On the limit behavior of the solution of a Cauchy problem for the equation of heat conduction with a random right part. (Russian) Zbl 0577.60062

Divnich, N. T.

Teor. Veroyatn. Mat. Stat. 29, 34-37 (1983).

The author studies the stochastic differential equation \[ -u_ t(t,x)+a^ 2/2\cdot u''_{xx}(t,x)=F(x,w(t),\dot w(t)), \] where \(t\geq 0\), \(u(0,x)=g(x)\), \(F=F(x,y,z)\) and g are real functions, and W is a Wiener process of the white noise type with \(\int^{t_ 2}_{t_ 1}\dot w(t)dt=w(t_ 2)-w(t_ 1)\) for \(t_ 1\leq t_ 2\). The case \(F(x,y,z)=f(x)q(y)\) or \(F(x,y,z)=f(x)q(y)z\) is considered. Conditions for the convergence of the distribution function of u(t,x)/t and \(u(t,x)/\sqrt{t}\) for \(t\to \infty\) to functionals of W are obtained.

Reviewer: C.Baldauf

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60B10	Convergence of probability measures
60H15	Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

Cauchy problem for the equation of heat conduction; convergence of the distribution