The dynamics of solutions of the Cauchy problem for a parabolic type stochastic equation with power nonlinearities (the stochastic term is linear). (English. Ukrainian original) Zbl 0998.60023

Mel’nyk, S. A.

Theory Probab. Math. Stat. 64, 129-137 (2002); translation from Teor. Jmovirn. Mat. Stat. 64, 110-117 (2001).

Almost sure upper and lower bounds are constructed for the solution \(u(t,x)\) of the following stochastic equation \[ du(t,x)=(a(u^{\sigma+1}_{xx}+bu^{\beta}))dt+cud W(t),\quad t\in[0,T),\;x\in R^1, \] with the initial condition \(u(0,x)=u_0(x)\), where \(W(t)\) is a standard Brownian motion, \(a,b,c,\beta\) and \(\sigma\) are positive numbers, \(1\leq \beta \leq \sigma +1\), \(u_0(x)\) is non-negative bounded function vanishing on \(x\in (-\infty,l)\) and \(x\in (l,\infty)\) for some \(l>0\). In special cases, when solutions can be obtained in explicit form, an asymptotic behaviour of \(u(t,x)\) as \(t\to \infty\) is investigated.

Reviewer: N.M.Zinchenko (Kyïv)

MSC:

60F10	Large deviations
62F05	Asymptotic properties of parametric tests

Keywords:

Cauchy problem; stochastic equation; polynomial nonlinearities; asymptotic behaviour