Analysis of a Brownian particle moving in a time-dependent drift field (Q2774117)

The probability theory problem considered by the authors leads to a parabolic equation \(\frac 12 Q_{XX}-TQ_{X}=Q_{T}\) for \(X>0\), \(T>0\), with the boundary conditions \(\frac 12 Q_{X}(0,T)-TQ(0,T)=0\), \(Q(X,T_{0})=\delta (X-X_{0})\). The first result (Theorem 1) is an explicit expression of the solution \(Q(X,T;X_{0},T_{0})\) involving integrals, the Airy functions. The other results express the asymptotic behavior of the solution: for \(X_{0}=0\) and \(T_{0}\rightarrow -\infty\), for \(X_{0}\) and \(T_{0}\rightarrow +\infty\), for a bounded \(T_{0}\) and \(X_{0}=0\), for \(X_{0}\rightarrow \infty\) and \(T_{0}+(2X_{0})^{1/2}\rightarrow -\infty\), for \(X_{0}\rightarrow \infty\) and \(T_{0}+(2X_{0})^{1/2}\rightarrow +\infty\), for \(X_{0}\rightarrow \infty\) and bounded \((T_{0}+(2X_{0})^{1/2})X_{0}^{1/4}\) (Theorems 2-7). In each theorem, the asymptotic is different in different regions in which \((X,T)\) is allowed to vary (there are 9,7,5,5,8,8 regions, respectively). As an example, the nine regions in the case \(X_{0}=0\) and \(T_{0}\rightarrow -\infty\) are: bounded \((T-T_{0})T_{0}^{2}\) and \(X|T_{0}|\), \(X,T\rightarrow \infty\) and \(T<2T_{0}+(2X+T_{0}^{2})^{1/2}\), \(X,T\rightarrow \infty\) and bounded \([X-\frac 12 (T-T_{0})(T-3T_{0})]|T_{0}|^{-1/2}\), \(X,T\rightarrow \infty\) and \(2T_{0}+(2X+T_{0}^{2})^{1/2}<T<(2X)^{1/2}\), bounded \(X\) and \(T\), \(X,T\rightarrow \infty\) and \(T>(2X)^{1/2}\), \(X,T\rightarrow \infty\) and bounded \((X-\frac 12 T^{2})/T\), bounded \(X\) and \(T\rightarrow \infty\), bounded \(XT\) and \(T\rightarrow \infty\). There are some similarities of the regions in the other cases with this case, but not complete. NEWLINENEWLINENEWLINEThe results are interpreted in terms of the movement of the considered Brownian particle. Some regions (as 2,4,6 in Theorem 2) are shown to cover most of the domain of \((X,T)\). In the last section the authors indicate an alternative (geometric optics) method, considering an approximate solution \(Ke^{\psi}\) in which \(\psi_{T}+T\psi_{X}-\frac 12 \psi_{X}^{2}=0\), \(K_{T}+(T-\psi_{X})K_{X}=\frac 12 \psi_{XX}K\), the equation in \(\psi\) being solved by the method of characteristics, distinguishing characteristics starting from \((X_{0},T_{0})\), starting from \(X=0\) with \(dX/dT>0\) and tangent to \(X=0\). Relations with the first approach are discussed.