# Strangely sweeping one-dimensional diffusion

Annales Polonici Mathematici (1993)

- Volume: 58, Issue: 1, page 37-45
- ISSN: 0066-2216

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topRyszard Rudnicki. "Strangely sweeping one-dimensional diffusion." Annales Polonici Mathematici 58.1 (1993): 37-45. <http://eudml.org/doc/262362>.

@article{RyszardRudnicki1993,

abstract = {Let X(t) be a diffusion process satisfying the stochastic differential equation dX(t) = a(X(t))dW(t) + b(X(t))dt. We analyse the asymptotic behaviour of p(t) = ProbX(t) ≥ 0 as t → ∞ and construct an equation such that $lim sup_\{t→∞\} t^\{-1\} ∫_0^t p(s) ds = 1$ and $lim inf_\{t→∞\}t^\{-1\} ∫_0^t p(s) ds = 0$.},

author = {Ryszard Rudnicki},

journal = {Annales Polonici Mathematici},

keywords = {diffusion process; parabolic equation; sweeping semigroup of Markov operators; stochastic differential equation; speed measure; Feller boundary classification},

language = {eng},

number = {1},

pages = {37-45},

title = {Strangely sweeping one-dimensional diffusion},

url = {http://eudml.org/doc/262362},

volume = {58},

year = {1993},

}

TY - JOUR

AU - Ryszard Rudnicki

TI - Strangely sweeping one-dimensional diffusion

JO - Annales Polonici Mathematici

PY - 1993

VL - 58

IS - 1

SP - 37

EP - 45

AB - Let X(t) be a diffusion process satisfying the stochastic differential equation dX(t) = a(X(t))dW(t) + b(X(t))dt. We analyse the asymptotic behaviour of p(t) = ProbX(t) ≥ 0 as t → ∞ and construct an equation such that $lim sup_{t→∞} t^{-1} ∫_0^t p(s) ds = 1$ and $lim inf_{t→∞}t^{-1} ∫_0^t p(s) ds = 0$.

LA - eng

KW - diffusion process; parabolic equation; sweeping semigroup of Markov operators; stochastic differential equation; speed measure; Feller boundary classification

UR - http://eudml.org/doc/262362

ER -

## References

top- [1] I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer, Berlin 1972. Zbl0242.60003
- [2] A. K. Gushchin and V. P. Mikhailov, The stabilization of the solution of the Cauchy problem for a parabolic equation with one space variable, Trudy Mat. Inst. Steklov. 112 (1971), 181-202 (in Russian).
- [3] T. Komorowski and J. Tyrcha, Asymptotic properties of some Markov operators, Bull. Polish Acad. Sci. Math. 37 (1989), 221-228. Zbl0767.47012
- [4] R. Rudnicki, Asymptotical stability in L¹ of parabolic equations, J. Differential Equations, in press. Zbl0815.35034
- [5] Z. Schuss, Theory and Applications of Stochastic Differential Equations, Wiley, New York 1980.