Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I
A nonlinear stochastic partial differential equation in the Stratonovich form \[ du = \bigl \{ Au + f(t,x,u,\sigma ^ *(x)Du)\bigr \}\,dt + \sum ^ {d}_ {i=1} g_ {i}(t,x,u)\circ\,dB^ {i}_ {t}, \quad (t,x)\in (0,T)\times \mathbb R^ {m}, \tag{1} \] with initial condition \(u(0,\cdot ) = u_ 0\) on \(\mathbb R^ {m}\) is studied. In (1), \(B\) denotes the standard \(d\)-dimensional Wiener process and \(A\) is a second-order differential operator \[ A = \sum ^ {m}_ {i,j=1} \sum ^ {k}_ {l=1}\sigma _ {il}(x) \sigma _ {jl}(x)\partial ^ 2_ {x_ {i}x_ {j}} + \sum ^ {m}_ {i=1} \beta _ {i}(x)\partial _ {x_ {i}}. \] It is supposed that \(u_ 0\) is a continuous function of a polynomial growth, \(\sigma \), \(\beta \) are Lipschitz continuous functions, \(g\) is a continuous function sufficiently smooth in both the second and the third variables, and \(f\) is a progressively measurable random field, almost surely uniformly Lipschitz continuous in all variables. By extending the well known deterministic concept, the authors propose a new definition of a stochastic viscosity solution to (1). A transformation converting (1) into a partial differential equation with random coefficient (but with no martingale term) is found such that stochastic viscosity solutions to (1) are mapped in a one-to-one way to pathwise viscosity solutions of this random PDE. By means of this transformation and the theory of backward doubly stochastic differential equations, existence of stochastic viscosity solutions of (1) is established. [For part II see below, Zbl 1053.60064.]
stochastic partial differential equations
backward doubly stochastic differential equations
scientific article; zbMATH DE number 2119495
scientific article