Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local times for systems of non-linear stochastic heat equations   We consider $u(t,x)=(u_1(t,x),\cdots,u_d(t,x))$ the solution to a system of
non-linear stochastic heat equations in spatial dimension one driven by a
$d$-dimensional space-time white noise. We prove that, when $d\leq 3$, the
local time $L(\xi,t)$ of $\{u(t,x)\,,\;t\in[0,T]\}$ exists and $L(\bullet,t) $
belongs a.s. to the Sobolev space $ H^{\alpha}(\mathbb{R}^d)$ for
$\alpha<\frac{4-d}{2}$, and when $d\geq 4$, the local time does not exist. We
also show joint continuity and establish H\"{o}lder conditions for the local
time of $\{u(t,x)\,,\;t\in[0,T]\}$. These results are then used to investigate
the irregularity of the coordinate functions of $\{u(t,x)\,,\;t\in[0,T]\}$.
Comparing to similar results obtained for the linear stochastic heat equation
(i.e., the solution is Gaussian), we believe that our results are sharp.
Finally, we get a sharp estimate for the partial derivatives of the joint
density of $(u(t_1,x)-u(t_0,x),\cdots,u(t_n,x)-u(t_{n-1},x))$, which is a new
result and of independent interest.

24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph