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On a degenerate elliptic‐parabolic equation occurring in the theory of probability
Author(s) -
Brezis H.,
Rosenkrantz W.,
Singer B.,
Lax Peter D.
Publication year - 1971
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160240305
Subject(s) - mathematics , columbia university , degenerate energy levels , algebra over a field , pure mathematics , physics , sociology , media studies , quantum mechanics
In this paper we present results on the differentiability properties of solutions to the following singular parabolic partial differential equation : r >-g J Y q(', t , = t U X X (X J t> +-uX(x3 t> 1 the solution U satisfies the initial condition U(x, 0) = f (x) and the boundary condition f (0) = 0 = U,(O, t). Such equations occur in the theory of probability. For example, when y = $(n-I), equation (1) is the backward differential equation corresponding to the radial component of n-dimensional Brownian motion (see [ S ]). For other values of y , equation (1) is the backward differential equation corresponding to a stochastic process which is the limit of a sequence of random walks (see [5]). It is easily checked that the function satisfies the following degenerate elliptic-parabolic equation: V(x, t) = U (d i , i t) (2) m, 0) = g(x) =f(l/.> Y where a = y + 4 > 0. I n [I], Feller investigated a class of degenerate elliptic-parabolic equations which includes our equation (2). However, Feller discussed only the existence and uniqueness of solutions to equation (Z), he did not study the differentiability properties of their solutions and this is our main concern here.