Markov processes and diffusion equations on unbounded intervals (Q5935722)

The authors study the differential operator \(A:D(A)\to W_2^0\) defined by NEWLINE\[NEWLINEAu(x) = \alpha(x)u''(x) \text{ if } x>0 \quad \text{and}\quad Au(x)=0 \text{ if }x=0,NEWLINE\]NEWLINE where \(\alpha\) is a continuous function on \([0,\infty)\), differentiable at \(0\) and such that \(0<\alpha_0\leq\alpha(x)/x\leq\alpha_1 (x\geq 0)\) for some \(\alpha_0,\alpha_1\in {\mathbb R}\), \(D(A)\) denotes the subspace of all functions \(u\in W_2^0\cap C^2([0,\infty))\) such that NEWLINE\[NEWLINE\lim_{x\to 0+} \alpha(x)u''(x) = \lim_{x\to\infty}{\alpha(x)u''(x)\over {1+x^2}} = 0NEWLINE\]NEWLINE It is shown that this operator generates a Feller semigroup that is the transition semigroups associated with a suitable Markov processes on \([0;\infty)\). Moreover the authors evaluate the expected value and the variance of the random variables describing the position of the processes. An approximation formula (in the weak topology) of the distribution of the position of the processes at every time is found, provided the distribution of the initial position is given and possesses a finite moment of order two.