%I #12 Jun 03 2017 15:34:17

%S 3,8,8,2,3,8,2,9,4,7,0,6,7,8,5,5,2,9,3,5,3,9,6,4,1,5,4,4,4,6,7,4,7,7,

%T 5,4,2,0,9,5,0,4,0,3,5,5,0,5,1,8,5,9,9,0,8,3,2,4,1,2,1,2,3,9,6,1,3,9,

%U 0,6,7,8,5,7,3,9,5,3,7,2,3,1,0,3,4,9,4,5,9,6,7,5,0,3,5,1,9,6,5,0

%N Decimal expansion of lambda(1), a constant associated with the asymptotic upper tail of the distribution of the first hitting time T_{1,0} for an Ornstein-Uhlenbeck process across the level 1, starting at 0.

%C Following Steven Finch, it is assumed that the values of the parameters of the stochastic differential equation dX_t = -rho (X_t - mu) dt + sigma dW_t, satisfied by the process, are mu = 0, rho = 1 and sigma^2 = 2.

%H Steven R. Finch, <a href="/A249417/a249417.pdf">Ornstein-Uhlenbeck Process</a>, May 15, 2004, p. 7. [Cached copy, with permission of the author]

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/ParabolicCylinderFunction.html">Parabolic Cylinder Function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process">Ornstein-Uhlenbeck process</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Parabolic_cylinder_function">Parabolic cylinder function</a>

%F lambda(a) = lim_{t->infinity} (1/t)*log(P(T_{a,0}>t)).

%F lambda(a) is the zero of D_{-lambda}(-a) closest to 0, where D_nu(x) is the parabolic cylinder function or Weber function.

%e -0.38823829470678552935396415444674775420950403550518599...

%t lambda[a_?NumericQ] := x /. FindRoot[ParabolicCylinderD[-x, -a] == 0, {x, 0}, WorkingPrecision -> 100]; RealDigits[lambda[1]] // First

%Y Cf. A249417, A249418, A249445, A249449.

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Oct 29 2014