Defining coarsenings of valuations

We study the question which henselian fields admit definable henselian valuations (with or without parameters). We show that every field which admits a henselian valuation with non-divisible value group admits a parameter-definable (non-trivial) henselian valuation. In equicharacteristic $0$, we give a complete characterization of henselian fields admitting a parameter-definable (non-trivial) henselian valuation. We also obtain partial characterization results of fields admitting 0-definable (non-trivial) henselian valuations. We then draw some Galois-theoretic conclusions from our results.