On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients

Abstract

We consider the existence and pathwise uniqueness of the stochastic heat equation with a multiplicative colored noise term on R d for d ≥ 1. We focus on the case of non-Lipschitz noise coefficients and singular spatial noise correlations. In the course of the proof a new result on Hölder continuity of the solutions near zero is established.

Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case

“…With this notation we can state the following standard existence result whose proof is a minor modification of Theorem 1.2 of (MPS06) and is given in the next Section. Theorem 1.1 Let X 0 ∈ C tem , and let b, σ : R + × R 2 → R satisfy (1.2), (1.3), and (1.4).…”

“…Hence, the better the regularity one has forũ near its zero set, the "weaker" the hypotheses required for pathwise uniqueness. It was shown in (MPS06) that at the points x whereũ(t, x) is "small",ũ(t, ·) is Hölder continuous with any exponent ξ such that ξ < 1 − α 2 1 − γ ∧ 1.…”

“…For finite-dimensional sde's this was the setting for Yamada and Watanabe's extension (YW71) of Itô's pathwise uniqueness results to Hölder (1/2) continuous coefficients, and so our plan will be to carry over their approach to our infinite dimensional setting. This programme was already carried out in the context of coloured noise in (MPS06), but the methods used there when specialized to white noise given nothing beyond the classical Lipschitz uniqueness. In fact for coloured noise in higher dimensions the results in (MPS06) did not even come close to the known results on pathwise uniqueness for Lipschitz continuous coefficients (Dal99)-see the discussion after Remark 1.5 in (MPS06).…”

“…This programme was already carried out in the context of coloured noise in (MPS06), but the methods used there when specialized to white noise given nothing beyond the classical Lipschitz uniqueness. In fact for coloured noise in higher dimensions the results in (MPS06) did not even come close to the known results on pathwise uniqueness for Lipschitz continuous coefficients (Dal99)-see the discussion after Remark 1.5 in (MPS06). This is what led to our belief that there was room for substantial improvement in the methods of (MPS06) and hence to the present work.…”

On pathwise uniqueness of stochastic evolution equations in Hilbert spaces

“…The classical methods to study the pathwise uniqueness of the solution are under some strictly regular conditions on the coefficients, like the Lipschitz continuity or dissipativity. Recently, many authors also devoted themselves to studying such problems with weak regular coefficients, see [6] for the finite dimensional case and [13,19,20] for the infinite dimensional case.…”