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Asymptotic behavior of parabolic equations arising from one-dimensional null-recurrent diffusions. (English) Zbl 0957.35023

The authors study the asymptotic behavior, as \(t\to\infty\), of \(U(x,t)\) and \(V(x,t)\), the solutions of the Cauchy problems: \[ \partial U/ \partial t=a(x) \partial^2U/ \partial x^2+b(x) \partial U/\partial x,\quad U(x, 0)= \varphi(x), \] and \[ \partial V/\partial t=a(x) \partial^2V/ \partial x^2+ b(x)\partial V/ \partial x+f(x), \quad V(x,0)= \varphi(x). \] The solution of these problems admit a stochastic representation \(U(x,t)= E\varphi(X^x(t))\) and \(V(x,t)= E\varphi(X^x(t)) +E\int^t_0f(X^x(s))ds\), where \(X^x(t)\) is a solution of the stochastic differential equation \(dX(t)= b(X(t))dt +\sqrt {2a (X(t))} dW(t)\), \(X(0)=x\), \(W(t)\) is a standard Brownian motion. The case of the null-recurrent process \(X^x(t)\) is considered. Convergence under suitable scaling of \(U(x,t)\) and \(V(x,t)\), when \(\varphi(x)\) and \(f(x)\) are integrable or not integrable with respect to the invariant measure, is proved.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
60H30 Applications of stochastic analysis (to PDEs, etc.)
35C15 Integral representations of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
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References:

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